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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,

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Presentation on theme: "AŽD Praha Safety Code Assessment in QSC-model Štěpán Klapka, Lucie Kárná, Magdaléna Harlenderová AŽD Praha s.r.o., Department of research and development,"— Presentation transcript:

1 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, Prague 10, Czech Republic EURO – Zel 2010

2 2 Contents Introduction New version - FprEN 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 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 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 T n codewords: elements of C. Usually T=GF(2 m ). In this case every symbol from GF(2 m ) can be substituted by its linear expansion and given 2 m -nary (n,k)-code can be analysed as a binary (nm,km)-code. most popular non-binary codes: Reed-Solomon (RS) codes

5 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 A i : number of codewords with exactly i nonzero symbols P i : probability that there are exactly i wrong symbols in the word.

6 6 Binary Symmetrical Channel (BSC)  BSC: model based on the bit (binary symbol) transmission  The probability p e that the bit changes its value during the transmission (bit error rate) is the same for both possibilities (0 → 1, 1 → 0). 

7 7 Q-nary Symmetrical Channel (QSC)  QSC: model based on the q- symbols transmission  e : probability that a symbol changes value during the transmission 

8 8 Undetected Errors Probability (BSC/QSC)  BSC model – P ud (1/2)  QSC model – P ud ((q-1)/q) 

9 9 Good and proper codes  ”good” q-nary linear (n,k)-code: inequality P ud ( e ) < q k-n is valid for every e  [0,(q-1)/q].  ”proper” q-nary linear (n,k)-code: function P ud ( e ) is monotone for e  [0,(q-1)/q].  Unfortunately goodness and properness are relatively rare conditions.  example: perfect codes, MDS codes

10 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) = x x x 2 + x RS codes are Maximum Distance Separable codes (MDS) => they are ”proper” in the QSC model.

11 11 RS code x x x 2 + x + 214

12 12 RS code x x x 2 + x 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=( ) w_2=( ) w_3=( )

13 13 RS code x x x 2 + x x

14 14 RS code x x x 2 + x Binary weight spectrum nA5A5 A6A6 A7A7 A8A x3= x3= x2= x4=

15 15 RS code x x x 2 + x + 214

16 16 RS code x x x 2 + x + 214

17 17 RS code x x x 2 + x Q-nary weight 5 nA5A5 5(40)255 6(48) (104) (128) (136) (208) (328) (336) (2040)

18 18 RS code x x x 2 + x SUMMARY QSC/BSC QSC model – proper code for codeword length255 BSC model – not good code for all codeword length

19 19 Conclusions The analysis of the probability P ud 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(2 m ), 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.

20 20 Safety Code Assessment in QSC-model Thank You for Your attention!


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