  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 $K_i$.
