Algebraic construction of sparse matrices with large girth

In this correspondence, we present a method for constructing sparse matrices that have a compact description and whose associated bipartite graphs have large girth. Based on an arbitrary seed matrix of nonnegative integers a new matrix is constructed which replaces each entry of the seed matrix with a sum of permutation matrices. Algebraic conditions that lead to short cycles in the associated bipartite graph are analyzed and methods to achieve large girth in two special cases are presented. In one, all the permutation matrices are circulants; in the other they are all affine permutation matrices. When used to define a low-density parity-check (LDPC) code the compact description should lead to efficient implementation and the large girth to good error correction performance. The method is adaptable to a variety of rates, and a variety of row and column degrees