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November  2009, 3(4): 349-361. doi: 10.3934/amc.2009.3.349

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

Ilias S. Kotsireas 1, , Christos Koukouvinos 2, and  Dimitris E. Simos 2,

1. 

Department of Physics and Computer Science, Wilfrid Laurier University, University Avenue West, Waterloo, Ontario N2L 3C5, Canada
2. 

Department of Mathematics, National Technical University of Athens, Zografou 15773, Athens, Greece

Received  May 2009 Revised  October 2009 Published  November 2009

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.
Keywords: construction., MDS codes, circulant matrices, Self-dual codes, NMDS codes, diophantine equations.
Mathematics Subject Classification: Primary: 94B05, 94B25; Secondary: 05B2.
Citation: Ilias S. Kotsireas, Christos Koukouvinos, Dimitris E. Simos. MDS and near-MDS self-dual codes over large prime fields. Advances in Mathematics of Communications, 2009, 3 (4) : 349-361. doi: 10.3934/amc.2009.3.349
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