Hadamard matrices related to a certain series of ternary self-dual codes

05/31/2022

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by   Makoto Araya, et al.

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In 2013, Nebe and Villar gave a series of ternary self-dual codes of length 2(p+1) for a prime p congruent to 5 modulo 8. As a consequence, the third ternary extremal self-dual code of length 60 was found. We show that the ternary self-dual code contains codewords which form a Hadamard matrix of order 2(p+1) when p is congruent to 5 modulo 24. In addition, it is shown that the ternary self-dual code is generated by the rows of the Hadamard matrix. We also demonstrate that the third ternary extremal self-dual code of length 60 contains at least two inequivalent Hadamard matrices.