Transcript 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,

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
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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New version - FprEN 50159
 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
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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
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(RS) codes
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
Ai Pi
Pud  
i 1  n 
i
 q  1
i
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).
 n i
ni
Pi    pe 1  pe 
i
n

Pud  pe    Ai p 1  pe 
i 1
i
e
ni
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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
 n i
ni
Pi    1   
i

i
  
n i
 1   
Pud     Ai 
i 1
 q 1
n
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Undetected Errors Probability (BSC/QSC)
 BSC model – Pud(1/2)
i
1 n 1 1
Pud     Ai    
 2  i 1  2   2 
n i
1
 
 2
2k  1
Ai  n

2
i 1
n n
 QSC model – Pud((q-1)/q)
i
 q 1 n  1   1 
   Ai    
Pud 
 q  i 1  q   q 

n i
1
  
q
n
q k  1 2km  1
Ai 
 nm

n
q
2
i 1
n
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Good and proper codes
 ”good” q-nary linear (n,k)-code:
inequality Pud(e) < qk-n is valid
every e  [0,(q-1)/q].
for
 ”proper” q-nary linear (n,k)-code:
function Pud(e) is monotone
e  [0,(q-1)/q].
for
 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.
i d
 n
j  i  1 i  d  j
q
Ai   q  1  1 
j 0
i
 j 
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RS code x4 + 54x3 + 143x2 + x + 214
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RS code x4 + 54x3 + 143x2 + x + 214
Codewords with binary weight 7
w_1=(32, 35, 4,
w_2=(64, 70,
8,
w_3=(128, 140, 16,
32,
1)
64,
128,
2)
4)
w_1=(00100000 00100011 00000100 00100000 00000001)
w_2=(01000000 01000110 00001000 01000000 00000010)
w_3=(10000000 10001100 00010000 10000000 00000100)
Pud  pe   3 p 1  pe 
7
e
33
Pud 7 / 40  113/ 232
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RS code x4 + 54x3 + 143x2 + x + 214
10000x
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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
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RS code x4 + 54x3 + 143x2 + x + 214
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RS code x4 + 54x3 + 143x2 + x + 214
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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
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RS code x4 + 54x3 + 143x2 + x + 214
SUMMARY QSC/BSC
QSC model – proper code for codeword length255
BSC model – not good code for all codeword length
max Pud  pe 

32
2
1,00000000
0000000000
0000000000
7364176794
88
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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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