The discriminant problem for Artin-Schreier extensions: Explicit results

Given a global function field K of characteristic p, for all effective divisors $\mathfrak{D}$ in the divisor group of K we count the number of cyclic extensions $F\supset K$ of degree p where the relative discriminant ${\mathrm{\text{Disc}}}_K(F)=(p-1)\mathfrak{D}$.     

A note on the distribution of weights of fixed-rank matrices over the binary field

Let M be a random $m\times n$ rank-r matrix over the binary field ${\mathbb{F}}_2$, and let $\mathrm{wt}(\mathbf{M})$ be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as $m,n\to +\infty$ with r fixed and $m/n$ tending to a constant, we have that$\frac{\mathrm{wt}(\mathbf{M})-\frac{1-2^{-r}}{2}mn}{\sqrt{\frac{2^{-r}(1-2^{-r})}{4}(m+n)mn}}$ converges in distribution to a standard normal random variable.