Weight distribution of cosets of small codes with good dual properties
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by
Louay Bazzi
2014
Abstract
The bilateral minimum distance of a binary linear code is the maximum d
such that all nonzero codewords have weights between d and n-d. Let
Q⊂{0,1}^n be a binary linear code whose dual has bilateral minimum
distance at least d, where d is odd. Roughly speaking, we show that the
average L_∞-distance -- and consequently the L_1-distance -- between
the weight distribution of a random cosets of Q and the binomial distribution
decays quickly as the bilateral minimum distance d of the dual of Q
increases. For d = Θ(1), it decays like n^-Θ(d). On the other
d=Θ(n) extreme, it decays like and e^-Θ(d). It follows that,
almost all cosets of Q have weight distributions very close to the to the
binomial distribution. In particular, we establish the following bounds. If the
dual of Q has bilateral minimum distance at least d=2t+1, where t≥ 1
is an integer, then the average L_∞-distance is at most
{(en/2t)^t(2t/n)^t/2, √(2) e^-t/10}. For the average L_1-distance, we conclude
the bound {(2t+1)(en/2t)^t(2t/n)^t/2-1,√(2)(n+1)e^-t/10},
which gives nontrivial results for t≥ 3. We given applications to the
weight distribution of cosets of extended Hadamard codes and extended dual BCH
codes. Our argument is based on Fourier analysis, linear programming, and
polynomial approximation techniques.
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