An enumeration of the equivalence classes of self-dual matrix codes

Katherine  Morrison

doi:10.3934/amc.2015.9.415

Article Contents

2015, Volume 9, Issue 4: 415-436. Doi: 10.3934/amc.2015.9.415

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An enumeration of the equivalence classes of self-dual matrix codes

Katherine Morrison^1,

1.
School of Mathematical Sciences, University of Northern Colorado, 501 20th St, CB 122, Greeley, CO 80639

Received: October 2013

Revised: May 2015

Published: November 2015

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Abstract

As a result of their applications in network coding, space-time coding, and coding for criss-cross errors, matrix codes have garnered significant attention; in various contexts, these codes have also been termed rank-metric codes, space-time codes over finite fields, and array codes. We focus on characterizing matrix codes that are both efficient (have high rate) and effective at error correction (have high minimum rank-distance). It is well known that the inherent trade-off between dimension and minimum distance for a matrix code is reversed for its dual code; specifically, if a matrix code has high dimension and low minimum distance, then its dual code will have low dimension and high minimum distance. With an aim towards finding codes with a perfectly balanced trade-off, we study self-dual matrix codes. In this work, we develop a framework based on double cosets of the matrix-equivalence maps to provide a complete classification of the equivalence classes of self-dual matrix codes, and we employ this method to enumerate the equivalence classes of these codes for small parameters.

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Mathematics Subject Classification: Primary: 94B60, 11T71.

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