Petersburg State Transport University
“Automation and Remote Control on Railways”
department
Properties of code with summation for logical
circuit test organization
A. Blyudov, D. Efanov, V. Sapozhnikov, Vl. Sapozhnikov
Functional control system of the combinational circuit
codemnSgggfff km ,...... 2121
operating outputs
Test
f1(x)
f2(x)
fm(x)
f(x)
g(x)
x
g1(x)
g2(x)
gk(x)
G
g1(x)
g2(x)
gk(x)
MC
check bits
informational bits
Properties of code with summation for logical circuit test
organization, EWDTS`20122
Berger code
00000 0000100010
00011
001000010100110
00111
01000 0100101010
01011
01100
0110101110
01111
10000
1000110010
10011
10100
1010110110
10111
11000
1100111010
11011
11100
1110111110
11111
Groups by weight of code words – isolated groups of code words
r=0 r=1 r=2 r=3 r=4 r=5
S(8,5)-code, m=5
undetectable error
r
mNr
in group
00001 001
01000 001
сheck bits
informational bits
S(n,m)-codes
Properties of code with summation for logical circuit test
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Berger codeThe errors of an even multiplicity, which have the number of distortions of the informational bits of the type 10 equal to the distortions 01, cannot be detected in the S(n,m)-code.
Formula for calculating the number of undetected errors
d – multiplicity of undetectable error;m – length of informational wordr – weight of informational word
1,
2
2
2 22
mm
d
dm
dr
m drm
dm
r
mN
In S(8,5)-code 160 undetectable errors with d=2 & 60 undetectable errors with d=2
Properties of code with summation for logical circuit test
organization, EWDTS`20124
Berger codeIn the S(n,m)-code the share of undetectable errors βd doesn’t depend on the number of informational bits and it is constant
22 d
dd
d
The Berger code has a great number of undetectable errors. Particularly, it doesn’t detect 50% of distortions of multiplicity 2
Properties of code with summation for logical circuit test
organization, EWDTS`20125
Modulo codes with summationDecreasing a number of check bits in the code with summation could be reached if the calculation of “ones” in the informational vector is carried out by some modulo M (M=2i<2k
, i=1,2,…) SM(n,m)-codes
There are modulo codes, besides the S(n,m)-code, for any value of m.
11log2 m
Properties of code with summation for logical circuit test
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All the errors of information bits of the odd multiplicity are detected in the modulo codes, but the errors of the even multiplicity may not be detected.
The share of the undetectable errors βd for a modulo code doesn’t depend on the number of informational bits and it is a constant value.
Any S2(n,m)-code (a parity code) doesn’t detect 100% of errors of informational bits of any even multiplicity.
Any S4(n,m)-code doesn’t detect 50% of errors of informational bits of any even multiplicity.
The SM(n,m)-code has the same number of errors of multiplicity d<M like the S(n,m)-code.
Any SM(n,m)-code with M≥4 doesn’t detect 50% of errors of informational bits of multiplicity 2.
The SM(n,m)-code with M≥8 has more errors of multiplicity d≥M than the S(n,m)-code.
The SM(n,m)-code has the value of characteristic βd equal or less than the same characteristic for the SM'(n,m)-code (M'>M) for every d.
Modulo codes with summation
The properties of modulo codes
Properties of code with summation for logical circuit test
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Modulo codes with summation
Values of the characteristic βd
Properties of code with summation for logical circuit test
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Modified modulo codes with summation
1. The S(n,m)-code is formed for the current m.2. The modulo M is fixed: M=2,4,8,…,2b ; b=]log2(m+1)[-2.3. For the current informational word the number of “ones” q is counted, which is converted to the number W=q(mod M).4. The special coefficient α is determined.5. If ( ), then α=0, otherwise α=1.6. Then the resultant weight of an informational word is counted. 7. The check word is a binary notation of V.
0...1 pmm xxx 2log2 Mp
The method of forming a RSM(n,m)-code is:RSM(n,m)-codes
Properties of code with summation for logical circuit test
organization, EWDTS`20129
Classification of codes with summation
Codes with summation of “ones”
Non-modified codes Modified codes
S(n,m)
S2(n,m)
S4(n,m)
RS(n,m)
RS2(n,m)
RS4(n,m)
S2k-1(n,m) RS2k-1(n,m)
Properties of code with summation for logical circuit test
organization, EWDTS`201210
Classification of codes with summation
The analysis tables for different codes allows to deduce the following conclusions:1. The RS(n,m)-codes have the best characteristics to detect errors. For example, the RS(21,16)-code has about by two times less undetectable errors in comparing with the Berger code, and this concerns to the errors of any multiplicity.2. Any modulo modified code with M≥4 has the better detection characteristics than any non-modified code.3. The modulo non-modified codes have an advantage in detecting errors among all the codes with the number of check bits d=2.
Properties of code with summation for logical circuit test
organization, EWDTS`201211
Conclusion
The table of codes, in which the main characteristics of all the available codes with summation with a current number of informational bits are given, allows to choose a code more reasonably at organizing the check of a combinational circuit. Herewith there is a possibility to consider the properties of a control circuit, for example, the possibility of appearing errors of the determined multiplicity at the outputs of the circuit.
A. Blyudov, D. Efanov, V. Sapozhnikov, Vl. Sapozhnikov
Petersburg State Transport University“Automation and Remote Control on Railways” department
Properties of code with summation for logical circuit test
organization, EWDTS`201212