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Using methods of statistical physics, we study the average number and kernelsize of general sparse random matrices over GF(q), with a given connectivityprofile, in the thermodynamical limit of large matrices. We introduce a mappingof $GF(q)$ matrices onto spin systems using the representation of the cyclicgroup of order q as the q-th complex roots of unity. This representationfacilitates the derivation of the average kernel size of random matrices usingthe replica approach, under the replica symmetric ansatz, resulting in saddlepoint equations for general connectivity distributions. Numerical solutions arethen obtained for particular cases by population dynamics. Similar techniquesalso allow us to obtain an expression for the exact and average number ofrandom matrices for any general connectivity profile. We present numericalresults for particular distributions.

Typical kernel size and number of sparse random matrices over GF(q) - a statistical physics approach