Binomial moments of the distance distribution: bounds andapplications
Ashikhmin, A.
Barg, A.
Los Alamos Nat. Lab., NM;
This paper appears in: Information Theory, IEEE Transactions on
Publication Date: Mar 1999
Volume: 45,
Issue: 2
On page(s): 438-452
ISSN: 0018-9448
References Cited: 34
CODEN: IETTAW
INSPEC Accession Number: 6198524
Digital Object Identifier: 10.1109/18.748994
Current Version Published: 2002-08-06
Abstract
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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