  Using methods of statistical physics, we study the average number and kernel
size of general sparse random matrices over GF(q), with a given connectivity
profile, in the thermodynamical limit of large matrices. We introduce a mapping
of $GF(q)$ matrices onto spin systems using the representation of the cyclic
group of order q as the q-th complex roots of unity. This representation
facilitates the derivation of the average kernel size of random matrices using
the replica approach, under the replica symmetric ansatz, resulting in saddle
point equations for general connectivity distributions. Numerical solutions are
then obtained for particular cases by population dynamics. Similar techniques
also allow us to obtain an expression for the exact and average number of
random matrices for any general connectivity profile. We present numerical
results for particular distributions.
