USNA Coding theory, Cryptography, and Number Theory Conference

USNA Mathematics Department

Title:

Abstract: The p-rank of an integer matrix A is the rank of A over a field of characteristic p. Thus the p-rank of A equals the dimension of the linear code (i.e., vector space) generated by the rows of A over the integers modulo p. Design theorists use p-ranks to distinguish among non-isomorphic block designs.

We discuss p-ranks of matrices of combinatorial interest with an emphasis on the 2-ranks of incidence matrices of Hadamard designs. In particular, we give elementary proofs of generalizations of theorems of de Caen; MacWilliams and Mann; Jungnickel; Hamada and Ohmori; and Wallis.