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

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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

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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

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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

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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

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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

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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