A Heuristic Procedure with Guided Reproduction for Constructing Cocyclic Hadamard Matrices

Álvarez, Víctor; Frau, M. D.; Osuna, Amparo

doi:10.1007/978-3-642-04921-7_16

Cited by 4 publications

(7 citation statements)

References 11 publications

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“…This allows them to design better search algorithms such as "guided reproduction" genetic algorithms [5], and to isolate "centrally distributed" subtypes of coboundaries [2] with higher likelihood (experimentally confirmed) of forming CHMs, and thus of providing experimental evidence supporting the Cocyclic Hadamard Conjecture (RP 38). RPs 39,40,41,42 and 43 RPs 39, 40, 41 and 42 [34, Chapter 6.5.1] each ask if a specific construction technique which gives HMs is cocyclic.…”

Section: Computational Search For Chms: Rps 2(a) 6 and 38mentioning

confidence: 98%

Hadamard matrices and their applications: Progress 2007–2010

Horadam

2010

Cryptogr. Commun.

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Section: Computational Search For Chms: Rps 2(a) 6 and 38mentioning

confidence: 98%

Hadamard matrices and their applications: Progress 2007–2010

Horadam

2010

Cryptogr. Commun.

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“…Example 2 Let us consider the case t = 5, k = 2 and v = 684646 (its binary representation gives the 2-vector v = (10100|11100|10011|00110)). We are going to calculate the full set of s-vectors orthogonal to v, for 0 ≤ s ≤ 2 = ⌊ 5 2 ⌋ (recall that the remaining orthogonal vectors are obtained by simply interchanging the 0-bits and 1-bits, since they are the negation of the vectors just calculated).…”

Section: Propositionmentioning

confidence: 99%

Searching for partial Hadamard matrices

Álvarez,

Armario,

Frau

et al. 2012

Preprint

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Three algorithms looking for pretty large partial Hadamard matrices are described. Here "large" means that hopefully about a third of a Hadamard matrix (which is the best asymptotic result known so far, [8]) is achieved. The first one performs some kind of local exhaustive search, and consequently is expensive from the time consuming point of view. The second one comes from the adaptation of the best genetic algorithm known so far searching for cliques in a graph, due to Singh and Gupta [21]. The last one consists in another heuristic search, which prioritizes the required processing time better than the final size of the partial Hadamard matrix to be obtained. In all cases, the key idea is characterizing the adjacency properties of vertices in a particular subgraph G t of Ito's Hadamard Graph ∆(4t) [18], since cliques of order m in G t can be seen as (m + 3) × 4t partial Hadamard matrices.

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“…Actually, constructing Hadamard matrices is a difficult problem of optimization. That being so, different heuristics have been proposed to look for Hadamard matrices (see [2,6,5] for instance), but they all seem to run in exponential time O (2 t ).…”

Section: Introductionmentioning

confidence: 99%

ACS Searching for D 4t -Hadamard Matrices

Álvarez

Armario

Frau

et al. 2010

Lecture Notes in Computer Science

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An Ant Colony System (ACS) looking for cocyclic Hadamard matrices over dihedral groups D4t is described. The underlying weighted graph consists of the rooted trees described in [1], whose vertices are certain subsets of coboundaries. A branch of these trees defines a D4t-Hadamard matrix if and only if two conditions hold: (i) Ii = i − 1 and, (ii) ci = t, for every 2 ≤ i ≤ t, where Ii and ci denote the number of ipaths and i-intersections (see [3] for details) related to the coboundaries defining the branch. The pheromone and heuristic values of our ACS are defined in such a way that condition (i) is always satisfied, and condition (ii) is closely to be satisfied.

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A Heuristic Procedure with Guided Reproduction for Constructing Cocyclic Hadamard Matrices

Cited by 4 publications

References 11 publications

Hadamard matrices and their applications: Progress 2007–2010

Hadamard matrices and their applications: Progress 2007–2010

Searching for partial Hadamard matrices

ACS Searching for D 4t -Hadamard Matrices

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