TL;DR
This paper investigates the existence of Hadamard matrices modulo 5, establishing precise conditions for their existence and solving the 5-modular Hadamard conjecture.
Contribution
It introduces modular symmetric designs and characterizes the existence of 5-modular Hadamard matrices, resolving the conjecture for this case.
Findings
Existence of 5-modular Hadamard matrices for n not congruent to 3, 7 mod 10 or 6, 11.
Complete characterization of 5-modular Hadamard matrices.
Resolution of the 5-modular Hadamard conjecture.
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.
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