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# MDS and near-MDS self-dual codes over large prime fields

• In this paper, we are interested in the construction of maximum distance separable (MDS) self-dual codes over large prime fields that arise from the solutions of systems of diophantine equations. Using this method we con- struct many self-dualMDS (or near-MDS) codes of lengths up to 16 over various prime fields $GF(p)$, where $p$ = 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193 and 197. In addition, a number of optimal codes are presented for many lengths up to 40 over small prime fields $GF(p)$. Furthermore, our results on the minimum weight of self-dual codes over prime fields give a better bound than the Pless-Pierce bound obtained from a modified Gilbert-Varshamov bound.
Mathematics Subject Classification: Primary: 94B05, 94B25; Secondary: 05B20.

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