Hadamard matrices modulo 5

Abstract

In this paper we introduce modular symmetric designs and use them to study the existence of Hadamard matrices modulo 5. We prove that there exist 5-modular Hadamard matrices of order n if and only if n != 3, 7 (mod 10) or n != 6, 11. In particular, this solves the 5-modular version of the Hadamard conjecture.