A class of cyclic $(v;k_1,k_2,k_3;\lambda)$ difference families with $v \equiv 3 \pmod{4}$ a prime

TL;DR

This paper constructs new cyclic difference families for primes congruent to 3 mod 4, enabling the creation of large skew-Hadamard matrices, including the first known of orders 956 and 1324.

Contribution

It introduces new cyclic difference families for primes v ≡ 3 mod 4, leading to the first known skew-Hadamard matrices of orders 956 and 1324.

Findings

Constructed cyclic difference families for primes v ≡ 3 mod 4.

Generated skew-Hadamard matrices of orders 956 and 1324.

Demonstrated applications in combinatorial design theory.

Abstract

We construct several cyclic $(v;k_1,k_2,k_3;\lambda)$ difference families with $v\equiv3 \pmod{4}$ a prime and $\lambda=k_1+k_2+k_3-(3v-1)/4$. Such families can be used in conjunction with the well-known Paley-Todd difference sets to construct skew-Hadamard matrices of order $4v$. Our main result is that we have constructed for the first time the example of skew-Hadamard matrices of orders $4\cdot239=956$ and $4\cdot331=1324$.