Abstract. In this paper we propose a conjecture concerning partial sums of an arbitrary finite subset of an abelian group, that naturally arises investigating simple Heffter systems. Then, we show its connection with related open problems and we present some results about the validity of these conjectures.

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 Zv of order t. A Ht(m, n; s, k) Heffter array over Zv relative to J is an m × n partially filled array with elements in Zv such that: (i) each row contains s filled cells and each column contains k filled cells; (ii) for every x ∈ Z 2nk+t \ J, either x or −x appears in the array; (iii) the elements in every row and column sum to 0. In particular, here we study the existence for t = k of integer (i.e. the entries are chosen in ± 1, . . . , 2nk+t 2 and the sums are zero in Z) square relative Heffter arrays.

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A subset of an abelian group is sequenceable if there is an ordering $(x_1, \ldots, x_k)$ of its elements such that the partial sums $(y_0, y_1, \ldots, y_k)$, given by $y_0 = 0$ and $y_i = \sum_{j=1}^i x_j$ for $1 \leq i \leq k$, are distinct, with the possible exception that we may have $y_k = y_0 = 0$.   We demonstrate the sequenceability of subsets of size $k$ of $\mathbb{Z}_n \setminus \{ 0 \}$ when $n = mt$ in many cases, including when $m$ is either prime or has all prime factors larger than $k! /2$ for $k \leq 11$ and $t \leq 5$ and for $k=12$ and $t \leq 4$.  We obtain similar, but partial, results for $13 \leq k \leq 15$.   This represents progress on a variety of questions and conjectures in the literature concerning the sequenceability of subsets of abelian groups, which we combine and summarize into the conjecture that if a subset of an abelian group does not  contain $0$ then it is sequenceable.

In this paper we define a new class of partially filled arrays, called λ-fold relative Heffter arrays, that are a generalisation of the Heffter arrays introduced by Archdeacon in 2015. After showing the connection of this new concept with several other ones, such as signed magic arrays, graph decompositions and relative difference families, we determine some necessary conditions and we present existence results for infinite classes of these arrays. In the last part of the paper we also show that these arrays give rise to biembeddings of multigraphs into orientable surfaces and we provide infinite families of such biembeddings. To conclude, we present a result concerning pairs of λ-fold relative Heffter arrays and covering surfaces.

Strong difference families are an interesting class of discrete structures which can be used to derive relative difference families. Relative difference families are closely related to 2-designs, and have applications in constructions for many significant codes, such as optical orthogonal codes and optical orthogonal signature pattern codes. In this paper, with a careful use of cyclotomic conditions attached to strong difference families, we improve the lower bound on the asymptotic existence results of (F p ×F q , F p ×{0}, k, λ)-DFs for k ∈ {p, p + 1}. We improve Buratti's existence results for 2-(13q, 13, λ) designs and 2-(17q, 17, λ) designs, and establish the existence of seven new 2-(v, k, λ) designs for (v, k, λ)

Frame difference families, which can be obtained via a careful use of cyclotomic conditions attached to strong difference families, play an important role in direct constructions for resolvable balanced incomplete block designs. We establish asymptotic existences for several classes of frame difference families. As corollaries new infinite families of 1-rotational (pq + 1, p + 1, 1)-RBIBDs over F + p × F + q are derived, and the existence of (125q + 1, 6, 1)-RBIBDs is discussed. We construct (v, 8, 1)-RBIBDs for v ∈ {624, 1576, 2976, 5720, 5776, 10200, 14176, 24480}, whose existence were previously in doubt. As applications, we establish asymptotic existences for an infinite family of optimal constant composition codes and an infinite family of strictly optimal frequency hopping sequences.

Strong difference families of special types are introduced to produce new relative difference families from the point of view of both asymptotic existences and concrete examples. As applications, group divisible designs of type 30 u with block size 6 are discussed, r-rotational balanced incomplete block designs with block size 6 are derived for r ∈ {6, 10}, and several classes of optimal optical orthogonal codes with weight 5, 6, 7, or 8 are obtained.