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AŽD Praha
Safety Code Assessment in QSC-model
Štěpán Klapka, Lucie Kárná , Magdaléna Harlenderová
AŽD Praha s.r.o., Department of research and development,
Address: Žirovnická 2/3146, 106 17 Prague 10, Czech Republic
e-mail: [email protected], [email protected], [email protected]
EURO – Zel 2010
2
Contents
Introduction New version - FprEN 50159
Non-binary linear codes The probability of undetected errors Binary Symmetrical Channel (BSC) q-nary Symmetrical Channel (QSC) Good and proper codes Reed-Solomon code example Conclusion
3
Merging two parts of the former standard (for open and close transmission systems)
Modifications of the standard Common terminology Classification of transmission systems
three categories of transmission systems are defined More precise requirements for safety codes
standard recommends BSC and QSC model
New version - FprEN 50159
4
Non-binary linear codes
T: finite field with q elements (code alphabet).
q-nary linear (n,k)-code: k-dimensional linear subspace C of the space Tn
codewords: elements of C.
Usually T=GF(2m). In this case every symbol from GF(2m) can be substituted by its linear expansion and given 2m-nary (n,k)-code can be analysed as a binary (nm,km)-code.
most popular non-binary codes: Reed-Solomon (RS) codes
5
Undetected Errors
Structure of undetected errors all undetected errors of a linear (n,k)-code =
all nonzero codewords of the code
Probability of an undetected error
n
i i
iiud
qi
n
PAP
1 1
Ai: number of codewords with exactly i nonzero symbols
Pi: probability that there are exactly i wrong symbols in the word.
6
Binary Symmetrical Channel (BSC)
BSC: model based on the bit (binary symbol) transmission
The probability pe that the bit changes its value during the transmission (bit error rate) is the same for both possibilities (0→1, 1→0).
ine
iei pp
i
nP
1
ine
ie
n
iieud ppApP
11
7
Q-nary Symmetrical Channel (QSC)
QSC: model based on the q- symbols transmission
e: probability that a symbol changes value during the transmission
inii i
nP
1
in
in
iiud qAP
111
8
Undetected Errors Probability (BSC/QSC)
BSC model – Pud(1/2)
QSC model – Pud((q-1)/q)
nm
km
n
kn
ii
ninin
iiud q
qA
qqqA
q
qP
2
1211111
11
n
kn
ii
ninin
iiud AAP
2
12
2
1
2
1
2
1
2
1
11
9
Good and proper codes
”good” q-nary linear (n,k)-code: inequality Pud(e) < qk-n is valid for every e [0,(q-1)/q].
”proper” q-nary linear (n,k)-code: function Pud(e) is monotone for e [0,(q-1)/q].
Unfortunately goodness and properness are relatively rare conditions.
example: perfect codes, MDS codes
10
Example
Objective: to show how different results is possible to get in QSC and BSC models
Example: RS code on GF(256) with generator polynomial:
g(x) = x4 + 54x3 + 143x2 + x + 214.
RS codes are Maximum Distance Separable codes (MDS) => they are ”proper” in the QSC model.
di
j
jdiji q
j
iq
i
nA
0
111
1212
RS code x4 + 54x3 + 143x2 + x + 214
Codewords with binary weight 7
w_1=(32, 35, 4, 32, 1)w_2=(64, 70, 8, 64, 2) w_3=(128, 140, 16, 128, 4) w_1=(00100000 00100011 00000100 00100000 00000001)w_2=(01000000 01000110 00001000 01000000 00000010)w_3=(10000000 10001100 00010000 10000000 00000100)
337 13 eeeud pppP 322/11340/7 udP
1414
RS code x4 + 54x3 + 143x2 + x + 214
Binary weight spectrum
n A5 A6 A7 A8
40 0 0 3 0
48 0 0 2x3=6 0
104 0 0 9x3=27 36
128 0 2 53 265
136 0 2x2=4 72 477
208 0 34 796 17604
328 4 559 18920 710551
336 2x4=8 633 22418 863144
2040 66198 23033470 6729268440 1708427500185
1717
RS code x4 + 54x3 + 143x2 + x + 214
Q-nary weight 5
n A5
5(40) 255
6(48) 1530
13(104) 328185
16(128) 1113840
17(136) 1577940
26(208) 16773900
41(328) 191096490
42(336) 216920340
255(2040) 2202559325505
1818
RS code x4 + 54x3 + 143x2 + x + 214
SUMMARY QSC/BSC
QSC model – proper code for codeword length255
BSC model – not good code for all codeword length
322
max eud pP
887364176794000000000000000000001,00000000
19
Conclusions
The analysis of the probability Pud in the BSC model cannot be replaced by the analysis in the QSC model.
The QSC model could be a suitable alternative when a character oriented transmission is used.
The QSC and BSC models of a communication channel are rather abstract criteria of the linear code structure than the mathematical models, which could describe a real transmission system.
For the code over the GF(2m), it is possible to use the both models.
Without an a priori information about the transmission channel there is no reason to prefer any one from these models.