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Classification of skew‐Hadamard matrices of order 32 and association schemes of order 31
Author(s) -
Hanaki Akihide,
Kharaghani Hadi,
Mohammadian Ali,
TayfehRezaie Behruz
Publication year - 2020
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21706
Subject(s) - mathematics , hadamard transform , skew , combinatorics , hadamard's maximal determinant problem , order (exchange) , complex hadamard matrix , permutation (music) , association scheme , equivalence (formal languages) , hadamard matrix , hadamard's inequality , backtracking , matrix (chemical analysis) , hadamard product , discrete mathematics , algorithm , computer science , mathematical analysis , telecommunications , finance , economics , physics , materials science , acoustics , composite material
Using a backtracking algorithm along with an essential change to the rows of representatives of known 13 710 027 equivalence classes of Hadamard matrices of order 32, we make an exhaustive computer search feasible and show that there are exactly 6662 inequivalent skew‐Hadamard matrices of order 32. Two skew‐Hadamard matrices are considered SH ‐equivalent if they are similar by a signed permutation matrix. We determine that there are precisely 7227 skew‐Hadamard matrices of order 32 up to SH ‐equivalence. This partly settles a problem posed by Kim and Solé. As a consequence, we provide the classification of association schemes of order 31.