So far, HOP has been defined with the constraint that each table size is at least four. In this talk, I generalize the problem to allow for tables of size two. A solution to the generalized HOP with $s$ tables of size $2$ and $t$ round tables of sizes $2m_1, 2m_2, \ldots, 2m_t$ is equivalent to a decomposition of the multigraph $K_{2n} +(\gamma-1)I$, for an appropriate integer $\gamma$, into subgraphs consisting of disjoint $I$-alternating cycles of lengths $2m_1,2m_2,\ldots,2m_t$ and $s$ copies of $K_2$. I will present a general approach to this problem, and recent solutions to several cases.

A $d$-CFF$(t,n)$ is a $t\times n$ array such that for each subset of $d+1$ columns, every possible weight-$1$ binary tuple occurs in at least one of the rows. CFFs are used in combinatorial group testing to determine up to $d$ defective items in a collection of $n$ items by pooling items to be tested together in $t$ tests; the defective items are deduced from the test results.

A covering array (CA) of size $N$, strength $t$, $k$ factors and alphabet size $v$ is an $N\times k$ array such that for each subset of $t$ columns, every possible $t$-tuple of the alphabet occurs in at least one of the rows. CAs give effective test suits for software testing giving a good coverage of the parameter space.

Both types of array require that ``coverage" must occur in every subset of columns of a fixed size. The number of rows/tests must be minimized and it grows as the logarithm of the number of columns. In the hypergraph-dependent versions of these problems, edges of a hypergraph specify in which columns ``coverage" is required. The hypergraph model for CAs has been studied since Meagher\&Stevens (2005). The hypergraph model for CFFs was introduced by Idalino and Moura (IWOCA2022), motivated by applications in pandemic screening, where fewer tests are needed by using knowledge about connected communities.

We will overview known results and future work on hypergraph-dependent covering arrays.

For an affine group $G$, we will demonstrate how the structure of the conjugacy classes of the general linear group $GL(n, q)$ can be used to obtain the parameters of the $G$-invariant designs constructed by this method, and we will explicitly compute the parameters for small values of $n$. We will also demonstrate how fixed-points of primitive groups can be used to find designs and obtain what we call "reduced designs" which enable us to determine the automorphism groups of the designs in many cases.

In this talk, we discuss the directed version for complete symmetric equipartite digraphs. Thus, we are interested in decomposing $K_{n[m]}^\ast$, the complete symmetric equipartite digraph with $n$ parts of size $m$, into spanning subdigraphs, each a disjoint union of $k$ directed cycles of lengths $t_1, t_2,\ldots, t_k$ (where $t_1+t_2+ \ldots +t_k=mn$). Such a decomposition models a seating arrangement of $mn$ participants, consisting of $n$ delegations of $m$ participants each, at $k$ tables of sizes $t_1, t_2,\ldots, t_k$ so that each participant sits to the right of each participant from a different delegation exactly once. Recent solutions to extensive cases of this problem for uniform cycle lengths will be presented.

This is joint work with Nevena Franceti\'{c}.

This family of examples illustrates two concepts that we shall attempt to convince the audience are interesting: Firstly, an infinite family of regular weights with so-called minimal closure; secondly the explicit covering schemes derived from these weights that coincide with the well known double covers of folded distance-regular graphs.

If time permits, we will show that the Hamming scheme H(n-1,2) lends itself to a similar construction involving the folded $n$-cube.

$\bullet$ the number of nonzero entries in each row (resp. column) of the array be not constant, and

$\bullet$ the entries of the array, in absolute value, belong to an arbitrarily chosen subset $S$ of a group $G$, not necessarily abelian.

We show that generalized Heffter arrays (GHA) can be used to construct orthogonal path or cycle decompositions and biembeddings of Cayley graphs onto orientable surfaces. The structural properties of the latter depend on the sum of the entries in each row and column of the GHA (with respect to a given ordering). Preferable are those satisfying the further strong property of being simple: for each row and each column, the partial sums (of the non-zero entries) are pairwise distinct, and only the total sum is possibly zero. We show that simple GHAs over cyclic groups can be easily built by means of near alternating sign matrices [2]. Further results and future works will be discussed.

This is joint work with Lorenzo Mella. \newline

References \newline [1] L. Mella and T. Traetta. Constructing generalized Heffter arrays via near alternating sign matrices, submitted. \newline [2] R.A. Brualdi and H.K. Kim. A Generalization of Alternating Sign Matrices, J. Combin. Des. 23 (2015), 204-215.