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8/6/2019 Upper Bounds for Ring Linear Codes
http://slidepdf.com/reader/full/upper-bounds-for-ring-linear-codes 1/32
Upper Bounds
forRing-Linear
Codes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
Vitaly
Skachek
Upper Bounds for Ring-Linear Codes
Eimear Byrne1, Marcus Greferath1, Axel Kohnert2,
Vitaly Skachek1
1Claude Shannon Institute andSchool of Mathematical Sciences
University College DublinIreland
2Dept MathematicsUniversity of Bayreuth
Germany
May 19 2009
8/6/2019 Upper Bounds for Ring Linear Codes
http://slidepdf.com/reader/full/upper-bounds-for-ring-linear-codes 2/32
Upper Bounds
forRing-Linear
Codes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
Vitaly
Skachek
Outline
Codes over finite fields
Code optimalityBounds for codes for the Hamming weight
Ring-linear coding
The homogeneous weight
Bounds on the size of a code for the homogeneous weight
8/6/2019 Upper Bounds for Ring Linear Codes
http://slidepdf.com/reader/full/upper-bounds-for-ring-linear-codes 3/32
Upper Bounds
forRing-Linear
Codes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
Vitaly
Skachek
Notation
F = GF (q ), q = p m some prime p
R is a finite ring with identity
R̂ := HomZ(R ,C×) the characters on (R , +)
χ ∈ R̂ is a character on (R , +)
C is a code of length n and minimum distance d
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-Linear
Codes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
Vitaly
Skachek
Using Codes for Error Correction
One parameter that indicates the error-correcting capability of a code is its minimum distance.
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-Linear
Codes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Using Codes for Error Correction
One parameter that indicates the error-correcting capability of a code is its minimum distance.The higher the minimum distance, the more errors that can bedetected and corrected by the receiver.
8/6/2019 Upper Bounds for Ring Linear Codes
http://slidepdf.com/reader/full/upper-bounds-for-ring-linear-codes 6/32
Upper Bounds
forRing-Linear
Codes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Using Codes for Error Correction
One parameter that indicates the error-correcting capability of a code is its minimum distance.The higher the minimum distance, the more errors that can bedetected and corrected by the receiver.
8/6/2019 Upper Bounds for Ring Linear Codes
http://slidepdf.com/reader/full/upper-bounds-for-ring-linear-codes 7/32
Upper Bounds
forRing-Linear
Codes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Using Codes for Error Correction
One parameter that indicates the error-correcting capability of a code is its minimum distance.The higher the minimum distance, the more errors that can bedetected and corrected by the receiver.
8/6/2019 Upper Bounds for Ring Linear Codes
http://slidepdf.com/reader/full/upper-bounds-for-ring-linear-codes 8/32
Upper Bounds
forRing-Linear
Codes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Code Optimality
The Main Coding Problem:
1 For fixed length n and minimum distance d , what is themaximum size of any code over R ?i.e., what is AR (n, d )?
2 For a fixed length n and minimum distance d , what is the
maximum size of any linear code over R ?i.e., what is B R (n, d )?
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-Linear
Codes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Some Distance Functions
Definition (Hamming Metric)
Let u, v ∈ R n. The Hamming distance between u and v is thenumber of components where u and v differ, i.e.
dHam(u, v) = |{i : u i = v i }|
u = [0, 0, 1, 1, 3, 3], v = [1, 2, 2, 1, 1, 3] ∈ Z4
d Ham(u, v) = 4.
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Some Distance Functions
Definition (Lee Metric)
Let u , v ∈ Zm. The Lee distance between u and v is theabsolute value modulo m of u − v , i.e.
dLee(u , v ) = |u − v |m =
u − v if u − v ∈ {0, ..., m/2}v − u otherwise
If u, v ∈ Znm then dLee(u, v) =
i =1..n |u i − v i |m.
u = [0, 0, 1, 1, 3, 3], v = [1, 2, 2, 1, 1, 3] ∈ Z4
d Lee(u, v) = 1 + 2 + 1 + 2 = 6
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Some Bounds for Codes over Finite Fields
Singleton: |C | ≤ Aq (n, d ) ≤ q n−d +1
Hamming: |C | ≤ Aq (n, d ) ≤ q n
V q (n, d −12
),
Plotkin: |C | ≤ Aq (n, d ) ≤ d d −γ n
, γ = q −1q
, if n < d γ
Gilbert-Varshamov: Aq (n, d ) ≥ q n
V q (n,d −1)
Elias-Bassalygo bound
Mc-Eliece-Rodemich-Rumsey-Welch boundLinear Programming bound
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Asymptotic Representations
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Codes over Finite Rings
Definition
An code of length n over R is a nonempty subset of R n. A(left) linear code of length n over R is a left R -submodule of R n.
We will usually assume that R is a finite Frobenius ring.
Many of the foundational results of classical coding theory (e.g.the MacWilliams’ theorems) can be extended to the finite ringcase when R is Frobenius.
[Wood, Honold, Nechaev, Greferath, Schmidt..]
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Finite Frobenius Rings
For a finite ring R , R̂ is an R − R bimodule via
χr (x ) = χ(rx ), r χ(x ) = χ(xr )
for all x , r ∈ R , χ ∈ R̂ .R is a finite Frobenius ring iff
R R R R̂
Then R R̂ = R χ for some (left) generating character χ
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Finite Frobenius Rings
Let R and S be finite Frobenius rings, let G be a finite group.The following are examples of Frobenius rings.
integer residue rings Zm
Galois rings
principal ideal rings
R × S
the matrix ring M n(R )the group ring R [G ]
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Homogeneous Weights
Definition
A weight w : R −→ Q is (left) homogeneous , if w (0) = 0 and
1 If Rx = Ry then w (x ) = w (y ) for all x , y ∈ R .
2 There exists a real number γ such that
y ∈Rx
w (y ) = γ |Rx | for all x ∈ R \ {0}.
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Examples of Homogeneous Weights
ExampleOn every finite field Fq the Hamming weight is a homogeneousweight of average value γ = q −1
q .
Example
On Z4 the Lee weight is homogeneous with γ = 1.
x 0 1 2 3
w Lee(x ) 0 1 2 1
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Examples of Homogeneous Weights
ExampleOn Z10 the following weight is homogeneous with γ = 1:
x 0 1 2 3 4 5 6 7 8 9
whom(x ) 0 1 54 1 5
4 2 54 1 5
4 1
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Examples of Homogeneous Weights
Example
On the ring R of 2 × 2 matrices over GF(2) the weight
w : R −→ R, X →
0 : X = 0,2 : X singular, X = 0,1 : otherwise,
is a homogeneous weight of average value γ = 32 .
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Examples of Homogeneous Weights
Example
On a local Frobenius ring R with q -element residue field theweight
w : R −→ R, x →
0 : x = 0,q
q −1 : x ∈ soc (R ), x = 0,
1 : otherwise,
is a homogeneous weight of average value γ = 1.
Which finite rings admit a homogeneous weight?
Up to the choice of γ , every finite ring admits a uniquehomogeneous weight .
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Homogeneous Weights of FFRs
Theorem (Honold)
Let R be a finite Frobenius ring with generating character χ.
Then the homogeneous weights on R are precisely the functions
w : R −→ R, x → γ
1 −1
|R ×|
u ∈R ×
χ(xu )
where γ is a real number.
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Bounds
forRing-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Bounds on AR (n, d ) for the Homogeneous Weight
The following bounds have been found for codes over FFRs forthe homogeneous weight.
Sphere-packing (Hamming)
Sphere-covering (Gilbert-Varshamov)
Plotkin-like bounds
Elias-like bounds
Singleton-like boundLinear programming bound
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Boundsfor
Ring-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
A Key Lemma
Lemma
Let C ≤ R R n
be a linear code, and let x ∈ R n
. Then1
|C |
c ∈C
w (x + c ) = γ |supp(C )| +
i ∈supp(C )
w (x i ).
C
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Boundsfor
Ring-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Residual Codes
Definition
Let C ≤R R n, c ∈ R n. Res (C , c ) := {(x i ) : x ∈ C , c i = 0}.Example
Let C be the Z4-linear code generated by
1 0 0 0 3 1 2 10 1 0 0 1 2 3 10 0 1 0 3 3 3 20 0 0 1 2 3 1 1
.
Let c = [0, 0, 0, 2, 0, 2, 2, 2]. Then Res (C , c ) is generated by
1 0 0 30 1 0 10 0 1 3
0 0 0 2
.
R id l C d
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Boundsfor
Ring-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Residual Codes
Theorem
Let C ≤ R R n have minimum homogeneous weight d, and let c ∈ C satisfy (c ) := w Ham < d
γ . Then Res (C , c ) has
length n − (c ),
minimum homogeneous weight d ≥ d − γ(c ),
|Res (C , c )| =|C |
|Rc |and
|C | ≤ |Rc | d − γ(c )d − γ n
.
B d B ( d) f h H W i h
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Boundsfor
Ring-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Bounds on B R (n, d ) for the Homogeneous Weight
Corollary (BGKS)
Let C ≤ R R
n
be a linear code of minimum homogeneous weight d and minimum Hamming weight where ≤ n ≤ d γ
. Then
|C | ≤ |R |d − γ
d − γ n.
B d B ( d) f h H W i h
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Boundsfor
Ring-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Bounds on B R (n, d ) for the Homogeneous Weight
Corollary (BGKS)
Let C ≤ R R n be a linear code of minimum homogeneous
weight d and minimum Hamming weight where < n ≤ d γ
.Let Q be the maximum size of any minimal ideal of R. Then
|C | ≤ Q d − γ
d − γ n.
A Pl tki O ti l C d
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Upper Boundsfor
Ring-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
A Plotkin Optimal Code
ExampleLet R = F
2×22 . Let C be the length 16m − 1 Simplex Code over
R . Then |C | = 16m,
d = |R |mγ = 16mγ,
:= dHam(C ) = 16m −16m
4=
3
416m.
R has 3 minimal ideals, each of size Q = 4 and so
|C | ≤ Q d − γd − γ n
= 416mγ − 3
4 16mγ
16mγ − (16m − 1)γ = 4
16m
4= 16m.
B d B ( d) f th H W i ht
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Upper Boundsfor
Ring-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Bounds on B R (n, d ) for the Homogeneous Weight
Singleton-like bounds:
Theorem (BGKS)
Let C ≤ R R n be an [n, d ] linear code and suppose that n ≤ d γ
.Then
n − |R | − 1
|R |
d
γ
≥
log|R | |C | − 1
.
Theorem (BGKS)
Let C be an [n, d ] code over R satisfying n ≤d γ and (C ) < n.
Let P := max{|Ra| : a ∈ R n, Ra ≤ C , (a) < n}. Then
n −
P − 1
P
d
γ
≥ logP |C | − logP |R | .
A Class of MDS Codes
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Upper Boundsfor
Ring-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
A Class of MDS Codes
Example
Let R be a chain ring of length 2. Then R × = R \rad R and|R | = q 2. Let U := R 2\rad R 2, let P := {xR : x ∈ U }. Then
|P| = q 2
+ q .Let C < R R n be the length n := q 2 + q code with 2 × ngenerator matrix whose columns are the distinct elements of P .
Clearly (c ) < n for each c ∈ C .
C is free of rank 2 and the maximal cyclic submodules of C have size P := |R | = q 2.
Let r = logP |C | − 1 = logq 2 q 4 − 1 = 1.
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Upper Boundsfor
Ring-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
Example (cont.)
Setting γ = 1, each word xG of C has weight
w (xG ) =
q 2 + q if x ∈ U q 3
q −1 if x ∈ radR 2,
=⇒ n −
P − 1
P d
= n −
q 2 − 1
q 2(q 2 + q )
= n − q 2 + q − 1 −1
q
= q 2 + q − q 2 − q + 1 = 1 = r .
References
8/6/2019 Upper Bounds for Ring Linear Codes
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Upper Boundsfor
Ring-LinearCodes
Eimear Byrne,Marcus
Greferath,Axel Kohnert,
VitalySkachek
References
E. Byrne, M. Greferath and M. E. O’Sullivan, The linear programming bound for codes over finite Frobenius rings,Designs, Codes and Cryptography, Vol. 42 , 3 (2007), pp.289 - 301.
I. Constantinescu and W. Heise, A metric for codes over
residue class rings of integers , Problemy PeredachiInformatsii 33 (1997), no. 3, 22–28.
M. Greferath and M. E. O’Sullivan, On Bounds for Codes over Frobenius Rings under Homogeneous Weights ,Discrete Mathematics 289 (2004), 11–24.
T. Honold, A characterization of finite Frobenius rings,Arch. Math. (Basel), 76 (2001), 406–415.
J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121
(1999), 555–575.