The dimensions of codes

A family of LDPC codes, called LU ( 3 , q ) codes, has been constructed from q-regular bipartite graphs. Recently, P. Sin and Q. Xiang determined the dimensions of these codes in the case that q is a power of an odd prime. They also obtained a lower bound for the dimension of an LU ( 3 , q ) code when q is a power of 2. In this paper we prove that this lower bound is the exact dimension of the LU ( 3 , q ) code. The proof involves the geometry of symplectic generalized quadrangles, the representation theory of Sp ( 4 , q ) , and the ring of polynomials.