Related papers: Tight Heffter arrays from finite fields

A tight Heffter array H(m,n) is an m x n matrix with nonzero entries from Z_{2mn+1} such that i) the sum of the elements in each row and each column is 0, and ii) no element from {x,-x\ appears twice. We prove that H(m,n) exist if and only…

Square relative non-zero sum Heffter arrays, denoted by $\mathrm{N}\mathrm{H}_t(n;k)$, have been introduced as a variant of the classical concept of Heffter array. An $\mathrm{N}\mathrm{H}_t(n; k)$ is an $n\times n$ partially filled array…

The shiftable Heffter arrays are naturally generalized to the shiftable Heffter spaces. We present a recursive construction which starting from a single shiftable Heffter space leads to infinitely many other shiftable Heffter spaces of the…

Heffter arrays are combinatorial structures used to construct orthogonal cyclic cycle decompositions and biembeddings of complete graphs onto surfaces. A Heffter array $H(m,n;h,k)$ is an $m \times n$ partially filled array with distinct…

Let $m,n,s,k$ be four integers such that $3\leq s \leq n$, $3\leq k\leq m$ and $ms=nk$. Set $d=\gcd(s,k)$. In this paper we show how one can construct a Heffter array $H(m,n;s,k)$ starting from a square Heffter array $H(nk/d;d)$ whose…

Let $F_q$ be a field with $q$ elements, where $q$ is a power of a prime number $p\geq 5$. For any integer $m\geq 2$ and $a\in F_q^*$ such that the polynomial $x^m-a$ is irreducible in $F_q[x]$, we combine two different methods to construct…

Relative Heffter arrays, denoted by $\mathrm{H}_t(m,n; s,k)$, have been introduced as a generalization of the classical concept of Heffter array. A $\mathrm{H}_t(m,n; s,k)$ is an $m\times n$ partially filled array with elements in…

In this paper, we make significant progress on a conjecture proposed by Dan Archdeacon on the existence of integer Heffter arrays $H(m,n;s,k)$ whenever the necessary conditions hold, that is, $3\leqslant s \leqslant n$, $3\leqslant…

An integer Heffter array $H(m,n;s;k)$ is an $m\times n$ partially filled array whose entries are the elements of a subset $\Omega\subset \mathbb{Z}$ such that $\{\Omega,-\Omega\}$ is a partition of the set $\{1,2,\ldots,2nk\}$ and such that…

A $(v,k;r)$ Heffter space is a resolvable $(v_r,b_k)$ configuration whose points form a half-set of an abelian group $G$ and whose blocks are all zero-sum in $G$. It was recently proved that there are infinitely many orders $v$ for which,…

The notion of a Heffter array, which received much attention in the last decade, is equivalent to a pair of orthogonal Heffter systems. In this paper we study the existence problem of a set of $r$ mutually orthogonal Heffter systems for any…

We construct explicitly in any finite field of the form Fq[x]/(x^m-a) elements with multiplicative order at least 2^{(2m)^(1/2)}

Let $S$ be a subset of a group $G$ (not necessarily abelian) such that $S\,\cap -S$ is empty or contains only elements of order $2$, and let $\mathbf{h}=(h_1,\ldots, h_m)\in \mathbb{N}^m$ and $\mathbf{k}=(k_1, \ldots, k_n)\in \mathbb{N}^n$.…

Explicit embeddings of the group $\mathbb{Q}$ into a finitely presented group $\mathcal{Q}$ and into a $2$-generator finitely presented group $T_{\mathcal{Q}}$ are suggested. The constructed embeddings reflect questions mentioned by…

We construct orthogonal arrays OA$_{\lambda} (k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m / \lambda$ is as large as possible; these…

A Heffter array over an additive group $G$ is any partially filled array $A$ satisfying that: (1) each one of its rows and columns sum to zero in $G$, and (2) if $i\in G\setminus\{0\}$, then either $i$ or $-i$ appears exactly once in $A$.…

It is a theorem of Artin, Tits et al. that a finite simple group is determined by its order, with the exception of the groups (A_3(2), A_2(4)) and (B_n(q), C_n(q)) for n > 2, q odd. We investigate the situation for finite semisimple groups…

In this paper we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let $v=2nk+t$ be a positive integer, where $t$ divides $2nk$,…

In 2015, Archdeacon proposed the notion of Heffter arrays in view of its connection to several other combinatorial objects. In the same paper he also presented the following variant. A weak Heffter array $\mathrm{W}\mathrm{H}(m,n;h,k)$ is…

A square integer relative Heffter array is an $n \times n$ array whose rows and columns sum to zero, each row and each column has exactly $k$ entries and either $x$ or $-x$ appears in the array for every $x \in \mathbb{Z}_{2nk+t}\setminus…