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

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.