Cyclotomic Constructions of Skew Hadamard Difference Sets

We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields using unions of cyclotomic classes of order $N=2p_1^m$, where $p_1$ is a prime and $m$ a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers.Comment: 15 pages; corrected a few typos; to appear in J. Combin. Theory (A

Matrix constructions of divisible designs

AbstractWe present two new constructions of group divisible designs. We use skew-symmetric Hadamard matrices and certain strongly regular graphs together with (v, k, λ)-designs. We include many examples, in particular several new series of divisible difference sets

Implementing Hadamard Matrices in SageMath

Hadamard matrices are $(-1, +1)$ square matrices with mutually orthogonal rows. The Hadamard conjecture states that Hadamard matrices of order $n$ exist whenever $n$ is $1$, $2$, or a multiple of $4$. However, no construction is known that works for all values of $n$, and for some orders no Hadamard matrix has yet been found. Given the many practical applications of these matrices, it would be useful to have a way to easily check if a construction for a Hadamard matrix of order $n$ exists, and in case to create it. This project aimed to address this, by implementing constructions of Hadamard and skew Hadamard matrices to cover all known orders less than or equal to $1000$ in SageMath, an open-source mathematical software. Furthermore, we implemented some additional mathematical objects, such as complementary difference sets and T-sequences, which were not present in SageMath but are needed to construct Hadamard matrices. This also allows to verify the correctness of the results given in the literature; within the $n\leq 1000$ range, just one order, $292$, of a skew Hadamard matrix claimed to have a known construction, required a fix.Comment: pdflatex+biber, 32 page

Cubes of symmetric designs

We study $n$-dimensional matrices with $\{0,1\}$-entries ($n$-cubes) such that all their $2$-dimensional slices are incidence matrices of symmetric designs. A known construction of these objects obtained from difference sets is generalized so that the resulting $n$-cubes may have inequivalent slices. For suitable parameters, they can be transformed into $n$-dimensional Hadamard matrices with this property. In contrast, previously known constructions of $n$-dimensional designs all give examples with equivalent slices.Comment: 18 page