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

Abstract

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 these ternary self-dual codes contain codewords which form a Hadamard matrix of order $$2(p+1)$$ when p is congruent to 5 modulo 24. In addition, we show that the ternary self-dual codes found by Nebe and Villar are generated by the rows of the Hadamard matrices. We also demonstrate that the third ternary extremal self-dual code of length 60 contains at least two inequivalent Hadamard matrices.