On the weight distribution of random binary linear codes

Abstract

We investigate the weight distribution of random binary linear codes. For $0<\hat{I}»<1$ and $n\to \infty$ pick uniformly at random $\hat{I}»n$ vectors in ${\mathbb{F}}_2^n$ and let $C\le {\mathbb{F}}_2^n$ be the orthogonal complement of their span. Given $0<{\hat{I}}^3<1/2$ with $0<\hat{I}»<h({\hat{I}}^3)$ let $X$ be the random variable that counts the number of words in $C$ of Hamming weight ${\hat{I}}^{3}n$. In this paper we determine the asymptotics of the moments of $X$ of all orders $o(\frac{n}{\log n})$.