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# Construction of new self-dual codes over $GF(5)$ using skew-Hadamard matrices

• In this paper, we give optimal self-dual codes over $GF(5)$ for lengths $24$, $40$, $48$ and $56$. In particular, new inequivalent $[48, 24]$ and $[56, 28]$ self-dual codes over $GF(5)$ whose minimum weights are $14$ and $16$, are constructed using skew-Hadamard matrices of order $24$ and $28$, thus improving the only known quadratic double circulant self-dual codes of length $48$ and $56$. Moreover, $[80, 40]$ and $[88, 44]$ self-dual codes whose minimum weights are $17$ and $19$ over $GF(5)$, are constructed for the first time. These codes are derived from skew-Hadamard matrices of order $40$ and $44$, respectively. Finally, a new $[56, 28, 17]$ self-dual code is constructed over $GF(7)$ having the highest minimum weight among $[56, 28]$ self-dual codes. This new optimal code is constructed from a skew-Hadamard-matrix of order $28$, for the first time.
Mathematics Subject Classification: Primary: 94B05, 05B20; Secondary: 94B25.

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