Weight distribution of random linear codes and Krawchouk polynomials

TL;DR

This paper extends the analysis of the weight distribution in random linear codes by connecting the moments of the count of codewords of a fixed weight to the norms of Krawchouk polynomials, advancing understanding of their asymptotic behavior.

Contribution

It extends previous moment estimates of weight distributions in random linear codes to higher moments, linking these to Krawchouk polynomial norms.

Findings

Normalized moments are determined by Krawchouk polynomial norms.

Extended asymptotic estimates up to linear order moments.

Provides deeper insight into the structure of random linear codes.

Abstract

For $0 < \lambda < 1$ and $n \rightarrow \infty$ pick uniformly at random $\lambda n$ vectors in $\{0,1\}^n$ and let $C$ be the orthogonal complement of their span. Given $0 < \gamma < \frac12$ with $0 < \lambda < h(\gamma)$, let $X$ be the random variable that counts the number of words in $C$ of Hamming weight $i = \gamma n$ (where $i$ is assumed to be an even integer). Linial and Mosheiff determined the asymptotics of the moments of $X$ of all orders $o\left(\frac{n}{\log n}\right)$. In this paper we extend their estimates up to moments of linear order. Our key observation is that the behavior of the suitably normalized $k^{th}$ moment of $X$ is essentially determined by the $k^{th}$ norm of the Krawchouk polynomial $K_i$.