Binomial moments of the distance distribution: bounds and applications

We study a combinatorial invariant of codes which counts the number of ordered pairs of codewords in all subcodes of restricted support in a code. This invariant can be expressed as a linear form of the components of the distance distribution of the code with binomial numbers as coefficients. For th...

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Vydáno v:IEEE transactions on information theory Ročník 45; číslo 2; s. 438 - 452
Hlavní autoři: Ashikhmin, A., Barg, A.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York IEEE 01.03.1999
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:
Asymptotic properties
Binary codes
Binomial distribution
Binomials
Codes
Combinatorial analysis
Computer programming
Computer science
Error detection
Information theory
Invariants
Laboratories
Linear programming
Lower bounds
Optimization
Postal services
Software
Upper bound
Vectors
ISSN:0018-9448, 1557-9654
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Shrnutí:We study a combinatorial invariant of codes which counts the number of ordered pairs of codewords in all subcodes of restricted support in a code. This invariant can be expressed as a linear form of the components of the distance distribution of the code with binomial numbers as coefficients. For this reason we call it a binomial moment of the distance distribution. Binomial moments appear in the proof of the MacWilliams (1963) identities and in many other problems of combinatorial coding theory. We introduce a linear programming problem for bounding these linear forms from below. It turns out that some known codes (1-error-correcting perfect codes, Golay codes, Nordstrom-Robinson code, etc.) yield optimal solutions of this problem, i.e., have minimal possible binomial moments of the distance distribution. We derive several general feasible solutions of this problem, which give lower bounds on the binomial moments of codes with given parameters, and derive the corresponding asymptotic bounds. Applications of these bounds include new lower bounds on the probability of undetected error for binary codes used over the binary-symmetric channel with crossover probability p and optimality of many codes for error detection. Asymptotic analysis of the bounds enables us to extend the range of code rates in which the upper bound on the undetected error exponent is tight.
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ISSN:0018-9448
1557-9654
DOI:10.1109/18.748994