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.
翻译:2013年,Nebe 和 Villar 给出了一系列自成一体的永久性代码,其长度为2美元(p+1美元),与美元等值为5美元,与美元等值为8美元。结果,发现了第三种自成一体的自成一体的自成一体代码,其长度为60美元。我们显示,自成一体的自成一体代码含有代号,它构成Hadamard 顺序矩阵2美元(p+1美元),而美元与5美元等值为24美元。此外,还表明,自成一体的自成一体代码是由Hadamard矩阵的行生成的。我们还表明,第三种自成一体的自成一体代码60美元,其中至少含有两个等值的Hadamard矩阵。