arXiv subjects
Publications and source records attributed to J. A. Armario.
We introduce almost supplementary difference sets (ASDS). For odd $m$, certain ASDS in ${\mathbb Z}_m$ that have amicable incidence matrices are equivalent to quaternary sequences of odd length $m$ with optimal autocorrelation. As one consequence, if $2m-1$ is a prime power, or $m \equiv 1 \mod 4$ is prime, then ASDS of this kind exist. We also explore connections to optimal binary sequences and group cohomology.
Generalized perfect binary arrays (GPBAs) were used by Jedwab to construct perfect binary arrays. A non-trivial GPBA can exist only if its energy is $2$ or a multiple of $4$. This paper introduces generalized optimal binary arrays (GOBAs) with even energy not divisible by $4$, as analogs of GPBAs. We give a procedure to construct GOBAs based on a characterization of the arrays in terms of $2$-cocycles. As a further application, we determine negaperiodic Golay pairs arising from generalized optimal binary sequences of small length.
We introduce the notion of quasi-orthogonal cocycle. This is motivated in part by the maximal determinant problem for square $\{\pm 1\}$-matrices of size congruent to $2$ modulo $4$. Quasi-orthogonal cocycles are analogous to the orthogonal cocycles of algebraic design theory. Equivalences with new and known combinatorial objects afforded by this analogy, such as quasi-Hadamard groups, relative quasi-difference sets, and certain partially balanced incomplete block designs, are proved.
Hadamard ideals were introduced in 2006 as a set of nonlinear polynomial equations whose zeros are uniquely related to Hadamard matrices with one or two circulant cores of a given order. Based on this idea, the cocyclic Hadamard test enable us to describe a polynomial ideal that characterizes the set of cocyclic Hadamard matrices over a fixed finite group $G$ of order $4t$. Nevertheless, the complexity of the computation of the reduced Gröbner basis of this ideal is $2^{O(t^2)}$, which is excessive even for very small orders. In order to improve the efficiency of this polynomial method, we take advantage of some recent results on the inner structure of a cocyclic matrix to describe an alternative polynomial ideal that also characterizes the mentioned set of cocyclic Hadamard matrices over $G$. The complexity of the computation decreases in this way to $2^{O(n)}$, where $n$ is the number of $G$-coboundaries. Particularly, we design two specific procedures for looking for $\mathbb{Z}_t \times \mathbb{Z}_2^2$-cocyclic Hadamard matrices and $D_{4t}$-cocyclic Hadamard matrices, so that larger cocyclic Hadamard matrices (up to $t \leq 31$) are explicitly obtained.
An $n$ by $n$ skew-symmetric type $(-1,1)$-matrix $K=[k_{i,j}]$ has $1$'s on the main diagonal and $\pm 1$'s elsewhere with $k_{i,j}=-k_{j,i}$. The largest possible determinant of such a matrix $K$ is an interesting problem. The literature is extensive for $n\equiv 0 \mod 4$ (skew-Hadamard matrices), but for $n\equiv 2\mod 4$ there are few results known for this question. In this paper we approach this problem constructing cocyclic matrices over the dihedral group of $2t$ elements, for $t$ odd, which are equivalent to $(-1,1)$-matrices of skew type. Some explicit calculations have been done up to $t=11$. To our knowledge, the upper bounds on the maximal determinant in orders 18 and 22 have been improved.