Random codes and matrices

Abstract

Results of our study are two fold. From the code theoretical point of view our study yields the expectations and the covariances of the coefficients of the weight enumerator of a random code. Particularly interesting is that, the coefficients of the weight enumerator of a code with random parity check matrix are uncorrelated. We give conjectures for the triple correlations of the coefficients of weight enumerator of random codes. From the random matrix theory point of view we obtain results in the rank distribution of column submatrices. We give the expectations and the covariances between the ranks (q −rank) of such submatrices over Fq. We conjecture the counterparts of these results for arbitrary submatrices. The case of higher correlations gets drastically complicated even in the case of three submatrices. We give a formula for the correlation of ranks of three submatrices and a conjecture for its closed form.