2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order, and their related Hadamard matrices and codes

Authors:
Iliya Bouyukliev;Veerle Fack;Joost Winne
Affiliations:
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Veliko Tarnovo, Bulgaria 5000;Research Group on Combinatorial Algorithms and Algorithmic Graph Theory, Department of Applied Mathematics and Computer Science, Ghent University, Ghent, Belgium 9000;Research Group on Combinatorial Algorithms and Algorithmic Graph Theory, Department of Applied Mathematics and Computer Science, Ghent University, Ghent, Belgium 9000
Venue:
Designs, Codes and Cryptography
Year:
2009

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Abstract

We present the full classification of Hadamard 2-(31,15,7), Hadamard 2-(35, 17,8) and Menon 2-(36,15,6) designs with automorphisms of odd prime order. We also give partial classifications of such designs with automorphisms of order 2. These classifications lead to related Hadamard matrices and self-dual codes. We found 76166 Hadamard matrices of order 32 and 38332 Hadamard matrices of order 36, arising from the classified designs. Remarkably, all constructed Hadamard matrices of order 36 are Hadamard equivalent to a regular Hadamard matrix. From our constructed designs, we obtained 37352 doubly-even [72,36,12] codes, which are the best known self-dual codes of this length until now.