Classifying Cocyclic Butson Hadamard Matrices

Egan, Ronan; Flannery, Dane; Catháin, Padraig Ó

doi:10.1007/978-3-319-17729-8_8

Cited by 12 publications

(20 citation statements)

References 16 publications

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“…q = 11: The Fourier matrix F 11 is unique [18]. q = 12: A BH (5,12) does not exist since all 5 × 5 complex Hadamard were shown to be equivalent to F 5 in [14]. A BH (11,12) does not exist by Theorem 4.10. q = 13: The Fourier matrix F 13 is unique [18].…”

Section: Results and Case Studiesmentioning

confidence: 99%

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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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Section: Results and Case Studiesmentioning

confidence: 99%

“…There does not exist a BH(13, 10) matrix. r BH (11,6) BH (11,10) BH (11,12) BH (11,14) BH (11, 15) 1 1 1 1 1 1 2 5 5 32 4 3 3 499 0 168564 2091 584 4 33655 7950174 2572 94 5 42851 561071 14 22 6 171 578 0 0 7 0 0 Table 6. The nonexistence of BH(11, q) matrices for various q.…”

Section: Classification Of Bhmentioning

confidence: 99%

Orderly generation of Butson Hadamard matrices

Lampio

Östergård²,

Szöllősi

2019

Math. Comp.

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“…Given H ∈ BH(4, 8) of Example 1.1. Then [1,3,5,7], [1,5,1,5], [1,7,5,3], [2,2,2,2], [2,4,6,0], [2,6,2,6], [2,0,6,4], [3,3,3,3], [3,5,7,1], [3,7,3,7], [3,1,7,5], [4,4,4,4], [4,6,0,2], [4,0,…”

Section: Bh-codes and Propelinear Codesmentioning

confidence: 99%

“…x ∈ C 1 (1,2,3,4,5,6,7,8) x ∈ C 2 (1, 3, 5, 7) (2,4,6,8) x ∈ C 3 (1,4,7,2,5,8,3,6) x ∈ C 4 (1, 5)(2, 6)(3, 7)(4, 8)…”

Section: Letmentioning

confidence: 99%

Quasi-Hadamard Full Propelinear Codes

et al. 2018

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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 k th roots of unity, we can construct a larger Butson matrix over the ℓ th roots of unity for any ℓ dividing k, provided that any prime p dividing k also divides ℓ. We prove that a Z p s-additive code with p a prime number is isomorphic as a group to a BH-code over Z p s and the image of this BH-code under the Gray map is a BH-code over 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.

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“…We remark that the machinery set up in this paper has been applied successfully in a recent classification of Butson Hadamard matrices of order n over pth roots of unity, for p prime and np ≤ 100 [5].…”

Section: Introductionmentioning

confidence: 95%