Safety code assessment in qsc model t p n klapka lucie k rn magdal na harlenderov
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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]

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Safety code assessment in qsc model t p n klapka lucie k rn magdal na harlenderov l.jpg

Safety Code Assessment in QSC-modelŠtěpánKlapka, 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


Contents l.jpg
Contents

  • IntroductionNew 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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New version - FprEN 50159

  • Mergingtwo parts of the former standard (for open and close transmission systems)

  • Modifications of the standard

    • Common terminology

    • Classification of transmission systemsthree categories of transmission systems are defined

    • More precise requirements for safety codesstandard recommends BSC and QSC model


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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

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).


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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


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Undetected Errors Probability (BSC/QSC)

BSC model – Pud(1/2)

QSC model – Pud((q-1)/q)


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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 fore [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.


Rs code x 4 54 x 3 143 x 2 x 214 l.jpg
RS code x4+54x3+143x2+x+214

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Rs code x 4 54 x 3 143 x 2 x 21412 l.jpg
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)

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RS code x4+54x3+143x2+x+214

10000x


Rs code x 4 54 x 3 143 x 2 x 21414 l.jpg
RS code x4+54x3+143x2+x+214

Binary weight spectrum

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Rs code x 4 54 x 3 143 x 2 x 21415 l.jpg
RS code x4+54x3+143x2+x+214

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Rs code x 4 54 x 3 143 x 2 x 21416 l.jpg
RS code x4+54x3+143x2+x+214

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Rs code x 4 54 x 3 143 x 2 x 21417 l.jpg
RS code x4+54x3+143x2+x+214

Q-nary weight 5

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Rs code x 4 54 x 3 143 x 2 x 21418 l.jpg
RS code x4+54x3+143x2+x+214

SUMMARY QSC/BSC

QSC model – proper codefor codeword length255

BSC model – not good code for all codeword length

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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.


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Safety Code Assessment in QSC-model

Thank You for Your attention!

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