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Let $m,n,s,k$ be four integers such that $1\leqslant s \leqslant n$, $1\leqslant k\leqslant m$ and $ms=nk$. A signed magic array $SMA(m,n; s,k)$ is an $m\times n$ partially filled array whose entries belong to the subset $\Omega\subset \mathbb{Z}$, where $\Omega=\{0,\pm 1, \pm 2,\ldots, \pm (nk-1)/2\}$ if $nk$ is odd and $\Omega=\{\pm 1, \pm 2, \ldots, \pm nk/2\}$ if $nk$ is even, satisfying the following requirements: $(a)$ every $\omega \in \Omega$ appears once in the array; $(b)$ each row contains exactly $s$ filled cells and each column contains exactly $k$ filled cells; $(c)$ the sum of the elements in each row and in each column is $0$. In this paper we construct these arrays when $n$ is even and $s,k\geqslant 5$ are odd coprime integers. This allows us to give necessary and sufficient conditions for the existence of an $SMA(m,n; s,k)$ for all admissible values of $m,n,s,k$.

A $\Gamma$-magic rectangle set $\mathrm{MRS}_\Gamma (a, b; c)$ is a collection of $c$ arrays of size $a\times b$ whose entries are the elements of an abelian group $\Gamma$ of order $abc$, each one appearing once and in a unique array in such a way that the sum of the elements of each row is equal to a constant $\omega \in \Gamma$ and the sum of the elements of each column is equal to a constant $\delta \in \Gamma$. In this paper we provide new evidences for the validity of a conjecture proposed by Sylwia Cichacz and Tomasz Hinc on the existence of an $\mathrm{MRS}_\Gamma(a,b;c)$. We also generalize this problem, describing constructions of $\Gamma$-magic rectangle sets, whose elements are partially filled arrays.

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, $3\leqslant s \leqslant n$, $3\leqslant k\leqslant m$, $ms=nk$ and $nk\equiv 0,3 \pmod 4$. By constructing integer Heffter array sets, we prove the conjecture in the affirmative whenever $k\geqslant 7\cdot \gcd(s,k)$ is odd and $s\neq 3,5,6,10$.

In this paper we introduce a special class of partially filled arrays. A magic partially filled array $\mathrm{MPF}_\Omega(m,n; s,k)$ on a subset $\Omega$ of an abelian group $(\Gamma,+)$ is a partially filled array of size $m\times n$ with entries in $\Omega$ such that $(i)$ every $\omega \in \Omega$ appears once in the array; $(ii)$ each row contains $s$ filled cells and each column contains $k$ filled cells; $(iii)$ there exist (not necessarily distinct) elements $x,y\in \Gamma$ such that the sum of the elements in each row is $x$ and the sum of the elements in each column is $y$. In particular, if $x=y=0_\Gamma$, we have a zero-sum magic partially filled array ${}^0\mathrm{MPF}_\Omega(m,n; s,k)$. Examples of these objects are magic rectangles, $\Gamma$-magic rectangles, signed magic arrays, (integer or non integer) Heffter arrays. Here, we give necessary and sufficient conditions for the existence of a magic rectangle with empty cells, i.e., of an $\mathrm{MPF}_\Omega(m,n;s,k)$ where $\Omega=\{1,2,\ldots,nk\}\subset\mathbb{Z}$. We also construct zero-sum magic partially filled arrays when $\Omega$ is the abelian group $\Gamma$ or the set of its nonzero elements.

In this paper, we prove the existence of a relation between the prime divisors of the order of a Sylow normalizer and the degree of characters having values in some small cyclotomic fields. This relation is stronger when the group is solvable.

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 elements belong to $d$ consecutive diagonals. As an example of application of this method, we prove that there exists an integer $H(m,n;s,k)$ in each of the following cases: $(i)$ $d\equiv 0 \pmod 4$; $(ii)$ $5\leq d\equiv 1 \pmod 4$ and $n k\equiv 3\pmod 4$; $(iii)$ $d\equiv 2 \pmod 4$ and $nk\equiv 0 \pmod 4$; $(iv)$ $d\equiv 3 \pmod 4$ and $n k\equiv 0,3\pmod 4$. The same method can be applied also for signed magic arrays $SMA(m,n;s,k)$ and for magic rectangles $MR(m,n;s,k)$. In fact, we prove that there exists an $SMA(m,n;s,k)$ when $d\geq 2$, and there exists an $MR(m,n;s,k)$ when either $d\geq 2$ is even or $d\geq 3$ and $nk$ are odd. We also provide constructions of integer Heffter arrays and signed magic arrays when $k$ is odd and $s\equiv 0 \pmod 4$.

Label the vertices of the complete graph $K_v$ with the integers $\{ 0, 1, \ldots, v-1 \}$ and define the length of the edge between $x$ and $y$ to be $\min( |x-y| , v - |x-y| )$. Let $L$ be a multiset of size $v-1$ with underlying set contained in $\{ 1, \ldots, \lfloor v/2 \rfloor \}$. The Buratti-Horak-Rosa Conjecture is that there is a Hamiltonian path in $K_v$ whose edge lengths are exactly $L$ if and only if for any divisor $d$ of $v$ the number of multiples of $d$ appearing in $L$ is at most $v-d$. We introduce "growable realizations," which enable us to prove many new instances of the conjecture and to reprove known results in a simpler way. As examples of the new method, we give a complete solution when the underlying set is contained in $\{ 1,4,5 \}$ or in $\{ 1,2,3,4 \}$ and a partial result when the underlying set has the form $\{ 1, x, 2x \}$. We believe that for any set $U$ of positive integers there is a finite set of growable realizations that implies the truth of the Buratti-Horak-Rosa Conjecture for all but finitely many multisets with underlying set $U$., Comment: 26 pages

Let $m,n,s,k$ be integers such that $4\leq s\leq n$, $4\leq k \leq m$ and $ms=nk$. Let $\lambda$ be a divisor of $2ms$ and let $t$ be a divisor of $\frac{2ms}{\lambda}$. In this paper we construct magic rectangles $MR(m,n;s,k)$, signed magic arrays $SMA(m,n;s,k)$ and integer $\lambda$-fold relative Heffter arrays ${}^\lambda H_t(m,n;s,k)$ where $s,k$ are even integers. In particular, we prove that there exists an $SMA(m,n;s,k)$ for all $m,n,s,k$ satisfying the previous hypotheses. Furthermore, we prove that there exist an $MR(m,n;s,k)$ and an integer ${}^\lambda H_t(m,n;s,k)$ in each of the following cases: $(i)$ $s,k \equiv 0 \pmod 4$; $(ii)$ $s\equiv 2\pmod 4$ and $k\equiv 0 \pmod 4$; $(iii)$ $s\equiv 0\pmod 4$ and $k\equiv 2 \pmod 4$; $(iv)$ $s,k\equiv 2 \pmod 4$ and $m,n$ both even.

The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singleton's defect. We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution. From these relations we are able to derive known identities (in an easier way) for interesting cases, such as extremal codes, Hermitian codes, MDS and NMDS codes. Moreover, we are able to present for the first time the weight distribution of AMDS codes. We also discuss the link between our results and the Pless equations., Comment: Accepted for publication in Finite Fields and their Applications (2021) 101933

In this paper we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset $A$ of $\mathbb{Z}_n\setminus \{0\}$ of size $k$ such that $\sum_{z\in A} z\not= 0$, it is possible to find an ordering $(a_1,\ldots,a_k)$ of the elements of $A$ such that the partial sums $s_i=\sum_{j=1}^i a_j$, $i=1,\ldots,k$, are nonzero and pairwise distinct. This conjecture is known to be true for subsets of size $k\leq 11$ in cyclic groups of prime order. Here, we extend such result to any torsion-free abelian group and, as a consequence, we provide an asymptotic result in $\mathbb{Z}_n$. We also consider a related conjecture, originally proposed by Ronald Graham: given a subset $A$ of $\mathbb{Z}_p\setminus\{0\}$, where $p$ is a prime, there exists an ordering of the elements of $A$ such that the partial sums are all distinct. Working with the methods developed by Hicks, Ollis and Schmitt, based on the Alon's combinatorial Nullstellensatz, we prove the validity of such conjecture for subsets $A$ of size $12$.

The conjecture, still widely open, posed by Marco Buratti, Peter Horak and Alex Rosa states that a list $L$ of $v-1$ positive integers not exceeding $\left\lfloor \frac{v}{2}\right\rfloor$ is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set $\{0,1,\ldots,v-1\}$ if and only if, for every divisor $d$ of $v$, the number of multiples of $d$ appearing in $L$ is at most $v-d$. In this paper we present new methods that are based on linear realizations and can be applied to prove the validity of this conjecture for a vast choice of lists. As example of their flexibility, we consider lists whose underlying set is one of the following: $\{x,y,x+y\}$, $\{1,2,3,4\}$, $\{1,2,4,\ldots,2x\}$, $\{1,2,4,\ldots,2x,2x+1\}$. We also consider lists with many consecutive elements., Comment: In this version we describe new methods that allowed us to improve the results of the previous "A new result on the problem of Buratti, Horak and Rosa"

Let $v=2ms+t$ be a positive integer, where $t$ divides $2ms$, and let $J$ be the subgroup of order $t$ of the cyclic group $\mathbb{Z}_v$. An integer Heffter array $H_t(m,n;s,k)$ over $\mathbb{Z}_v$ relative to $J$ is an $m\times n$ partially filled array with elements in $\mathbb{Z}_v$ such that: (a) each row contains $s$ filled cells and each column contains $k$ filled cells; (b) for every $x\in \mathbb{Z}_v \setminus J$, either $x$ or $-x$ appears in the array; (c) the elements in every row and column, viewed as integers in $\pm\left\{ 1, \ldots, \left\lfloor \frac{v}{2}\right\rfloor \right\}$, sum to $0$ in $\mathbb{Z}$. In this paper we study the existence of an integer $H_t(m,n;s,k)$ when $s$ and $k$ are both even, proving the following results. Suppose that $4\leq s\leq n$ and $4\leq k \leq m$ are such that $ms=nk$. Let $t$ be a divisor of $2ms$. (a) If $s,k \equiv 0 \pmod 4$, there exists an integer $H_t(m,n;s,k)$. (b) If $s\equiv 2\pmod 4$ and $k\equiv 0 \pmod 4$, there exists an integer $H_t(m,n;s,k)$ if and only if $m$ is even. (c) If $s\equiv 0\pmod 4$ and $k\equiv 2 \pmod 4$, then there exists an integer $H_t(m,n;s,k)$ if and only if $n$ is even. (d) Suppose that $m$ and $n$ are both even. If $s,k\equiv 2 \pmod 4$, then there exists an integer $H_t(m,n;s,k)$., Comment: In this version, we also construct non-square relative Heffter arrays

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 $\mathbb{Z}_v$, where $v=2nk+t$, whose rows contain $s$ filled cells and whose columns contain $k$ filled cells, such that the elements in every row and column sum to zero and, for every $x\in \mathbb{Z}_v$ not belonging to the subgroup of order $t$, either $x$ or $-x$ appears in the array. In this paper we show how relative Heffter arrays can be used to construct biembeddings of cyclic cycle decompositions of the complete multipartite graph $K_{\frac{2nk+t}{t}\times t}$ into an orientable surface. In particular, we construct such biembeddings providing integer globally simple square relative Heffter arrays for $t=k=3,5,7,9$ and $n\equiv 3 \pmod 4$ and for $k=3$ with $t=n,2n$, any odd $n$., Comment: arXiv admin note: text overlap with arXiv:1906.03932

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 $2nk$, and let $J$ be the subgroup of $\mathbb{Z}_v$ of order $t$. A $H_t(m,n; s,k)$ Heffter array over $\mathbb{Z}_v$ relative to $J$ is an $m\times n$ partially filled array with elements in $\mathbb{Z}_v$ such that: (a) each row contains $s$ filled cells and each column contains $k$ filled cells; (b) for every $x\in \mathbb{Z}_v\setminus J$, either $x$ or $-x$ appears in the array; (c) the elements in every row and column sum to $0$. Here we study the existence of square integer (i.e. with entries chosen in $\pm\left\{1,\dots,\left\lfloor \frac{2nk+t}{2}\right\rfloor \right\}$ and where the sums are zero in $\mathbb{Z}$) relative Heffter arrays for $t=k$, denoted by $H_k(n;k)$. In particular, we prove that for $3\leq k\leq n$, with $k\neq 5$, there exists an integer $H_k(n;k)$ if and only if one of the following holds: (a) $k$ is odd and $n\equiv 0,3\pmod 4$; (b) $k\equiv 2\pmod 4$ and $n$ is even; (c) $k\equiv 0\pmod 4$. Also, we show how these arrays give rise to cyclic cycle decompositions of the complete multipartite graph.