Weight Distribution of Cosets of Small Codes With Good Dual Properties

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 L1-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 Linfin;-distance is at most min{(e ln (n/2t))t (2t/n)(t/2), √2e-(t/10)}. For the average L1-distance, we conclude the bound min{(2t + 1)(e ln (n/2t))t (2t/n)(t/2)-1, √2(n + 1)e-(t/10)}, which gives nontrivial results for t ≥ 3. We give 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. © 1963-2012 IEEE.