Weight distribution of random linear codes and Krawchouk polynomials

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 $K_i$.