Tight Heffter arrays from finite fields

TL;DR

This paper introduces explicit constructions of tight Heffter arrays over finite fields, extending classical notions and providing optimal arrays with rich symmetry properties for many parameter pairs.

Contribution

It presents new direct constructions of elementary abelian tight Heffter arrays over finite fields, including optimal arrays with maximal symmetry groups, for a broad class of parameters.

Findings

Explicit constructions for elementary abelian tight Heffter arrays over finite fields.

Most constructed arrays are nearly always optimal in terms of their symmetry group.

The methods apply to all admissible pairs with two distinct odd primes dividing the array dimensions.

Abstract

After extending the classic notion of a tight Heffter array H$(m,n)$ to any group of order $2mn+1$, we give direct constructions for elementary abelian tight Heffter arrays, hence in particular for prime tight Heffter arrays. If $q=2mn+1$ is a prime power, we say that an elementary abelian H$(m,n)$ is ``over $\mathbb{F}_q$" since, for its construction, we exploit both the additive and multiplicative structure of the field of order $q$. We show that in many cases a direct construction of an H$(m,n)$ over $\mathbb{F}_q$, say $A$, can be obtained very easily by imposing that $A$ has rank 1 and, possibly, a rich group of {\it multipliers}, that are elements $u$ of $\mathbb{F}_q$ such that $u A=A$ up to a permutation of rows and columns. An H$(m,n)$ over $\mathbb{F}_q$ will be said {\it optimal} if the order of its group of multipliers is the least common multiple of the odd parts of $m$ and $n$, since this is the maximum possible order for it. The main result is an explicit construction of a rank-one H$(m,n)$ -- reaching almost always the optimality -- for all admissible pairs $(m,n)$ for which there exist two distinct odd primes $p$, $p'$ dividing $m$ and $n$, respectively.