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: klapka.stepan@azd.cz, karna.lucie@azd.cz, harlenderova.magdalena@azd.cz

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

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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

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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

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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.

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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

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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

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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

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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

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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

1111

RS code x4 + 54x3 + 143x2 + x + 214

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

13

RS code x4 + 54x3 + 143x2 + x + 214

10000x

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

1515

RS code x4 + 54x3 + 143x2 + x + 214

1616

RS code x4 + 54x3 + 143x2 + x + 214

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

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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.

2020

Safety Code Assessment in QSC-model

Thank You for Your attention!