Uniqueness of Butson Hadamard matrices of small degrees

Hirasaka, Mitsugu; Kim, Kyoung-Tark; Mizoguchi, Yoshihiro

doi:10.1016/j.jda.2015.05.009

Cited by 3 publications

(9 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section we briefly report on our computational results regarding the BH(21, 3) matrices. The classification of BH (18,3) matrices was reported earlier in [15] and independently in [28], while several examples of BH (21,3) matrices were reported in [2].…”

Section: Results and Case Studiesmentioning

confidence: 83%

“…q = 3: Complete classification is available up to n ≤ 21, see Section 4.3. The case BH (18,3) was reported in [15] and independently in [28]. Several cases of BH(21, 3) were found by Brock and Murray as reported in [2] along with additional examples.…”

Section: Results and Case Studiesmentioning

confidence: 84%

“…To the best of our knowledge BH(14, 10) matrices were not known prior to this work, and Example 2.1 shows a new discovery. 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].…”

Section: Results and Case Studiesmentioning

confidence: 99%

“…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]. q = 14: Several nonexistence results are known.…”

Section: Results and Case Studiesmentioning

confidence: 99%

“…Existence follows from the existence of BH(n, 4) matrices. q = 17: The Fourier matrix F 17 was shown to be unique in [18] by computers. Examples of matrices corresponding to the cases marked by "E" in Table 2 can be obtained from either by viewing a matrix H ∈ BH(n, q) as a member of BH(n, r) with some r which is a multiple of q; or by considering the Kronecker product of two smaller matrices [19,Lemma 4.2].…”

Section: Results and Case Studiesmentioning

confidence: 99%

See 4 more Smart Citations

Orderly generation of Butson Hadamard matrices

Lampio

Östergård²,

Szöllősi

2019

Math. Comp.

View full text Add to dashboard Cite

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.

show abstract

Section: Results and Case Studiesmentioning

confidence: 83%

Section: Results and Case Studiesmentioning

confidence: 84%

Section: Results and Case Studiesmentioning

confidence: 99%

Section: Results and Case Studiesmentioning

confidence: 99%

Section: Results and Case Studiesmentioning

confidence: 99%

See 3 more Smart Citations

Orderly generation of Butson Hadamard matrices

Lampio

Östergård²,

Szöllősi

2019

Math. Comp.

View full text Add to dashboard Cite

show abstract

Construction and classification of group invariant Butson Hadamard matrices

Wong¹

View full text Add to dashboard Cite

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.

show abstract

Topics in cocyclic development of pairwise combinatorial designs

Egan¹

2015

Irish Math. Soc. Bull.

View full text Add to dashboard Cite

and submitted in July 2015. This thesis is a compilation of results dealing with cocyclic development of pairwise combinatorial designs.Motivated by a classification of the indexing and extension groups of the Paley Hadamard matrices due to de Launey and Stafford, we investigate cocyclic development of the so-called generalized Sylvester (or Drake) Hadamard matrices. We describe the automorphism groups and derive strict conditions on possible indexing groups, addressing research problems of de Launey and Flannery in doing so.The shift action, discovered by Horadam, is a certain action of any finite group on the set of its 2-cocycles with trivial coefficients, which preserves both cohomological equivalence and orthogonality. We answer questions posed by Horadam about the shift action, in particular regarding its fixed points. One of our main innovations is the concept of linear shift representation. We give an algorithm for calculating the matrix group representation of a shift action, which enables us to compute with the action in a natural setting. We prove detailed results on reducibility, and discuss the outcomes of some computational experiments, including searches for orthogonal cocycles.Using the algorithms developed for shift representations, and other methods, we classify up to equivalence all cocyclic BH(n, p)s where p is an odd prime (necessarily dividing n) and np ≤ 100. This was 2010 Mathematics Subject Classification. 05B20, 20B25, 20J06.

show abstract

Uniqueness of Butson Hadamard matrices of small degrees

Cited by 3 publications

References 3 publications

Orderly generation of Butson Hadamard matrices

Orderly generation of Butson Hadamard matrices

Construction and classification of group invariant Butson Hadamard matrices

Topics in cocyclic development of pairwise combinatorial designs

Contact Info

Product

Resources

About