New linear codes over Zps via the trace map

The trace map has been used very successfully to generate cocyclic complex and Butson Hadamard matrices and simplex codes over Z4 and Z2s. We extend this technique to obtain new linear codes over Zps. It is worth nothing here that these codes are cocyclic but not simplex codes. Further we find that the construction method also gives Butson Hadamard matrices of order psm

Cocyclic butson Hadamard matrices and codes over Zn via the trace map

Over the past couple of years, trace maps over Galois fields and Galois rings have been used very succesfully o construct cocyclic Hadamard, complex Hadamard and Butson Hadamard matrices and subsequently to generate simplex codes over Z4, Z2 and ZP and new linear codes over ZP. Here we define a new map, the trace-like map and more generally the weighted map and extend these techniques to construct cocyclic Budson Hadamard matrices of order (nm) for all n and m and linear and non-linear codes over Zn

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-

A Heuristic Procedure with Guided Reproduction for Constructing Cocyclic Hadamard Matrices

A genetic algorithm for constructing cocyclic Hadamard matrices over a given group is described. The novelty of this algorithm is the guided heuristic procedure for reproduction, instead of the classical crossover and mutation operators. We include some runs of the algorithm for dihedral groups, which are known to give rise to a large amount of cocyclic Hadamard matrices.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM–296Junta de Andalucía P07-FQM-0298

Equivalences of Zt×Z22-cocyclic Hadamard matrices

One of the most promising structural approaches to resolving the Hadamard Conjecture uses the family of cocyclic matrices over Zt × Z2 2. Two types of equivalence relations for classifying cocyclic matrices over Zt × Z2 2 have been found. Any cocyclic matrix equivalent by either of these relations to a Hadamard matrix will also be Hadamard. One type, based on algebraic relations between cocycles over any fi- nite group, has been known for some time. Recently, and independently, a second type, based on four geometric relations between diagrammatic visualisations of cocyclic matrices over Zt × Z2 2, has been found. Here we translate the algebraic equivalences to diagrammatic equivalences and show one of the diagrammatic equivalences cannot be obtained this way. This additional equivalence is shown to be the geometric translation of matrix transposition

A Mixed Heuristic for Generating Cocyclic Hadamard Matrices

A way of generating cocyclic Hadamard matrices is described, which combines a new heuristic, coming from a novel notion of fitness, and a peculiar local search, defined as a constraint satisfaction problem. Calculations support the idea that finding a cocyclic Hadamard matrix of order 4 · 47 might be within reach, for the first time, progressing further upon the ideas explained in this work.Junta de Andalucía FQM-01