Weight distribution of cosets of small codes with good dual properties

TL;DR

This paper studies how the weight distribution of cosets of small binary linear codes with duals having good bilateral minimum distance closely approximates the binomial distribution, with decay rates depending on the dual distance.

Contribution

It provides new bounds on the closeness of coset weight distributions to the binomial distribution based on the dual code's bilateral minimum distance, using Fourier analysis and polynomial approximation.

Findings

Average $L_$-distance decays exponentially with dual distance.

Most cosets have weight distributions close to binomial.

Bounds are established for codes like extended Hadamard and BCH codes.

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 $L_\infty$-distance is at most $\min\{\left(e\ln{\frac{n}{2t}}\right)^{t}\left(\frac{2t}{n}\right)^{\frac{t}{2} }, \sqrt{2} e^{-\frac{t}{10}}\}$. For the average $L_1$-distance, we conclude the bound $\min\{(2t+1)\left(e\ln{\frac{n}{2t}}\right)^{t} \left(\frac{2t}{n}\right)^{\frac{t}{2}-1},\sqrt{2}(n+1)e^{-\frac{t}{10}}\}$, which gives nontrivial results for $t\geq 3$. 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.