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Abstract
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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