Problems linear block codes Error_Control_Coding_SHU LIN COSTELLO_2nd Edition_2004

8/12/2019 Problems linear block codes Error_Control_Coding_SHU LIN COSTELLO_2nd Edition_2004

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0.2. Problems

2.1 Construct the group under modulo-6 addition.

2.2 Construct the group under modulo-3 multiplication.

2.3 Let inbe a positive integer. If in is not a prime, prove that the set {1, 2, , m!1" is not a group

under modulo-m multiplication.

2.# Construct the prime field GF$l1% &ith modulo-11 addition and multiplication. 'ind all the primitiveelements, and determine the orders of other elements.

2.( Let m he a positive integer. If in is not prime, prove that the set $), 1, 2, , m-1" is not a field

under modulo-m addition and multiplication.

2.6 Consider the integer group G $), 1, 2, , 31" under modulo-32 addition. *ho& thatH+ {), #, ,

12, 16, 2), 2#, 2" forms a subgroup of G. ecompose G into cosets &ith respect toH $or moduloII%.

2.Let / be the characteristic of a 0alois field GF$q%. Let 1 be the unit element of GF$q%. *ho& that

the sums

form a subfield of GF$q%.

2. rove that ever finite field has a primitive element.

2. *olve the follo&ing simultaneous e4uations ofX. Y,Z, and 5 &ith modulo-2 arithmetic

2.1) *ho& thatX ( 7X 3 7 1is irreducible over GF$2%.

2.11 Letf$X% be a polnomial of degree 8over GF$2%. 9he reciprocal off$X%is defined as

a. rove thatf: $X%is irreducible over GF$2%if and onl iff$X%is irreducible over GF$2%.

b. rove thatf: $X%is primitive if and onl iff$X%is primitive.

2.12 'ind all the irreducible polnomials of degree ( over GF$2%.

2.13 Construct a table for GF$2

3

% based on the primitive polnomialp$X% + 1 7X7X 3.. ispla thepo&er, polnomial, and vector representations of each element.

etermine the order of each element.

2.1# Construct a table for GF$2(%based on the primitive polnomialp$X% + 1 7X 2 7 X(. Let abe a

primitive element of GF$2(%. 'ind the minimal polnomials of a3and a.

2.1( Let be an element in GF$2m%. Let ebe the smallest nonnegative integer such that 2; + 13. rovethat 2,, p2se 9able 2. to find the roots off$X% +X37 a6?2 7 aX7 a.

2.1 Let abe a primitive element in GF$2#%. ivide the polnomialf$X% + a3? 7 aX6 7 aX#7 a2X2

7 all X7 1 over GF$2#% b the polnomial g$X% +X#7 a3?2 7 a(? 7 1 over GF$2#%. 'ind the 4uotient

and the remainder $use 9able 2.%.

2.1 Let a be a primitive element in GF$2#%. >se 9able 2. to solve the follo&ing simultaneous

e4uations forX, Y, and @

2.2) Let A be a vector space over a fieldF. 'or an element c inF, prove that c) + ).

2.21 Let A be a vector space over a field F. rove that, for an c inF and an v in A, $-c%v+ c $-v%-$c

v%.

2.22 Let Sbe a subset of the vector space AB of all n-tuples over GF$2%. rove that S is a subspace of

AB if for an u and v in S, a7 v is in S.

2.23 rove that the set of polnomials over GF$2%&ith degree n ! 1 or less forms a vector space GF$2%

&ith dimension n.


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2.2# rove that G F$2m%is a vector space over GF$2%.

2.2( Construct the vector space V(of all (-tuples over GF$2%. 'ind a three-dimensional subspace and

determine its null space.

2.26 0iven the matrices

sho& that the ro& space of is the null space of riI, and vice versa.

2.2 Let *1 and *2 be t&o subspaces of a vector A. *ho& that the intersection of and *2 is also a

subspace in A.2.2 Construct the vector space of all 3-tuples over GF$3%. 'orm a t&o-dimensional subspace and its

dual space.


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0.3. Problems

3.11 Consider a sstematic $, #% code &hose parit-checD e4uations are

&here 1#), tit, u2, and u3, are message digits, and vo, vt, v2, and v2, are parit-checD digits. 'ind the

generator and parit-checD matrices for this code. *ho& analticall that the minimum distance of this

code is #.

3.2 Construct an encoder for the code given in roblem 3.1.

3.3 Construct a sndrome circuit for the code given in roblem 3.1.

3.# Let IE be the parit-checD matriF of an $a, k% linear code C that has both odd and even-&eight

code-&ords. Construct a ne& linear code Cl &ith the follo&ing parit-checD matriF

$Eote that the last ro& of 9hu consists of all s.%

a. *ho& that Ci is an $a7 1, k%linear code. C1is called an extension of C.

b. *ho& that ever code-&ord of C1has even &eight.

c. *ho& that C1can be obtained from Cb adding an eFtra parit-checD digit, denoted b vim, to the

left of each code-&ord v as follo&s $1% if v has odd &eight, then + 1, and $2% if v has even &eight,

then vo, + ). 9he parit-checD digit voc is called an overall parity-ceck digit.

3.( Let Cbe a linear code &ith both even- and odd-&eight code-&ords. *ho& that the number of even-

&eight code-&ords is e4ual to the number of odd-&eight code-&ords.

3.6 Consider an $n, k%linear code C &hose generator matriF Gcontains no Gero column. /rrange all

the code-&ords of C as ro&s of a 2k-b-n arra.

a. *ho& that no column of the arra contains onl Geros.

b. *ho& that each column of the arra consists of 2k-1 Geros and 2k-1 ones.

c. *ho& that the set of all code-&ords &ith Geros in a particular component position forms a subspace

of C. 5hat is the dimension of this subspaceH3. rove that the amming distance satisfies the triangle ine4ualitJ that is, let F, y, and G be three n-

tuples over G F$2%, and sho& that

3. rove that a linear code is capable of correcting ? or fe&er errors and simultaneousl detecting 1$1

K ?% or fe&er errors if its minimum distance !minK 2 7 1 7 1.

3. etermine the &eight distribution of the $, #% linear code given in roblem 3.1. Let the transition

probabilit of a *C bep+ 1)-2. Compute the probabilit of an undetected error of this code.

3.1) ecause the $, #% linear code given in roblem 3.1 has minimum distance #, it is capable of

correcting all the single-error patterns and simultaneousl detecting an combination of double errors.

Construct a decoder for this code. 9he decoder must be capable of correcting an single error anddetecting an double errors.

3.11 Let ' be the ensemble of all the binar sstematic $n, k%linear codes. rove that a nonGero binar

n-tuple v is contained in either eFactl 2$D-1%$


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3.1( 'or an binar $n, k%linear code &ith minimum distance $or minimum &eight% 2t 71or greater,

sho& that the number of parit-checD digits satisfies the follo&ing ine4ualit

.9he preceding ine4ualit gives an upper bound on the random-error-correcting capabilit t of an $n,

k%linear code. 9his bound is Dno&n as the Hammin" #o$n! M1#O. $Hint'or an $a, k%linear code &ith

minimum distance 2t 7 1 or greater, all the n-tuples of &eight t or less can be used as coset leaders in a

standard arra.%

3.16 *ho& that the minimum distance !rain of an $n, k%linear code satisfies the follo&ing ine4ualit

$Hint>se the result of roblem 3.6$b%. 9his bound is Dno&n as the%lotkin #o$n! M1-3O.%

3.1 *ho& that there eFists an $n, k%linear code &ith a minimum distance of at least

.$Hint>se the result of roblem 3.11 and the fact that the nonGero n-tuples of &eight c8- 1 or less can

be at most in

.$n, k%sstematic linear codes.%

3.1 *ho& that there eFists an $a, k%linear code &ith a minimum distance of at least !minthat satisfies

the follo&ing ine4ualit

$Hint *ee roblem 3.1. 9he second ine4ualit provides a lo&er bound on the minimum distance

attainable &ith an $a, k%linear code. 9his bound is Dno&n as the Varsarmov-Gil#ertbound M1-3O.%

3.1 Consider a rate- $n, n82%linear blocD code C &ith a generator matriF G. rove that C is self-dual if

C C9 + ).

3.2) evise an encoder for the $a, nP 1% *C code &ith onl one memor element $or flip-flop% and

one ?-QR gate $or modulo-2 adder%.


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(.12 'or a cclic code, if an error pattern e$X%is detectable, sho& that its itcclic shift e$+%$X%is also

detectable.

(.13 In the decoding of an $n, k%cclic code, suppose that the received polnomial r$?% is shifted into

the sndrome register from the right end, as sho&n in 'igure (.11. *ho& that &hen a received digit r,

is detected in error and is corrected, the effect of error digit e, on the sndrome can be removed b

feeding e, into the sndrome register from the right end, as sho&n in 'igure (.11.

(.1# Let v$?% be a code polnomial in a cclic code of length n. Let 1be the smallest integer such that

*ho& that if 1)), 8is a factor of n.

(.1(Let g$?% be the generator polnomial of an $n, k%cclic code C. *uppose C is interleaved to a

depth of rove that the interleaved code C%< is also cclic and its generator polnomial is "$X ,%.

(.16 Construct all the binar cclic codes of length 1(. $Hint>sing the fact that ?1( 71 has all the

nonGero elements of GF$2#%as roots and using 9able 2., factor ?1( 71 as a product of irreducible

polnomials.%

(.1 Let be a nonGero element in the 0alois field G F$2'1%, and )1. Let $1%$?% be the minimumpolnomial of p. Is there a cclic code &ith $X%as the generator polnomialH If our ans&er is es,find the shortest cclic code &ith $X%as the generator polnomial.

(.1 Let 1 and%32 be t&o distinct nonGero elements in GF$2'%. Let )1$X%and )2$?% be the minimalpolnomials of 2 and 2, respectivel. Is there a cclic code &ith "$X%)1$X% )2$?% as the generatorpolnomialH If our ans&er is es, find the shortest cclic code &ith "$X% )1 $X% ) $X% as thegenerator polnomial.

(.1 Consider the 0alois field GF$2m%, &hich is constructed based on the primitive polnomialp$X%of

degree in. Let abe a primitive element of GF$2'%&hose minimal polnomial isp$X%. *ho& that ever

code polnomial in the amming code generated b p$X%has a and its con=ugates as roots. *ho& that

an binar polnomial of degree 21< - 2 or less that has a as a root is a code polnomial in the

amming code generated bp$X%.

(.2) Let C1 and C2 be t&o cclic codes of length n that are generated b gi $X% an! "2$X%,respectivel. *ho& that the code polnomials common to both C1and C2 also form a cclic code C3.

etermine the generator polnomial of C3. If d1 and d2 are the minimum distances of C1and C2,

respectivel, &hat can ou sa about the minimum distance of C3H

(.21 *ho& that the probabilit of an undetected error for the distance-# cclic amming codes is upper

bounded b 2-)171%.

(.22 Let Cbe a $2


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v.. *ho& that all the double errors can be trapped.

b. Can all the error patterns of three errors be trappedH If not, ho& man error patterns of three errors

cannot be trappedH

c. evise a simple error-trapping decoder for this code.

(.26 roblem

a. evise a simple error-trapping decoder for the $23, 12% 0ola code.

b. o& man error patterns of double errors cannot be trappedH

c. o& man error patterns of three errors cannot be trappedH

(.2 *uppose that the $23, 12% 0ola code is used onl for error correction on a *C &ith transition

probabilitp. If Wasami;s decoder of 'igure (.1 is used for decoding this code, &hat is the probabilit

of a decoding errorH $Hint>se the fact that the $23, 12% 0ola code is a perfect code.%

(.2 >se the decoder of 'igure (.1 to decode the follo&ing received polnomials

/t each step in the decoding process, &rite do&n the contents of the sndrome register.

(.2 Consider the follo&ing binar polnomial

&here $X3

71% andp$X% are relativel prime, andp$X% is an irreducible polnomial of degree in &ith inK 3. Let is be the smallest integer such that g$?% divides X'7 1. 9hus, g$?% generates a cclic code of

length n.

a. *ho& that this code is capable of correcting all the single-error, double-ad=acent-error, and triple-

ad=acent-error patterns. $Hint*ho& that these error patterns can be used as coset leaders of a standard

arra for the code.%

b. evise an error-trapping decoder for this code. 9he decoder must be capable of correcting all the

single-error, double-ad=acent-error, and triple-ad=acent-error patterns. esign a combinational logic

circuit &hose output is 1 &hen the errors are trapped in the appropriate stages of the sndrome register.

c. *uppose thatp$X% + 1 7X7X#, &hich is a primitive polnomial of degree #. etermine the smallest

integer n such that g$?% + $X37 1%p$X% divides

(.3) Let C1 be the $3, 1% cclic code generated b gi $X% + 1 7 X 7 X2, and let C2 be the $, 3%

maFimum-length code generated b g2 $?% + 1 7 X 7 X2 7 X#. 'ind the generator and parit

polnomials of the cclic product of C1and C2. 5hat is the minimum distance of this product codeH

iscuss its error-correcting capabilit.

(.31 evise an encoding circuit for the $1(, (% 4uasi-cclic code given in SFample (.1#.


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0.6. Problems

6.1 Consider the 0alois field GF$2#% given b 9able 2.. 9he element + a is also a primitiveelement. Let go$X%be the lo&est-degree polnomial over GF$2% that has p2 p3, p# as its roots. 9his

polnomial also generates a double-error-correcting primitive C code of length 1(.

a. etermine go$X%.

b. 'ind the parit-checD matriF for this code.c. *ho& that go$X% is the reciprocal polnomial of the polnomial g$?% that generates the $1(, %

double-error-correcting C code given in SFample 6.1.

6.2 etermine the generator polnomials of all the primitive C codes of length 31.

>se the 0alois field GF$2(%generated bp$X% + 1 7X27X(.

6.3 *uppose that the double-error-correcting C code of length 31 constructed in roblem 6.2 is used

for error correction on a II*C. ecode the received polnomials rX1 $?% +X7X 3)and r2$X% + 1 7

X1%$2.

6.# Consider a t-error-correcting primitive binar C code of length n+ ! 1.

If 2t 7 1 is a factor of n, prove that the minimum distance of the code is eFactl 2t 7 1. $HintLet n+

1$2t 1%. *ho& that $X'7 1%8$X1 7 1%is a code polnomial of &eight 2t 7 1.%

6.( Is there a binar t-error-correcting C code of length 2sep$X% + 1 7X27X(

to generate GF$2(%.

6. evise a sndrome computation circuit for the binar double-error-correcting $31, 21% C code.

6. evise a Chien;s searching circuit for the binar double-error-correcting $31, 21% C code.

6.1) Consider the 0alois field GF$26%given b 9able 6.2. Let*+ a3, 1) + 2, and !+ (. etermine the

generator polnomial of the C code that has, 2%3 # 1( as its roots $the general form presentedat the end of *ection 6.1%. 5hat is the length of this codeH

6.11 Let 1) + -t and ! + 2t 7 2. 9hen &e obtain a C code of designed distance 2t 7 2 &hose

generator polnomial has and their con=ugates as all its roots.

a. *ho& that this code is a reversible cclic code.

b. *ho& that if t is odd, the minimum distance of this code is at least 2t 7 #.

$Hint*ho& that -$t71% and pY71 are also roots of the generator polnomial.%


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0.7. Problems

.1 Consider the triple-error-correcting R* code given in SFample .2. 'ind the code polnomial for

the message

.2 >sing the 0alois field GF$2(% given in /ppendiF /, find the generator polnomials of the double-

error-correcting and triple-error-correcting R* codes of length 31.

.3 >sing the 0alois field GF$26% given in 9able 6.2, find the generator polnomials of double-error-

correcting and triple-error-correcting R* codes of length 63.

.#Consider the triple-error-correcting * code of length 1( given in SFample .2. ecode the received

polnomial

using the erleDamp algorithm.

.( Continue roblem .#. ecode the received polnomial &ith the Suclidean algorithm.

.6 Consider the triple-error-correcting R* code of length 31 constructed in roblem .2. ecode the

received polnomial

using the Suclidean algorithm.

.Continue roblem .6. ecode the received polnomial in the fre4uenc domain using transformdecoding.

. 'or the same R* code of roblem .6, decode the follo&ing received polnomial &ith t&o

erasures

&ith the Suclidean algorithm.

.rove that the dual code of a '; * code is also a R* code.

.1) rove that the $2


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0.8. Problems

.1 Consider the triple-error-correcting R* code given in SFample .2. 'ind the code polnomial for

the message

.2 >sing the 0alois field GF$2(% given in /ppendiF /, find the generator polnomials of the double-

error-correcting and triple-error-correcting R* codes of length 31.

.3 >sing the 0alois field GF$26% given in 9able 6.2, find the generator polnomials of double-error-

correcting and triple-error-correcting R* codes of length 63.

.#Consider the triple-error-correcting * code of length 1( given in SFample .2. ecode the received

polnomial

using the erleDamp algorithm.

.( Continue roblem .#. ecode the received polnomial &ith the Suclidean algorithm.

.6 Consider the triple-error-correcting R* code of length 31 constructed in roblem .2. ecode the

received polnomial

using the Suclidean algorithm.

.Continue roblem .6. ecode the received polnomial in the fre4uenc domain using transformdecoding.

. 'or the same R* code of roblem .6, decode the follo&ing received polnomial &ith t&o

erasures

&ith the Suclidean algorithm.

.rove that the dual code of a '; * code is also a R* code.

.1) rove that the $2


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0.10. Problems

1).1 rove the sufficient condition for optimalit of a code-&ord given b $1).#%.

1).2 Consider the value 0$n, &i J 2, &t% given in $1).#6%.

1).3 iscuss 0$vt, &iJ a2, &t% in the case &here the code-&ord delivered b an algebraic decoder is

Dno&n. 5hat is the problem in tring to use this result for all received se4uencesH

o, iscuss 0$vi, &1J 2, &1% for vi + 0N decoding considers onl I ZdBB.B 7 1%82O erasures in thedB,JB ! 1 IL,s. SFplain &h not all dmiB ! 1 possible erasures are considered.

1).# Consider an $n, D% binar linear code &ith even minimum distance !min. *ho& that it is possible to

achieve the same error performance as for the conventional Chase algorithm-2 b erasing one given

position among the !min82least reliable positions $LRs% of the received se4uence and adding to thehard-decision decoding of the received seq$ence rall possible combinations of );s and 1;s in the

remaining L!min82 ! 1 LRs.

1).( Consider an error-and-erasure algebraic decoder that successfull decodes an input se4uence

&ith t errors and s erasures satisfing s 2t P dmiB and fails to decode other&ise. efine *e$a% as the set

of candidate code-&ords generated b the algorithm /e $a% presented in *ection 1).#. 'or a + 1, ,

MdB/iB121 ! 1, sho& that *e $a% c *e $a 7 1%.

1).6 In the WEI algorithm presented in *ection 1).6, sho& that an code-&ord v in Z $i% rather that

the one that has the smallest correlation discrepanc &ith the received seq$ence rcan be used for

evaluating 0 $v%. iscuss the implications of this remarD $advantages and dra&bacDs%.

1). In the RL* algorithm presented in *ection 1)., sho& that there eFists at most one $n - D%-

pattern that is not $n - D ! 1%-eliminated.

1). 'or the RL* algorithm presented in *ection 1)., determine the complete reduced list for the

$1(, 11, 3% amming code.

1). etermine the complete reduced list for the $, #, #% eFtended amming code. *ho& that this

complete list can be divided into t&o separate lists depending on &hether the sndrome s is a columnof the parit checD matriF . $int Sach list is composed of five distinct patterns%.

1).1) In the RL* algorithm presented in *ection 1)., prove that all n$v%-patterns &ith n$v% K n -, D

can be eliminated from all reduced lists. 'or n$v% P n$v%, determine an n$v%-pattern that =ustifies this

elimination.

1).11 Let C and C1be the t&o codes defined in *ection 1)..1. SFplain &h if A is the decoded code-

&ord in C1, then fri-lr2-1MAO is simpl the decoded code-&ord in C.

1).12 rove that the most reliable basis and the least reliable basis are information sets of a code and

its dual, respectivel.

1).13 rove that order-1 reprocessing achieves maFimum liDelihood decoding for the $, #, #% RNcode.

1).1# 5hich order of reprocessing achieves maFimum liDelihood decoding of an $n, n - 1, 2% single

parit-checD codeH ased on our ans&er, propose a much simpler method for achieving maFimum

liDelihood decoding of single parit-checD codes.

1).1( escribe the tpes of errors that can be corrected b Chase algorithm-2, but not b order-i

reprocessing.

1).16 /ssume that an rth-order RN code RN$r, in% is used for error control.

a. *ho& that all error patterns of &eight at most t, as &ell as all error patterns of &eight t 1 &ith one

error in a given position can be corrected.

b. /ssuming reliabilit values are available at the decoder, propose a simple modification of ma=orit-

logic decoding $Reed algorithm% of RN$r, in% RN codes in &hich the error performance can be

improved based on $a%.


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0.11. Problems

11.1 Consider the $3, 1, 2% nonsstematic feed-for&ard encoder &ith

a. ra& the encoder blocD diagram.

b. 'ind the time-domain generator matriF G.

c. 'ind the code-&ord v corresponding to the information se4uence u + $1 1 1 ) 1%.

11.2 Consider the $#, 3, 3% nonsstematic feed-for&ard encoder sho&n in 'igure 11.3.

a. 'ind the generator se4uences of this encoder.

b. 'ind the time-domain generator matriF G.

c. 'ind the code-&ord v corresponding to the information se4uence u + $11), )11, 1)1%.

11.3 Consider the $3, 1, 2% encoder of roblem 11.1.

a. 'ind the transform-domain generator matriF G$%.

b. 'ind the set of output se4uences $% and the code-&ord '%% corresponding to the information

se4uence

11.# Consider the $3, 2, 2% nonsstematic feed-for&ard encoder sho&n in 'igure 11.2.

a. 'ind the composite generator polnomials gi $% and g2 $%.

b. 'ind the code-&ord v$% corresponding to the set of information se4uences

11.( Consider the $3, 1, (% sstematic feed-for&ard encoder &ith

a. 'ind the time-domain generator matriF G.

b. 'ind the parit se4uences 8$1% and ;$2% corresponding to toe information se4uence a + $1 1 ) 1%.

11.6 Consider the $3, 2, 3% sstematic feed-for&ard encoder &ith

a. ra& the controller canonical form realiGation of this encoder. o& man dela elements arere4uired in this realiGationH

b. ra& the simpler observer canonical form realiGation that re4uires onl three dela elements.

11. Aerif the se4uence of elementar ro& operations leading from the nonsstematic feed-for&ard

realiGations of $11.3#% and $11.)% to the sstematic feedbacD realiGations of $11.66% and $11.1%.

11. ra& the observer canonical form realiGation of the generator matriF G;$% $11.6#% and determine

its overall constraint length v.

11. Consider the rate R + 2283nonsstematic feed-for&ard encoder generator matriF

a. ra& the controller canonical form encoder realiGation for 0$%, 5hat is the overall constraint

length vH

b. 'ind the generator matriF G;$% of the e4uivalent sstematic feedbacD encoder. Is 0 $% realiGableH

If not, find an e4uivalent realiGable generator matriF and dra& the corresponding minimal encoder

realiGation. Is this minimal realiGation in controller canonical form or observer canonical formH

5hat is the minimal overall constraint length vH

11.1) >se elementar ro& operations to convert the rate r + 2283generator matriF of $11.% tosstematic feedbacD form, and dra& the minimal observer canonical form encoder realiGation. 'ind

and dra& a nonsstematic feedbacD controller canonical form encoder realiGation &ith the same

number of states.

11.11 Redra& the observer canonical form realiGation of the $3, 2, 2% sstematic feedbacD encoder in

'igure 11.$b% using the notation of $11.2% and the relabeling scheme of 'igure 11.11.


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11.12 Consider the $3, 1, 2% sstematic feedbacD encoder sho&n in 'igure 11.6$c%. etermine the v + 2

termination bits re4uired to return this encoder to the all-Gero state &hen the information se4uence u +

$1)111%.

11.13 Consider the $#, 3, 3% nonsstematic feed-for&ard encoder realiGation in controller canonical

form sho&n in 'igure 11.3.

a. ra& the e4uivalent nonsstematic feed-for&ard encoder realiGation in observer canonical form, and

determine the number of termination bits re4uired to return this encoder to the all-Gero state. 5hat is

the overall constraint length of this encoder realiGationHb. Eo&, determine the e4uivalent sstematic feedbacD encoder realiGation in observer canonical form,

and find the number of termination bits re4uired to return this encoder to the all-Gero state. 5hat is the

overall constraint length of this encoder realiGationH

11.1# Consider the $2, 1, 2% nonsstematic feed-for&ard encoder &ith 0$% + M1 7 2 1 7 7 2 O.

a. 'ind the 0C of its generator polnomials.

b. 'ind the transfer function matriF G-1 $% of its minimum-dela feed-for&ard inverse.

11.1( Consider the $2, 1, 3% nonsstematic feed-for&ard encoder &ith 0$% + M1 7 2 1 7 7 2 7

3 O.

a. 'ind the 0C of its generator polnomials.

b. ra& the encoder state diagram.

c. 'ind a Gero-output &eight ccle in the state diagram.

d. 'ind an infinite-&eight information se4uence that generates a code-&ord of finite &eight.

e. Is this encoder catastrophic or non-catastrophicH

11.16 'ind the general form of transfer function matriF G-1$% for the feed-for&ard inverse of an $n,

D, v% sstematic encoder. 5hat is the minimum dela 1H

11.1 Aerif the calculation of the 5S' in SFample 11.13.

11.1 Aerif the calculation of the IQ5S' in SFample 11.12.

11.1 Consider the $3, 1, 2% encoder of roblem 11.1.

a. ra& the state diagram of the encoder.

b. ra& the modified state diagram of the encoder.

c. 'ind the 5S' /$?%.

d. ra& the augmented modified state diagram of the encoder.

e. 'ind the IQ5SS /$5, ?, L%.

11.2) >sing an appropriate soft&are pacDage, find the 5S' /$?% for the $#, 3, 3% encoder of 'igure11.3.

11.21 Consider the e4uivalent sstematic feedbacD encoder for SFample 11.1 obtained b dividing

each generator polnomial b g$X% $% + 1 7 2 7 3.

a. ra& the augmented modified state diagram for this encoder.

b. 'ind the IR5S' /$5, @%, the t&o lo&est input &eight C5S's, and the 5S' /$?% for this encoder.

c. Compare the results obtained in $b% &ith the IQ5S', C5S's, and 5S' computed for the

e4uivalent nonsstematic feed-for&ard encoder in SFample 11.1.

11.22 Aerif the calculation of the IQ5S' given in $11.12#% for the case of a terminated convolutional

encoder.

11.23 Consider the e4uivalent nonsstematic feed-for&ard encoder for SFample 11.1# obtained b

multipling 6$% in $11.1#)% b eK$% + 1 7 2.

a. ra& the augmented modified state diagram for this encoder.


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b. 'ind the IQ5S' /$5, ?, I.%, the three lo&est input &eight C5S's, and the 5S' /$?% for this

encoder.

c. Compare the results obtained in $b% &ith the IR5S', C5S's, and 5SS computed for the

e4uivalent sstematic feedbacD encoder in SFample 11.1#.

11.2# In SFample 11.1#, verif all steps leading to the calculation of the bit 5S' in $11.1(#%.

11.2( Consider the $2, 1, 2% sstematic feed-for&ard encoder &ith )$% + M1 17 2 O.

a. :ra& the augmented modified state diagram for this encoder.b. 'ind the IR5S' /$5, @, L%, the three lo&est input &eight C5S's, and the 5S' /$?% for this

encoder.

11.26 Recalculate the IQ5S' /$5, ?, L% in SFample 11.12 using the state variable approach of

SFample 11.1#.

11.2 Recalculate the 5SS /$?% in SFample 11.13 using the state variable approach of SFample

11.1#.

11.2 Consider the $3, 1, 2% code generated b the encoder of roblem 11.1.

a. 'ind the free distance df,-B.

b. lot the complete CI'.

c. 'ind the minimum distance dB,8B.

11.2 Repeat roblem 11.2 for the code generated b the encoder of roblem 11.1(.

11.3) rove that the free distance dire, is independent of the encoder realiGation, i.e., it is a code

propert.

b. rove that the C' d1 is independent of the encoder realiGationJ that is, it is a code propert.

$/ssume that the D F n sub-matriF Ghas full ranD.%

11.31 rove that for non-catastrophic encoders


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0.12. Problems

12.1 ra& the trellis diagram for the $3, 2, 2% encoder in SFample 11.2 and an information se4uence of

length h + 3 blocDs. 'ind the code-&ord corresponding to the information se4uence an + $11, )1, 1)%.

Compare the result &ith $11.16% in SFample 11.2.

12.2 *ho& that the path v that maFimiGes S8E )1 log $ri Z vi% also maFimiGes G-d + vU/ro1 Ci Mlog

cd, &here ci is an real number and c2 is an positive real number.

12.3 'ind the integer metric table for the NC of 'igure 12.3 &hen c1 + 1 and c2 + 1). >se the

Aiterbi algorithm to decode the receivedseq$encerof SFample 12.1 &ith this integer metric table and

the trellis diagram of 'igure 12.1. Compare our ans&er &ith the result of SFample 12.1.

12.# Consider a binar-input, -ar output NC &ith transition probabilities $11% given b the

follo&ing table

'ind the metric table and an integer metric table for this channel.

12.( Consider the $2, 1, 3% encoder of 'igure 11.1 &ith

a. ra& the trellis diagram for an information se4uence of length 8I + #.

b. /ssume a code-&ord is transmitted over the NC of roblem 12.#. >se the Aiterbi algorithm todecode the receivedseq$encer+ $1211, 12)1, )3)1, )113, 12), )311, )3)2%.

12.6 9he NC of roblem 12.# is converted to a *C b combining the soft-decision outputs )1, ).

)3, and )# into a single hard-decision output ), and the soft-decision outputs 11, 12, 13, and 1# into a

single hard-decision output 1. / code-&ord from the code of roblem 12.( is transmitted over this

channel. >se the Aiterbi algorithm to decode the hard-decision version of the received se4uence in

roblem 12.( and compare the result &ith roblem 12.(.

12. / code-&ord from the code of roblem 12.( is transmitted over a continuous-output /50E

channel. >se the Aiterbi algorithm to decode the $normaliGed b% received seq$encer+ $71.2, 7).3,

72.3#. -3.#2, -).1#, -2.#, -1.2, 7).23, 7)., -).63, -).)(, 72.(, -).11, -).((%.

12. Consider a binar-input, continuous-output /50E channel &ith signal-to-noise ratio )s80)+ )

d.

a. *Detch the conditional pdf;s of the $normaliGed b received signal ri given the transmitted bits vi +

[1.

b. Convert this channel into a binar-input, #-ar output smmetric NC b placing 4uantiGation

thresholds at the values ri + -1, ), and 7 1, and compute the transition probabilities for the resulting

NC.

c. 'ind the metric table and an integer metric table for this NC.

d. Repeat parts $b% and $c% using 4uantiGation thresholds ri + -2, ), and 7 2.

12. *ho& that $12.21% is an upper bound on d for d even.

12.1) Consider the $2, 1, 3% encoder of roblem 12.(. Svaluate the upper bounds on event-error

probabilit $12.2(% and bit-error probabilit $12.2% for a *C &ith transition probabilit

$int >se the 5S's derived for this encoder in SFample 11.12.%

12.11 Repeat roblem 12.1) using the approFimate eFpressions for $S% and i,$S% given b $12.26%

and $12.3)%.

12.12 Consider the $3, 1, 2% encoder of $12.1%. lot the approFimate eFpression $12.36% for bit-error

probabilit i, $S% on a *C as a function of S(80)in decibels. /lso plot on the same set of aFes the

approFimate eFpression $12.3% for 1,$S% &ithout coding. 9he coding gain $in decibels% is defined asthe difference bet&een the S(80)ratio needed to achieve a given bit-error probabilit &ith coding and

&ithout coding. lot the coding gain as a function of i, $S%. 'ind the value of )#80)for &hich the

coding gain is ) d, that is, the coding threshold.


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12.13 Repeat roblem 12.12 for an /50E channel &ith un-4uantiGed demodulator outputs, that is, a

continuous-output /50E channel, using the approFimate eFpression for i, $S% given in $12.#6%.

12.1# Consider using the $3, 1, 2% encoder of $12.1% on the NC of roblem 12.#. Calculate an

approFimate value for the bit-error probabilit b$S% based on the bound of $12.3b%. Eo&, convert the

NC to a *C, as described in roblem 12.6, compute an approFimate value for 1,$S% on this *C

using $12.2%, and compare the t&o results.

12.1( rove that the rate 1 + 182R + 182 4uicD-looD-in encoders defined b $12.(% are non-

catastrophic.12.16 Consider the follo&ing t&o nonsstematic feed-for&ard encoders $1% the encoder for the $2, 1,

% optimum code listed in 9able 12.1$c% and $2% the encoder for the $2, 1, % 4uicD-looD-in code listed

in 9able 12.2. 'or each of these codes find

a. the soft-decision asmptotic coding gain J

b. the approFimate event-error probabilit on a *C &ith p + ! 1)-2J

c. the approFimate bit-error probabilit on a *C &ith p + 1)-2J

d. the error probabilit amplification factor /.

12.1 >sing trial-and-error methods, construct a $2, 1, % sstematic feed-for&ard encoder &ith

maFimum di-re,. Repeat roblem 12.16 for this code.

12.1 Consider the $1(,% and $31,16% cclic C codes. 'or each of these codes find

a. the polnomial generator matriF and a lo&er bound on dfB, for the rate1+ 182R + 182convolutional

code derived from the cclic code using Construction 12.1.

b. the polnomial generator matriF and a lo&er bound on !freedfreefor the rate R + 18# convolutional

code derived from the cclic code using Construction 12.2.

$int !is at least one more than the maFimum number of consecutive po&ers of a that are roots of

h$?%.%

12.1 Consider the $2, 1, 1% sstematic feed-for&ard encoder &ith 0$% + M1 1 7 O.

a. 'or a continuous-output /50E channel and a truncated Aiterbi decoder &ith path memor r + 2,

decode the received seq$ence r+ $71.(33, 7).63), -).6#, -3.)13, 71.()6, 7).66#, -).#)1,

7).31(, 72.121. -).3)#, 71.#16, -2.)3#1, 7).1, -).3(1, 71.62(#, -1.16, 72.6(#, -1.)((%

corresponding to an information se4uence of length h + . /ssume that at each level the survivor &ith

the best metric is selected and that the information bit r time units bacD on this path is decoded.

b. Repeat $a% for a truncated Aiterbi decoder &ith path memor r + #.

c. Repeat $a% for a Aiterbi decoder &ithout truncation.

d. /re the final decoded paths the same in all cases H SFplain.

12.2) Consider the $3, 1, 2% encoder of roblem 11.1.

a. 'ind /i $5, ?, L%, /2$5, ?, L%, and /3$5, ?, L%.

b. 'ind rmiB.

c. 'ind d$r% and /do-% for r + ), 1, 2, , rmin U

d. 'ind an eFpression for lim,, d$G%.

12.21 / code-&ord from the trellis diagram of 'igure 12.1 is transmitted over a *C. 9o determine

correct smbol snchroniGation, each of the three 21-bit subse4uences of the se4uence

must be decoded, &here the t&o eFtra bits in r are assumed to be part of a preceding and8or a

succeeding code-&ord. ecode each of these subse4uences and determine &hich one is most liDel tobe the correctl snchroniGed received se4uence.

12.22 Consider the binar-input, continuous-output /50E channel of roblem 12..


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a. >sing the optimalit condition of $12.#%, calculate 4uantiGation thresholds for NCs &ith \ + 2, #,

and output smbols. Compare the thresholds obtained for \ + # &ith the values used in roblem

12..

b. 'ind the value of the hattachara parameter o for each of these channels and for a continuous-

output /50E channel.

c. 'iFing the signal energ + 1 and allo&ing the channel *ER..)s80)to var, determine the increase in

the *ER re4uired for each of the NCs to achieve

the same value of o as the continuous-output channel. 9his *ER difference is called the decibel lossassociated &ith receiver 4uantiGation. $Eote Changing the *ER also changes the 4uantiGation

thresholds.%

$int ou &ill need to &rite a computer program to solve this problem.%

12.23 Aerif that the t&o eFpressions given in $12.% for the modified metric used in the *QA/

algorithm are e4uivalent.

12.2# efine L$r% + In 1$r% as the log-liDelihood ratio, or L-value, of a received smbol r at the output

of an un-4uantiGed binar input channel. *ho& that the L-value of an /50E channel &ith binar

inputs [ ErSs and *ER)s80)is given b

12.2( Aerif that the eFpressions given in $12.% are correct, and find the constant c.

12.26 Consider the encoder, channel, and received se4uence of roblem 12.1.

a. >se the *QA/ &ith full path memor to produce a soft output value for each decoded information

bit.

b. Repeat $a% for the *QA/ &ith path memor r + #.

12.2 erive the eFpression for the bacD&ard metric given in $12.11%.

12.2 Aerif the derivation of $12.123% and sho& that /l + e 1 -1-,-L


20/39


21/39

0.13. Problems

13.1 Consider the $2, 1, 3% encoder Yv8id $% + m V1V p2 7 2 7 3O.

a. ra& the code tree for an information se4uence of length it + #.

b. 'ind the code-&ord corresponding to the, information se4uence an + $1 ) ) 1%.

13.2 'or a binar-input, )-au output smmetric 5IC &ith e4uall liDel input smbols, sho& that the

output smbol probabilities satisf $13.%.13.3 Consider the $2, 1, 3% encoder of roblem 13.1.

a. 'or a *C &ith p +, )#(, find an integer metric table for the 'ano metric.

b. ecode the received se4uence

using the stacD algorithm. Compare the number of decoding steps &ith the number re4uired b the

Aiterbi algorithm. Repeat $b% for the received se4uence

Compare the final decoded path &ith the results of roblem 12.6, &here the same received se4uence is

decoded using the Aiterbi algorithm.

13.# Consider the $2, 1, 3% encoder of roblem 13.1.a. 'or the binar-input, -ar output NC of roblem 12.#, find an integer metric table for the 'ano

metric. $int *cale each metric b an appropriate factor and round to the nearest integer.% *o decode

the received se4uence

using the stacD algorithm. Compare the Vfinal decoded path &ith the result of roblem 12.($b%, -&here

the same received se4uence is decoded using the Aiterbi algorithm.

13.( Consider the $2, 1, 3% encoder of roblem 13.1. 'or a binar-input, continuous-output /50E

channel &ith S, I0)+ 1, use the stacD algorithm and the /50E channel 'ano metric from $13.16% to

decode the receivedseq$encer+ $71.2, 7).3, 72.3#, -3.#2, -).1#, -2.2#, -1.2, 7).23, 7)., -).63,

-).)(, 72.(, -).11, -).((%. Compare the -final decoded path &ith the result of roblem 12., &here the

same received se4uence is decoded using the Aiterbi algorithm.

13.6 Repeat parts $b% and $c% of roblem 13.3 &ith the siGe of the stacD limited to 1) entries, 5hen the

starD is full, each additional entr causes the path on the bottom of the stacD to be discarded. 5hat is

the effect on the final decoded pathH

13. a. repeat SFample 13.( using the stacD-bucDet algorithm &ith a bucDet 4uantiGation interval of (.

/ssume the bucDet intervals are U 7 # to ), -1 to

.b. Repeat part $a% for a 4uantiGation interval of , &here the bucDet intervals are

.13. Repeat SFample 13. for the 'ano algorithm &ith threshold increments of / + ( and / + 1).

Compare the final decoded path and the number of computations to the results of SFamples 13. and

13.. /lso compare the final decoded path &ith the results of the stacD-bucDet algorithm in roblem13..

13. >sing a computer program, verif the results of 'igure 13.13, and plot p as a function of S1,80)$d% for R + 18( and R + #8(.

13.1) *ho& that the areto eFponent p satisfies p + oc and limn-,c p + ) for fiFed channel transition

probabilities. /lso sho& that a Rlap P ).

13.11 roblem

a. 'or a *C &ith crossover probabilit p, plot both the channel capacit C and the cut-off rate Ro as

functions of p. $Eote C + 1 p log2 p $1 - % log2 $1 - p%.%

b. repeat part $a% b plotting C and Ro as functions of the *ER St,80). 5hat is the *ER difference

re4uired to maDe C + Ro + 182H

13.12 roblem

a. Calculate Ro for the binar-input, -ar output NC of roblem 12.#.


22/39


23/39

decoding rule onl if all..1 orthogonal checD-sums include the same number of bits, that is, onl if n=

+ n2 + U + n

13.2 Consider an $n, D, in% convolutional code &ith minimum distance !min+ 2), 7 1. rove that

there is at least one error se4uence a &ith &eight 1' 7 1 in its first $in 7 1% blocDs for &hich a

feedbacD decoder &ill decode to) incorrectl.

13.2 Consider the $2, 1, 11% code of roblem 13.2).

a. 'ind the minimum distance clB,in..

b. Is this code self-orthogonalH

c. 'ind the maFimum number of orthogonal parit checDs that can be formed on eo$o%

d. Is this code completel orthogonaliGableH

13.2 Consider the $3, 1, 3% nonsstematic feed-for&ard encoder &ith 0B, $% + M1 7 73 1 7 3 1

7 7 2 O..

a. 'ollo&ing the procedure in SFample 13.1#, convert this code to a $3, 1, 3% sstematic feed-for&ard

encoder &ith the same $1,Bin.

b. 'ind the generator matriF >s $% of the sstematic feed-for&ard encoder.

c. 'ind the minimum distance dBB%,.

13.3) Consider the $2, 1, 6% code of SFample 13.1(.

a. estimate the bit-error probabilit b$S% of a feedbacD decoder &ith error-correcting capabilit tf-

on a *C &ith small crossover probabilit p.

b. repeat $a% for a feedbacD ma=orit-logic decoder &ith error-correcting capabilit &t.-

c. Compare the results of $a% and $b% for p + 1)-2.

13.31 Repeat roblem 13.3) for the $2, 1, (% code of SFample 13.16.

13.32 'ind and compare the memor orders of the follo&ing codes

a. the best rate1+ 182R + 182self-orthogonal code &ith dBBB + .

b. the best rate1+ 182R + 182orthogonaliGable code &ith dB,,B + .

c. the best rate1+ 182R + 182sstematic code &ith + .

d. the best rate1+ 182R + 182nonsstematic code &ith dfrB + .

13.33 Consider an $1, n ! 1, in% self-orthogonal code &ith Z3 orthogonal checD-sums on eo$i% *+ ), 1,

, n - 2. *ho& that !min+ Z 1, &here Z min$r%, =P,,V2% ZZ.

13.3# Consider the $2, 1, 1% self-orthogonal code in 9able 13.2$a%.

a. 'orm the orthogonal checD-sums on information error bit e9i%

b. ra& the blocD diagram of the feedbacD ma=orit-logic decoder for this code.

13.3( Consider an $n, 1, in% sstematic code &ith generator polnomials g)%$%, *+ 1, 2, , n ! 1.

*ho& that the code is self-orthogonal if and onl if the positive difference sets associated &ith each

generator polnomial are full and dis=oint.

13.36 'ind the effective decoding length n S for the $3, 1, 13% code of SFample 13.1.

13.3 Consider the $2, 1, 11% orthogonaliGable code in 9able 13.3$a%.

a. 'orm the orthogonal checD-sums on information error bit e9$%

b. ra& the blocD diagram of the feedbacD ma=orit-logic decoder for this code.


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0.14. Problems

1#.1 *uppose the $, #% RN code is decoded &ith the Aiterbi algorithm. etermine the number of real

operations $additions and comparisons% re4uired for the follo&ing trellises

a. 9he eight-section bit-level trellis.

b. 9he uniform four-section $t&o-bits per section% trellis sho&n in 'igure .1.

c. Qptimum sectionaliGation based on the Lafourcade-Aard algorithm.1#.2 *uppose the $, #% N code is decoded &ith the differential Aiterbi decoding algorithm based on

the uniform #-section trellis of the code. etermine the number of real operations re4uired to decode

the code.

1#.3 9he first-order RN code of length 16 is a $16, (% linear code &ith a minimum distance of .

ecode this code &ith the Aiterbi algorithm. etermine the number of real operations re4uired for the

decoding based on the follo&ing trellis sectionaliGations

a. 9he 16-section bit-level trellis.

b. 9he uniform eight-section trellis.

c. 9he uniform four-section trellis.

d. Qptimum sectionaliGation based on the Lafourcade-Aard algorithm.

1#.# ecode the $16, (% first-order RN code &ith the differential Aiterbi decoding algorithm based on

the uniform four-section trellis. 'or each section, determine the parallel components, the set of

branches leaving a state at the left end of a parallel component, arid the set of branches entering a stale

at the right end of

a component. ecompose each component into 2-state butterflies &ith doubl complementar

structure. etermine the total number of real operations re4uired to decode the code.

1#.( ecode the $, #% RN code &ith the trellis-based recursive NL algorithm. /t the beginning $or

bottom% of the recursion, the code is divided into four sections, and each section consists of 2 bits. 9hecomposite path metric table for each of these basic sections is constructed directl. evise a recursion

procedure to combine these metric tables to form metric tables for longer sections until the full length

of the code is reached $i.e., a procedure for combining metric tables%. 'or each combination of t&o

tables using the CombCN9$F, J G% procedure, construct the t&o-section trellis 9$$.r, vJ G%% for the

punctured code p.,,,,$C%. etermine the number of real operations re4uired to decode the code &ith the

R.NL-$11,A% algorithm.

1#.6 ecode the $16, (% N code &ith the RNL-$1,A% algorithm using uniform sectionaliGation. /t

the beginning, the code is divided into eight sections, of 2 bits each. evise a recursion procedure to

combine composite path metric tables. 'or each combination of t&o ad=acent metric tables, construct

the special t&o-section trellis for the corresponding punctured code. etermine the total number of realoperations re4uired to decode the code.

1#. Repeat roblem 1#.6 b dividing the code into four sections, # bits per section, at the beginning

of the recursion. Compare the computation compleFit of this recursion &ith that of the recursion

devised in roblem 1#.6.

1#. evise an iterative decoding algorithm based on a minimum-&eight trellis search using the

ordered statistic decoding &ith order-1 reprocessing $presented in *ection 1)..3)% to generate

candidate code-&ords for optimalit tests. /nalGe the computational compleFit of our algorithm.

9o reduce decoding computational compleFit, the order ishould be small, sa i+ ), 1, or 2. 9he

advantage of ordered statistic decoding over the Chase- decoding is that it never fails to 0enerate

candidate code-&ords.

1#. *imulate the error performance of the iterative decoding algorithm devised in roblem 1#. for

the $32, 16% RN code using order-1 reprocessing to generate 1 candidate code-&ords for testing and

search of the NL code-&ord. etermine the average numbers of real operations and decoding

iterations re4uired for various *ER.


25/39

1#.1) ecode the $32, 16% N code &ith N/ and NaF-log-Nap decoding algorithms based on a

uniform four-section trellis. *imulate and compare the error performances for t&o algorithms, and

compare their computational compleFities.

1#.11 9he $32, 16% RN code can be decomposed into eight parallel and structurall identical four-

section sub-trellises. ecode this code &ith the parallel NaF-log-N/ algorithm. Compute the number

of real operations re4uired to process a single sub-trellis and the total number of real operations

re4uired to decode the code. /lso determine the siGe of the storage re4uired to store the branch

metrics, state metrics, and the liDelihood ratios.


26/39

0.15. Problems

1(.1 rove that the concatenation of an $n1, D1% inner code &ith minimum distance d1 and an $n3, lo%

outer code &ith minimum distance d has a minimum distance of at least did2.

1(.2 rove the lo&er bound of the minimum distance of an di-level concatenated code given b 1(.12.

1(.3 Consider the concatenation of a R* outer code over GF$2m% and the binar $in 7 1, in, 2% single

parit-checD inner code. evise an error-erasure decoding for this concatenated code. Mint uring

the inner code decoding, if parit failure is detected in in 7 1 received bits, an erasure is declared. if no

parit failure is detected, the parit bit is removed to form a smbol in GF$21


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0.16. Problems

16.1 rove that the general rate R + 1turbo encoder sho&n in 'igure 16.1$a%, &here encoders 1 and 2are linear convolutional encoders $not necessar identical% separated b an arbitrar inter-leaver, is a

linear sstem.

16.2 'or the length W + 16 4uadratic inter-leaver of $16.%, determine all pairs of indices that are

interchanged b the permutation.

16.3 Consider a CC &ith t&o different constituent codes the $, #, 3% amming code and the $, #,

#% eFtended amming code. 'ind the C5S's, IR5S's, and 5S's of this code assuming a uniform

inter-leaver.

16.# 'ind the IR5S's and 5S's for SFample 16.(.

16.( Repeat SFample 16.( for the case h + #. 5hat is the minimum distance of the $#), 16% CC if a

# F # ro&-column $blocD% inter-leaver is usedH

16.6 Consider a CC &ith the $2#, 12, % eFtended 0ola code in sstematic form as the constituent

code.

a. 'ind the C5S's /u, $@%, & + 1, 2, , 12, of this code b generating the code-&ords of the $23, 12,

% 0ola code in sstematic form and then adding an overall parit checD.

/ssuming a uniform inter-leaver,

b. find the C5S's /rc $@%, & + 1, 2, , 12J

c. find the IR5S's /c $5, @% and C $5, @%J and

d. find the 5S's /c$?% and c $?%.

e. Eo&, consider a CC &ith the 12-repeated $2#, 12, % eFtended 0ola code sstematic form as the

constituent code. /ssume that the information bits are arranged in a s4uare arra and that a ro&-

column inter-leaver is usedJ that is, encoder 1 encodes across ro&s of the arra, and encoder 2 encodes

do&n columns of the arra. 'ind the parameters /, of the CC.

16. rove $16.3%.

16. 'ind the code&ord IR5S' and 5S' for the CCC in SFample 16.6.

16. 'ind the bit IN/8S's and 5SI9;s for the CCC in SFample 16.6.

16.1) Repeat SFample, 16.6 for the encoder &ith the reversed generators given in $16.62%.

16.11 Repeat SFample 16. for the encoder &ith the reversed generators given in $16.62%.

16.12 'ind of &eight- code-&ords inSFa-2-iple 16.3.

16.13 Repeat SFample 16,3 using the feed-for&ard encoder

for the second costituent code

16.1# Consider a rate R + 18# multiple turbo code $'CCY;% constituent encoder

separated b t&o random inter-leavers $see ;'igure 16.2%. /ssuming a uniform inter-leaver and large

blocD siGe W,

a. find the approFimate C/AS's $,% and. ,,, c $@% for & + 2, 3, #, (J

b. find the approFimate IR5S's / C, and s,; 8(c $5. @%J

c. find the approFimate ;t/8S's / C $?i and S $; $W%J and

d. sDetch the union bounds on ',,, $S% and i, $S% for W + 1))) and W + 1)))),

assuming a binar-input, un-4uantiGed-output /50E channel,

16.1( 'ind the minimum-v.8eight code-&ords corresponding to input &eights 2 and 3 for the CCCs,

&hose generators are given in 9able. 16.6. In each case determine the free distance dfree assuming

large W.


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29/39

0.17. Problems

1.1 oes the follo&ing matriF satisf the conditions of the parit-checD matriF of an 5C code given

b efinition 1.1H etermine the ranD of this matriF and give the co!e-2or!s of its null space.

1.2 'orm the transpose 9 of the parit-checD matriF given in roblem 1.1. Is a lo&-densit parit-

checD matriFH etermine the ranD of 119 and construct the code given b the null space of 9.

1.3 rove that the $n, 1% repetition code is an LC code. Construct a lo&-densit parit-checD

matriF for this code.

1.# Consider the matriF &hose columns are all the m-tuples of &eight 2. oes satisf the

conditions of the parit-checD matriF of an LC codeH etermine the ranD of and its null space.

1.( 9he follo&ing matriF is a lo&-densit parit-checD matriF. etermine the LC code given b

the null space of this matriF. 5hat is the minimum distance of this codeH

1.6 rove that the maFimum-length code of length 2;< ! 1 presented in *ection .3 is an LC code.

1. Construct the 9anner graph of the code given in roblem 1.1. Is the 9anner graph of this code

acclicH Zustif our ans&er.

1. Construct the 9anner graph of the code given in roblem 1.2. Is the 9anner graph of this codeacclicH Zustif our ans&er.

1. Construct the 9anner graph of the code given b the null space of the parit-checD matriF given in

roblem 1.(. oes the 9anner graph of this code contains ccles of length 6H etermine the number

of ccles of length 6 in the graph.

1.1) etermine the orthogonal checD-sums for ever code bit of the LC code given b the null

space of the parit-checD matriF of roblem 1.(.

1.11 rove that the minimum distance of the 0allager-LC code given in SFample 1.2 is 6.

1.12 etermine the generator polnomial of the t&o-dimensional tpe-I $), 3%th-order cclic S0-

LC code constructed based on the t&o-dimensional Suclidean geometr S0$2, 23%.1.13 etermine the parameters of the parit-checD matriF of the three-dimensional tpe-I $),2%th-

order cclic S0-LC code C$L,$3, ), 2%. etermine the generator polnomial of this code. 5hat are

the parameters of this codeH

1.1# etermine the parameters of the companion code of the S0-LC code given in roblem 1.13.

1.1( ecode the t&o-dimensional tpe-I $), 3%th-order cclic S0-LC code &ith one-step ma=orit-

logic decoding and give the bit- and blocD-error performance for the /50E channel &ith *W

signaling.

1.16 Repeat roblem 1.1( &ith ' decoding.

1.1 Repeat roblem 1.1( &ith &eighted ma=orit-logic decoding.

1.1 Repeat roblem 1.1( &ith &eighted ' decoding.

1.1 Repeat roblem 1.1( &ith */ decoding.

1.2) ecode the three-dimensional tpe-II $), 2%th-order 4uasi-cclic S0-LC code given in

roblem 1.1# &ith */ decoding, and give the bit- and blocD-error performance of the code for the

/50E channel &ith *W signaling.

1.21 Consider the parit-checD matriF $SlL, of the three-dimensional tpe-I $), 2%th order cclic

S0-LC code given in roblem 1.13. *plit each column of this

parit-checD matriF into five columns &ith rotating &eight distribution. 9he result is a ne& to, r-densit parit-checD matriF that gives an eFtended SC-LC code. ecode this code &ith */

decoding and give its bit- and blocD-error performances.

1.22 Construct a parit-checD matriF of the 0allager-LC code &ith the follo&ing parameters n +

6, p + #, and t + 3. Choose column -permutations for the sub-matrices such that JD. is no greater than 1.


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1.23 rove that the 9anner graph of a finite-geometr LC code contains ccles of length 6.

enumerate the number of ccles of length 6.

1.2# rove that the minimum distance of an S0-0allager LC code must be even. >se the result to

prove the lo&er bound on minimum distance given b $1.6%.

1.2( Construct an S0-0allager LC code using siF parallel bundles of lines in the t&o-dimensional

Suclidean geometr 1S)$2,2;% over 01-$2(%. Compute its bit- and blocD-error performances &ith */

decoding.

1.26 Construct a masDed S0-0allager SC code of length 1)2# b decomposing the incidencematrices of eight parallel bundles of lines in S0$2, 2;% into 32 F 32 -permutation matrices. 9o construct

such a code, set p + 32, and form an F 32 masDing matriF &ith column and ro& &eights # and 16,

respectivel, using four primitive -tuples over, 0'$2%. Compute the bit- and blocD-error performances

using */ decoding.

1.2 9he incidence vectors of the lines in S0$2, 2(% not passing through the origin form a single 1)23

F 1)23 circulant 0 &ith &eight 32. Construct a rate-182 4uasi-cclic code of length 1# b

decomposing 0 into a # F arra of 1)23 F 1)23 circulant permutation matrices. Compute the bit- and

blocD-error performance of the code &ith */ decoding.

2.2 rove that there eFist a primitive element a in 0 '$2#1% and an odd positive integer c less than

2#1 such that );6;1 7 1 + oc. esign a concatenated turbo coding sstem &ith a finite-geometr LCcode of our choice as the outer code. Construct the inner turbo code b using the second-order

$32B 16% RN code as the component code. 0ive the bit-error performance of our designed sstem.


31/39

0.18. Problems

1.1 rove e4uation $1.%.

1.2 'ind, as functions of the parameter d, the /S5Ss /2,;$?% and the NS5Ss for the t&o signal set

mappings sho&n in 'igure -1.2, and determine if the are uniform. /ssume each constellation has

unit average energ.

'I0>RS -1.2

1.3 etermine if an isometr eFists bet&een the subsets \$)% and \$1% for the t&o signal set

mappings in roblem 1.2.

1.# >se Lemma 1.1 to prove that for uniform mappings, 11,,, $?% can be computed b labeling the

error trellis &ith the /S5Ss and finding the transfer function of the modified state diagram.

1.( Construct a countereFample to sho& that Lemma 1.1 does not necessaril hold for rate R + D8$D

7 2% codes. *tate a rate R + D8$D 7 2% code lemma, similar to Lemma 1.1, specif the conditions for

uniformit, and prove the lemma.

.'I0>RS-1.

1).6 etermine the /S5Ss /2$?% and the NS5Ss g $?% for 0ra- and naturall mapped #-/N andsho& that the are both uniform mappings.

1. Consider mapping a rate R + 2283convolutional code into -/N using natural mapping.

a. etermine the /S5Ss /-H. $?% and the NS5Ss 3, 2 $?% for this mapping.

b. etermine if the mapping is uniform.

c. 'ind the coding gain $or loss% for the three #-state, rate R + 2283convolutional codes of SFample1.# compared &ith un-coded \*W.

d. Can ou find a #-state, rate R + 2283convolutional code &ith a better coding gain &hen used &ith

naturall mapped -/NH1. *ho& that the 0ra mapping of the -*W signal set sho&n in 'igure -1. is not uniform.

1.). Repeat SFample 1.#, finding the N'*S distances and asmptotic coding gains for three rate R

+ 2283trellis-coded -*W sstems, if natural mapping is replaced b the uniform mapping of 'igure1,2$a%. Compare the results &ith natural mapping.

1.1) Repeat SFample 1.( b finding a countereFample to the rate R + D8$D 1% code lemma for the

non-uniform signal set mapping in roblem 1.2$a%.

1.11 Repeat SFample 1.#, finding the N'*S distances and asmptotic coding gains for three rate R

+ 2283trellis-coded -*I* sstems, if natural mapping is replaced b the non-uniform 0ra-mapped

-*I* signal set in roblem 1.. $In this case, since the rate R + Dl$D 7 1% code lemma is notsatisfied, the distances bet&een all possible path pairs must be considered.% Compare the results &ith

natural mapping.

1.12 *ho& that set partitioning of the infinite t&o-dimensional integer lattice @2 results in a regular

mapping.

1.13 /ppl mapping b set partitioning to the 32-CRQ** signal constellation and determine the error

vectors efor &hich $1.26% is not satisfied &ith e4ualit.

1.1# Construct an eFample in &hich $1.2(% and $1.2% do not give the same result.

1.1( /ppl mapping b set partitioning to the -/N signal constellation and determine the N**s

/2, i+ ), 1, 2. 'ind the asmptotic coding gain and the average number of nearest neighbors/doB &hen the #-state code of 9able 1.6$a% is applied to -/N. Repeat for the one-dimensional

integer lattice @1.

1.16 Compute, as functions of the parameter d, the asmptotic coding gains of the 16-\/N codes in

9able 1.6$b% compared &ith the follo&ing un-coded constellations


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'I0>RS-1.16

$i% the -C :** constellation sho&n in 'igure -1.16$a% and $ii% the -\/N constellation sho&n in

'igure -1.16$b%. /ssume each constellation has unit average energ.

1.1 Calculate el,YB,$5, ?% for SFample 1..

1.1 /ppl mapping b set partitioning to the -\/N signal constellation sho&n in roblem 1.16

and determine the N**s i+ ), 1, 2. 'ind /ar, $5, ?% and /L%) $1#, ?% for the code of SFample

1. using this constellation.

1.1 Let 1$% + 1 7 7 2 7 3 7 in SFample 1.1) and recalculate $1.3% and $1.#)%. /re the

conditions for rotational invariance affectedH

1.2) erive general conditions on the number of terms in Ih$1% $% and RI< $% to satisf $1.#6%.

1.21 *ho& that $1.(2% is still satisfied &hen the rotated binar se4uences for naturall mapped

\*W given in $1.3% are substituted into the e4uation, and h$X% $% has an odd number of nonGero

terms.

1.22 Aerif that $1.(2% is satisfied for the encoder of $1.(3% &hen the rotated binar se4uences for

naturall mapped \*W given in $1.3% are substituted into the e4uation.

1.23 'ind minimal encoder realiGations for the )X rotationall invariant v + # and + ( nonlinear rate

1+ 182R + 182codes based on the parit-checD matrices

and

respectivel. *ho& that the v + ( case cannot be realiGed &ith 32 states.

1.2# erive general conditions on the number of nonGero terms in h$2% $%, h$1% $%, and 11)% $% to

satisf $1.(%.

1.2( *ho& that the #(X rotated binar code se4uences for naturall mapped -*W are 11.2%$given b

%..1$2% $% v$i% $% v$o% $%, ]vp% $% v$1% $% v$o% $.1=% and v$)%$% + v$)% $% 1$%.

1.26 *ho& that $1.61% is still satisfied &hen the rotated binar se4uences for naturall mapped -5

given in roblem 1 2( are substituted into the e4uation, and Ih$X% $% has an odd number of nonGeroterms.

1.2 *ho& ho& f$% in $1.6(b% can be re&ritten to correspond to the encoder realiGation sho&n in

'igure, 1.2(.

1.2 >se the method of Suclidean &eights to no& that dfree, + 3.(1( for 16-state, rate R + 2283,trellis-coded - *W &ith ll-i.$21 + (%, h$1% + $1 2%, and 1111$)% + $2 3%.

1.2 rove $1.%J that is, the partition level o e4uals the sum of the redundancies of the r linear

blocD codes that define the sub-code / p$)%.

1.3) ra& the appropriate signal set mappers, similar to 'igure 1.3), for the three 3 ? -*W

partitions of SFample 1.1.

1.31 ra& the complete encoder diagram for the -state, 2? 16-R C, + 3.( bits8smbol encoder listed

in 9able 1.1($b%, including differential encoding of appropriate input bits. >se $1.3% to eFpress the

first three 2 F16-*W encoder output signals in both unar and integer form, assuming the input

se4uence at + $no,% + $1)11))1. )))111). 11)111), %, and the encoder starts in the all-Gero state.

1.32 >se the approach of SFample 1.1 to determine the rotational invariance of the t&o 16-state, 3F

-u ;, + 2.33 ;bits8smbol encoders listed in 9able 1.1($a%.

1.33 ra& the augmented, modified state diagram and fl ii the /IQ5S's in SFample 1.2). Include

sDetches of ;I,$S% versus 80)and uncoiled Q*W, and estimate the real coding gain at a SR of 1)!(.

1.3# Repeat SFample 1.2) for $1.% a 2F 1.6-\/N sstem &ith i + 3.( bits8smbol and $2% trellis-coded using the -state code listed in 9able 1.1($c% &ith 4 + ).

1.3( /ssuming a distance of # bet&een neighboring signal points, calculate the average energies of

the 12-point signal set in 'igure 1.3# and of its shaped #- version as functions of 61, and compute

the shaping gain.


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1.36 /ssuming a distance of !bet&een neighboring signal points, compute the CSRs $compared &ith

2-/N% and the /Rs of the 1- signal sets 2-/N, #-/N, and -/N as functions of !.

1.3 ra& the encoder corresponding to the rate .R + 2283parit-checD matriF of $1.112%, andsho& that it is e4uivalent to the encoder in 'igure 1.3(.


34/39

0.19. Problems

1.1 rove that the minimum s4uared Suclidean distance of the 3-level -*W code given in SFample

1.1 is e4ual to #.

1.2 Construct a 3-level -*W code &ith the follo&ing three binar component codes $1% C1is the

$16, 1, 16% repetition codeJ $2% C2 is the $16, 11, #% second-order RN codeJ and $3% C3 is the $16, 1(, 2%

single parit code.

a. etermine the spectral efficienc of the code.

b. etermine the minimum s4uared Suclidean, smbol, and product distances of the code.

c. /nalGe the trellis compleFit of the code.

1.3 ecode the 3-level -*W code constructed in roblem 1.2 &ith a single-stage Aiterbi decoding,

and compute its error performance for an /50E channel.

1.# Replace the first component code C1in roblem 1.2 &ith the first-order $16, (, % RN code.

Construct a ne& 3-level -*W code. etermine its spectral efficienc, minimum s4uared Suclidean,

smbol, and product distances. /nalGe its trellis compleFit.

1.( ecode the code constructed in roblem 1.# &ith a three-stage soft-decision decoding. Sachcomponent code is decoded &ith Aiterbi decoding based on its trellis. Compute its error performance

for an /50E channel.

1.6 esign a single-level concatenated coded modulation sstem &ith the E/*/ standard $2((, 223%

R* code over GF$2% as the outer code and a 3-level -*W code of length 16 as the inner code. 9he

inner code is constructed using the follo&ing binar codes as the component codes $1% C1is the $16, 1,

16% repetition codeJ $2% C2 is the $16, 1(, 2% single-parit-checD codeJ and $3% C2 is the $16, 16, 1%

universal code. 5hat is the spectral efficienc of die overall sstemH ecode the inner code &ith a

single-stage Aiterbi decoding and the outer code &ith an algebraic decoding. Compute the error

performance of the sstem for an /50E channel.


35/39

0.20. Problems

2).1 *ho& that if an $n, D% cclic code is designed to correct all burst errors of length 1 or less and

simultaneousl to detect all burst errors of length d K 1 or less, the number of parit-checD digits of the

code must be at least 1 d.

2).2 evise an error-trapping decoder for an 8-burst-error-correcting cclic code. 9he received

polnomial is shifted into the sndrome register from the right end. escribe the decoding operation of

our decoder.

2).3 rove that the 'ire code generated b $2).#% is capable of correcting an error burst of length 1 or

less.

2).# 9he polnomial p$X% + 1 7 ? 7 X# is a primitive polnomial over GF$2%. 'ind the generator

polnomial of a 'ire code that is capable of correcting an single error burst of length # or less. 5hat

is the length of this codeH evise a simple error-trapping decoder for this code.

2).( evise a high-speed error-trapping decoder for the 'ire code constructed in roblem 2).#.

escribe the decoding operation.

2).6 >se a code from 9able 2).3 to derive a ne& code &ith burst-error-correcting capabilit 1 + (1,

length n + 2((, and burst-error-correcting efficienc G + 1. Construct a decoder for this ne& code.

2). Let g$?% be the generator polnomial of an $n, D% cclic code. Interleave this code to degree ?.

9he resultant code is a $.11, -% linear code. *ho& that this interleaved code is cclic and its generator

polnomial is g$? ?%.

2). *ho& that the urton code generated b g$?% + $?


36/39

0.21. Problems

21.1 >sing mathematical induction, sho& that the unDno&n elements of the matriF o can al&as be

chosen so that $21,11% is satisfied.

21.2 *ho& ho& to construct optimum phased-burst-error-correcting erleDamp-reparata codes &ith D

P n ! 1.

2.3 Consider the erleDamp-reparata code &ith it + 3.

a. 'ind in, 6, and g for this code.

b. 'ind the B)matriF.

c. 'ind the generator polnomials $% and e$%.

d. 'ind the )matriF.

e. ra& the complete encoder8decoder blocD diagram for this code.

21.# Consider the I&adare-Nasse cede &ith a + 2 and 2 + #.

a. 'ind, n, 6, and g for this code.

b. 'ind the generator polnomial g$1%$%.

a. 'ind the repeat distance of the information error bit e8$X%.

d. ra& the complete encoder8decoder blocD diagram for this code.

21.( / second class of I&adare-Nasse codes eFists &ith the follo&ing parameters

9he 1 ! 1 generator polnomials are given b $21.2)%, &here all% # $a - i%$#2 7 n - i- 3% 7 n ! 1, and

6$i% -% $#? 7 1 - i! 1% 7 n 72 - 2. Consider the code &ith a + 3 and ? + 3.

a. 'ind )1,6, and g for this code.

b. 'ind the generator polnomials g12% $% and g2$2%$%.

c. 'ind the repeat distance of the information error bits ei$)% and ei$1.%.

d. Construct a decoding circuit for this code.

21.6 Construct a general decoding circuit for the class of I&adare-Nasse codes in roblem 21.(. 'or

the t&o classes of I&adare-Nasse codes

a. compare the eFcess guard space re4uired beond the 0allager boundJ and

b. compare the number of register stages re4uired to implement a general decoder.

21. *ho& that for the I&adare-Nasse code of SFample 21.#, if iiMin7 $? 2,-2%11-11 -1 + (

consecutive error-free bits follo& a decoding error, the sndrome &ill return

to the all-Gero state.

21. Consider the $2, 1. (% double-error-correcting orthogonaliGable code from 9able 13.3 interleaved

to degree 2 + .

a. Completel characteriGe the multiple-burst-error-correcting capabilit and the associated guard-

space re4uirements of this interleaved code.

b. 'ind the maFimum single-burst length that can be corrected and the associated guard space.

c. 'ind the ratio of guard space to burst length for $b%.

d. 'ind the total memor re4uired in the interleaved decoder.

e. ra& a blocD diagram of the complete interleaved sstem.

21. Consider the interleaved encoder sho&n in 'igure 21.6$b%. /ssume that an information

se4uence 8to, 1,1, )2, U enters the encoder. 5rite do&n the string of encoded bits and verif that an

interleaving degree of + ( is achieved.


37/39

21.1) Consider the erleDamp-reparata code of roblem 21.3 interleaved to degree + .

a. 'ind the g8b ratio and compare it &ith the 0allager bound.

b. ra& a blocD diagram of the complete interleaved sstem.

21.11 Consider the n + 3 erleDamp-reparata code interleaved to degree ? + and the n + 3 I&adare-

Nasse code &ith ? + .

a. Compare the g8b ratios of the t&o codes.

b. Compare the number of register stages re4uired to implement the decoder in both cases.

21.12 Consider the $2, 1, % sstematic code &ith g$1% $% + 1 7 2 7 ( 7 .

a. Is this code self-orthogonalH 5hat is t34for this codeH

b. Is this a diffuse codeH 5hat is the burst-error-correcting capabilit b and the re4uired guard space gH

c. ra& a complete encoder8decoder blocD diagram for this code.

21.13 'or the diffuse code of 'igure 21., find the minimum number of error-free bits that must be

received follo&ing a decoding error to guarantee that the sndrome returns to the all-Gero state.

21.1# Consider using the $2, 1, 11% triple-error-correcting orthogonaliGable code from 9able 13.3 in the

0allager burst-finding sstem.

a. ra& a blocD diagram of the encoder.

b..ra& a blocD diagram of the decoder.

c. 5ith t/1 # + 1, choose N and L such that the probabilities of an undetected burst and of a false

return to the r-mode are less than 1)-2 and the g8b ratio is &ithin 1^ of the bound on


38/39

0.22. Problems

22.1VIn $22.(% &e sa& that the throughput of the go-bacD-E /R\ depends on the

channel blocD error rate + 1- - p%


39/39

22., esign a tpe-8I hbrid /R\ sstem using a rate-1convolutional code similar

to the sstem presented in *ection 22..

22. Let C be a half-rate invertible $2k, D% sstematic linear blocD code. Let u be an

information se4uence of D bits and f $aro be its corresponding parit se4uence.

rove that both $a,..f $a%% and $pa%, al% are code-&ords in C.

22,.1)Consider the R* outer code C2 defined in *ection 22.. rove that the parit &ord

RMv$?%O given b $22.3)% is also a code-&ord in C.

22,.11 esign a tpe-II hbrid /R\ sstem in &hich a R* code C2 over 0 '$2m% is used

for for&ard error correction, and a half-rate R* code C. obtained b shortening

CH is used for parit retransmission. 9his is simpl the hbrid sstem presented in

*ection 22. &ithout an inner code.

22.12 9he inner code C1of the hbrid sstem presented in *ection 22. can be

chosen as a binar $n, D% code designed for simultaneous error correction

and detection. esign a concatenated hbrid /R\ sstem &ith Cl as theinner code.

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Problems linear block codes Error_Control_Coding_SHU LIN COSTELLO_2nd Edition_2004