On permanents of Sylvester Hadamard matrices

It is well-known that the evaluation of the permanent of an arbitrary $(-1,1)$-matrix is a formidable problem. Ryser's formula is one of the fastest known general algorithms for computing permanents. In this paper, Ryser's formula has been rewritten for the special case of Sylvester Hadamard matrices by using their cocyclic construction. The rewritten formula presents an important reduction in the number of sets of $r$ distinct rows of the matrix to be considered. However, the algorithm needs a preprocessing part which remains time-consuming in general

On an inequivalence criterion for cocyclic Hadamard matrices

Given two Hadamard matrices of the same order, it can be quite difficult to decide whether or not they are equivalent. There are some criteria to determine Hadamard inequivalence. Among them, one of the most commonly used is the 4-profile criterion. In this paper, a reformulation of this criterion in the cocyclic framework is given. The improvements obtained in the computation of the 4-profile of a cocyclic Hadamard matrix are indicated.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM–296Junta de Andalucía P07-FQM-0298

Self-Dual codes from (−1,1)(-1,1)-matrices of skew symmetric type

Previously, self-dual codes have been constructed from weighing matrices, and in particular from conference matrices (skew and symmetric). In this paper, codes constructed from matrices of skew symmetric type whose determinants reach the Ehlich-Wojtas' bound are presented. A necessary and sufficient condition for these codes to be self-dual is given, and examples are provided for lengths up to 52

Generalized binary arrays from quasi-orthogonal cocycles

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.Junta de Andalucía FQM-01

On quasi-orthogonal cocycles

We introduce the notion of quasi-orthogonal cocycle. This is motivated in part by the maximal determinant problem for square {±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.Junta de Andalucía FQM-01

Boolean Functions and Permanents of Sylvester Hadamard Matrices

One of the fastest known general techniques for computing permanents is Ryser’s formula. On this note, we show that this formula over Sylvester Hadamard matrices of order 2m, Hm, can be carried out by enumerating m-variable Boolean functions with an arbitrary Walsh spectrum. As a consequence, the quotient per(Hm)/22m might be a measure of the “density” of m-variable Boolean functions with high nonlinearity

Self-Dual codes from (−1,1)-matrices of skew symmetric type

Previously, self-dual codes have been constructed from weighing matrices, and in particular from conference matrices (skew and symmetric). In this paper, codes constructed from matrices of skew symmetric type whose determinants reach the Ehlich- Wojtas’ bound are presented. A necessary and sufficient condition for these codes to be self-dual is given, and examples are provided for lengths up to 52.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM-016Junta de Andalucía P07-FQM-0298

Quasi-Hadamard Full Propelinear Codes

In this paper, we give a characterization of quasi-Hadamard groups in terms of propelinear codes. We define a new class of codes that we call quasi-Hadamard full propelinear codes. Some structural properties of these codes are studied and examples are provided.Junta de Andalucía FQM-016Ministerio de Economía y Competitividad TIN2016-77918-

Butson full propelinear codes

In this paper we study Butson Hadamard matrices, and codes over finite rings coming from these matrices in logarithmic form, called BH-codes. We introduce a new morphism of Butson Hadamard matrices through a generalized Gray map on the matrices in logarithmic form, which is comparable to the morphism given in a recent note of Ó Catháin and Swartz. That is, we show how, if given a Butson Hadamard matrix over the kth roots of unity, we can construct a larger Butson matrix over the ℓth roots of unity for any ℓ dividing k, provided that any prime p dividing k also divides ℓ. We prove that a Zps-additive code with p a prime number is isomorphic as a group to a BH-code over Zps and the image of this BH-code under the Gray map is a BH-code over Zp (binary Hadamard code for p=2). Further, we investigate the inherent propelinear structure of these codes (and their images) when the Butson matrix is cocyclic. Some structural properties of these codes are studied and examples are provided

Butson full propelinear codes

In this paper we study Butson Hadamard matrices, and codes over finite rings coming from these matrices in logarithmic form, called BH-codes. We introduce a new morphism of Butson Hadamard matrices through a generalized Gray map on the matrices in logarithmic form, which is comparable to the morphism given in a recent note of \'{O} Cath\'{a}in and Swartz. That is, we show how, if given a Butson Hadamard matrix over the $k^{\rm th}$ roots of unity, we can construct a larger Butson matrix over the $\ell^{\rm th}$ roots of unity for any $\ell$ dividing $k$, provided that any prime $p$ dividing $k$ also divides $\ell$. We prove that a $\mathbb{Z}_{p^s}$-additive code with $p$ a prime number is isomorphic as a group to a BH-code over $\mathbb{Z}_{p^s}$ and the image of this BH-code under the Gray map is a BH-code over $\mathbb{Z}_p$ (binary Hadamard code for $p=2$). Further, we investigate the inherent propelinear structure of these codes (and their images) when the Butson matrix is cocyclic. Some structural properties of these codes are studied and examples are provided.Comment: 24 pages. Submitted to IEEE Transactions on Information Theor