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 $Q\subset \{0,1\}^{n}$ 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_\infty $ -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 $d = \Theta (1)$ , it decays like $n^{-\Theta (d)}$ . On the other $d=\Theta (n)$ extreme, it decays like and $e^{-\Theta (d)}$ . 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 $t\geq 1$ is an integer, then the average $\vphantom {\sum ^{R^{R}}} L_\infty $ -distance is at most $\min \{ (e\ln {({n}/{2t})})^{t}({2t}/{n})^{({t}/{2})}, \sqrt {2} e^{-({t}/{10})}\}$ . For the average $L_{1}$ -distance, we conclude the bound $\min \{ (2t+1)(e\ln {({n}/{2t})})^{t} ({2t}/{n})^{({t}/{2})-1}, \sqrt {2}(n+1)e^{-({t}/{10})}\}$ , which gives nontrivial results for $t\geq 3$ . 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.