Classification of difference matrices over cyclic groups

Lampio, Pekka H.J.; Östergård, Patric R. J.

doi:10.1016/j.jspi.2010.09.023

Cited by 8 publications

(29 citation statements)

References 14 publications

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“…The definition of equivalence of BH(q, n) matrices given in Section 2 agrees with the equivalence relation ∼ = * defined in [10, Section 2] and the equivalence inducing group E * in [10,Section 5].…”

Section: Because Ofsupporting

confidence: 54%

“…The problem of checking equivalence of BH(q, m, n) matrices is solved by transforming it into a corresponding graph isomorphism problem in exactly the same way as was done with difference matrices over cyclic groups in [10,Section 3]. Each BH(q, m, n) matrix is mapped to a directed graph, called the equivalence graph of the matrix.…”

Section: Because Ofmentioning

confidence: 99%

“…The second method determines the number of BH(q, m, n) matrices from the number of BH(q, m − 1, n) matrices and number of ways these smaller matrices can be augmented to yield a BH(q, m, n) matrix. This double-counting method is the same as in [10,Section 5] except that the analytical formula for 2 × c difference matrices and the reduction used for 3 × c difference matrices are not valid for BH(q, 2, n) and BH(q, 3, n) matrices, respectively. With BH(q, n) matrices the double-counting starts one level earlier with BH(q, 1, n) matrices, which all belong to the same equivalent class of size q n .…”

Section: Because Ofmentioning

confidence: 99%

“…In what follows we motivate the formula (10). Consider a dephased complex Hadamard matrix H and a phasing matrix R whose first row and column is 0, and all other entries are a real variables.…”

Section: Parametrizing Complex Hadamard Matricesmentioning

confidence: 99%

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The quaternary complex Hadamard matrices of orders 10, 12, and 14

Lampio

Szöllősi²,

Östergård

2013

Discrete Mathematics

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A complete classification of quaternary complex Hadamard matrices of orders 10, 12 and 14 is given, and a new parametrization scheme for obtaining new examples of affine parametric families of complex Hadamard matrices is provided. On the one hand, it is proven that all 10 × 10 and 12 × 12 quaternary complex Hadamard matrices belong to some parametric family, but on the other hand, it is shown by exhibiting an isolated 14 × 14 matrix that there cannot be a general method for introducing parameters into these types of matrices.the BH(q, n) matrices makes it possible to escape equivalence classes and therefore to collect many inequivalent matrices into a single parametric family. Also the switching operation [16], a well-known technique in design theory, has this property.The existence of BH(q, n) matrices is wide open in general. Even the simplest case for q = 2 is undecided, as the famous Hadamard conjecture, stating that BH(2, 4k) matrices exist for every positive integer k, remains elusive despite continuous efforts [3]. Real Hadamard matrices, or BH(2, n) matrices, are completely classified up to n = 28, and there has been some promising advance in enumerating the case n = 32 very recently (see [7] and the references therein). For other values of q we have some constructions [2] and some nonexistence results [20]. Harada, Lam and Tonchev classified all 16 × 16 generalized Hadamard matrices over groups of order 4 and obtained new examples of BH(4, 16) matrices [5]. This particular result motivated the authors to investigate the existence of BH(4, n) matrices in more general and to start enumerating and classifying them for small n. The census of quaternary complex Hadamard matrices up to order 8 were carried out in [14]. The aim of this paper is to continue that work, and give a complete classification of the BH(4, n) matrices up to orders n = 14.This work has two major parts: first, we classify all BH(q, n) matrices using computeraided methods, and secondly, we define parametric families of complex Hadamard matrices by introducing parameters to the BH(q, n) matrices. Parametric families offer a compact way of representing a large number of BH(q, n) matrices, and they also yield information about complex Hadamard matrices.In the previous work [14] the authors collected all BH(4, n) matrices contained in various parametric families from the existing literature and then confirmed with an exhaustive computer search that those are the only examples. Here the situation is exactly the opposite, as it turns out that almost all BH(4, n) matrices of orders n = 12 and 14 found in the current work by the computer search are previously unknown. Therefore we are facing the inverse problem: we need to encode a given collection of BH(4, n) matrices by parametric families.The outline of the paper is as follows. In Section 2 we give some basic definitions and results that are used in later sections. Then in Section 3 we recall the computer-aided classification method of difference matrices from [10] and highlight the main diffe...

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Section: Because Ofsupporting

confidence: 54%

Section: Because Ofmentioning

confidence: 99%

Section: Because Ofmentioning

confidence: 99%

Section: Parametrizing Complex Hadamard Matricesmentioning

confidence: 99%

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The quaternary complex Hadamard matrices of orders 10, 12, and 14

Lampio

Szöllősi²,

Östergård

2013

Discrete Mathematics

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“…This is in contrast with the method of canonical augmentation employed earlier in [30] which relies on graph representation of the r × n rectangular orthogonal matrices with qth root entries on 3q(r + n) + r vertices. See [28], [29] for more on graph representation of Butson matrices.…”

Section: 2mentioning

confidence: 99%

Orderly generation of Butson Hadamard matrices

Lampio

Östergård²,

Szöllősi

2019

Math. Comp.

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In this paper Butson-type complex Hadamard matrices BH(n, q) of order n and complexity q are classified for small parameters by computer-aided methods. Our main results include the enumeration of BH(21, 3), BH(16, 4), and BH(14, 6) matrices. There are exactly 72, 1786763, and 167776 such matrices, up to monomial equivalence. Additionally, we show an example of a BH(14, 10) matrix for the first time, and show the nonexistence of BH(8, 15), BH(11, q) for q ∈ {10, 12, 14, 15}, and BH(13, 10) matrices.

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Quaternary complex Hadamard matrices of order 18

Östergård

Paavola

2020

J of Combinatorial Designs

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A technique for classifying Butson‐type Hadamard matrices over the complex 4th roots of unity is developed. The technique resembles the approach used by Kharaghani and Tayfeh‐Rezaie in their classification of the (real) Hadamard matrices, where the search is split into parts according to matrix types. A classification of the Butson‐type Hadamard matrices over the complex 4th roots of unity is carried out; the number of such objects up to Hadamard equivalence is 1 955 625.

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Classification of difference matrices over cyclic groups

Cited by 8 publications

References 14 publications

The quaternary complex Hadamard matrices of orders 10, 12, and 14

The quaternary complex Hadamard matrices of orders 10, 12, and 14

Orderly generation of Butson Hadamard matrices

Quaternary complex Hadamard matrices of order 18

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