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.
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Acknowledgements
This work was supported by JSPS KAKENHI Grant Nos. 19H01802, 20K03719 and 21K03350.
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Araya, M., Harada, M. & Momihara, K. Hadamard matrices related to a certain series of ternary self-dual codes. Des. Codes Cryptogr. 91, 795–805 (2023). https://doi.org/10.1007/s10623-022-01127-y
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DOI: https://doi.org/10.1007/s10623-022-01127-y