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