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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: klapka.stepan@azd.cz, karna.lucie@azd.cz, harlenderova.magdalena@azd.cz EURO – Zel 2010

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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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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

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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

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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.

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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).

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7 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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8 Undetected Errors Probability (BSC/QSC) BSC model – P ud (1/2) QSC model – P ud ((q-1)/q)

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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

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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 4 + 54x 3 + 143x 2 + x + 214. RS codes are Maximum Distance Separable codes (MDS) => they are ”proper” in the QSC model.

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11 RS code x 4 + 54x 3 + 143x 2 + x + 214

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12 RS code x 4 + 54x 3 + 143x 2 + 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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13 RS code x 4 + 54x 3 + 143x 2 + x + 214 10000x

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14 RS code x 4 + 54x 3 + 143x 2 + x + 214 Binary weight spectrum nA5A5 A6A6 A7A7 A8A8 400030 48002x3=60 104009x3=2736 1280253265 13602x2=472477 20803479617604 328455918920710551 3362x4=863322418863144 2040661982303347067292684401708427500185

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15 RS code x 4 + 54x 3 + 143x 2 + x + 214

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16 RS code x 4 + 54x 3 + 143x 2 + x + 214

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17 RS code x 4 + 54x 3 + 143x 2 + x + 214 Q-nary weight 5 nA5A5 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

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18 RS code x 4 + 54x 3 + 143x 2 + x + 214 SUMMARY QSC/BSC QSC model – proper code for codeword length255 BSC model – not good code for all codeword length

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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.

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

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