Orderly generation of Butson Hadamard matrices

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

doi:10.1090/mcom/3453

Cited by 18 publications

(22 citation statements)

References 42 publications

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“…where For k = p s , Lemma 3.1 of [16] gives a pattern that any row of L(H) has to follow. That is, any row x has to be a permutation of the vector (u,…”

Section: Constructing Butson Hadamard Matrices and Related Codesmentioning

confidence: 99%

“…Example 4.3. Let H 3,0 be the BH-code associated to F 4 ⊗ F 4 ∈ BH (16,4) and H 3,0 be its image by the Gray map which is known to be a nonlinear code (see [9, Table 1]). H 3,0 is full propelinear, with permutation group Π ∼ = The corresponding permutations ρ x π Φ(x) and ρ y π Φ(y) are as follows: Thus, H 3,0 can be endowed with a full propelinear structure with the group ρ x π Φ(x) , ρ y π Φ(y) of permutations, which is non-abelian of order 32.…”

Section: Propelinear Codes Via the Gray Mapmentioning

confidence: 99%

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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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“…where For k = p s , Lemma 3.1 of [16] gives a pattern that any row of L(H) has to follow. That is, any row x has to be a permutation of the vector (u,…”

Section: Constructing Butson Hadamard Matrices and Related Codesmentioning

confidence: 99%

Section: Propelinear Codes Via the Gray Mapmentioning

confidence: 99%

Quasi-Hadamard Full Propelinear Codes

et al. 2018

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“…One approach to the two conjectures is to completely classify the n BH (2, ) and the n BH(4, ) for small n and look for general constructions based on the results. The n BH (2, ) have been classified for n 32 ≤ and the n BH(4, ) for n 16 ≤ ; the cases with the largest parameters are settled in [6,7] and [12], respectively. We here extend these results by classifying the BH (4,18).…”

Section: Introductionmentioning

confidence: 99%

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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“…Meanwhile, Hirasaka, Kim and Mizoguchi [26] showed the uniqueness of BH(p, p) for primes p up to 17. Lampio,Östergård, Szöllősi enumerated Butson Hadamard matrices CONTENTS 5 for small n, h and classified BH (21,3), BH (14,6) matrices and also BH(n, 4) matrices for n ≤ 16 [37,38]. For a different type of Butson Hadamard matrices, Egan, Flannery and O Cathain [19] classified cocyclic BH(n, p) matrices for odd primes p and np < 100.…”

Section: One Of the Major Open Problems In Combinatorics Is The "Hadamentioning

confidence: 99%

“…37) and(4.38) follow from applying Lemmas 1.18 and 1.19 to (4.42). Finally, Lemmas 1.18 and 1.19 imply ε h = 1.…”

mentioning

confidence: 98%

Construction and classification of group invariant Butson Hadamard matrices

Wong¹

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provided the initial project direction with the construction of group invariant Butson Hadamard matrices by using translation planes, Theorem 3.1.  We expanded upon his ideas which resulted in the nested cyclic subgroups construction, recursive hyperplane construction, Theorems 4.2 and 5.2. I proved the correctness of the constructions.  Prof. Xiang gave the construction of group invariant Butson Hadamard matrices using Galois Rings, Theorem 6.2.  I discovered that the folding theorem can be applied to our construction to obtain more group invariant Butson Hadamard matrices. Prof. Bernhard refined the idea which resulted in the folding construction, Theorem 7.3.  The necessary conditions on the sum of roots of unity, Theorem 2.3, were first formulated by Prof. Schmidt. Together we proved the result.

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Orderly generation of Butson Hadamard matrices

Cited by 18 publications

References 42 publications

Quasi-Hadamard Full Propelinear Codes

Quasi-Hadamard Full Propelinear Codes

Quaternary complex Hadamard matrices of order 18

Construction and classification of group invariant Butson Hadamard matrices

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