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

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November 2015, 9(4): 415-436. doi: 10.3934/amc.2015.9.415

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

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

Received October 2013 Revised May 2015 Published November 2015

Abstract
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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.

Keywords: self-dual codes, isometries, equivalence maps, Matrix codes, double cosets., mass formula, orthogonal group.

Mathematics Subject Classification: Primary: 94B60, 11T7.

Citation: Katherine Morrison. An enumeration of the equivalence classes of self-dual matrix codes. Advances in Mathematics of Communications, 2015, 9 (4) : 415-436. doi: 10.3934/amc.2015.9.415

References:

[1]	A. Barra and H. Gluesing-Luerssen, MacWilliams extension theorems and local-global property for codes over Frobenius rings,, J. Pure Appl. Algebra, 219* (2015), 703. doi: 10.1016/j.jpaa.2014.04.026. Google Scholar*
[2]	M. Blaum, P. G. Farrell and H. C. A. van Tilborg, Array codes,, In V. Pless and W. C. Huffman, (1998), 1855. Google Scholar
[3]	P. Delsarte, Bilinear forms over a finite field with applications to coding theory,, Journal of Combinatorial Theory, 25 (1978), 226. doi: 10.1016/0097-3165(78)90015-8. Google Scholar
[4]	D. Dummit and R. Foote, Abstract Algebra,, Wiley, (2004). Google Scholar
[5]	D. Grant and M. Varanasi, Duality theory for space-time codes over finite fields,, Advances in Mathematics of Communications, 2 (2008), 35. doi: 10.3934/amc.2008.2.35. Google Scholar
[6]	D. Grant and M. Varanasi, The equivalence of space-time codes and codes defined over finite fields and Galois rings,, Advances in Mathematics of Communications, 2 (2008), 131. doi: 10.3934/amc.2008.2.131. Google Scholar
[7]	L. C. Grove, Classical Groups and Geometrical Algebra,, Graduate Studies in Mathematics, (2002). Google Scholar
[8]	G. Janusz, Parametrization of self-dual codes by orthogonal matrices,, Finite Fields and Their Applications, 13 (2007), 450. doi: 10.1016/j.ffa.2006.05.001. Google Scholar
[9]	R. Kötter and F. R. Kschischang, Coding for errors and erasures in random network coding,, IEEE Transactions on Information Theory, 54 (2008), 3597. doi: 10.1109/TIT.2008.926449. Google Scholar
[10]	F. J. MacWilliams, Combinatorial Problems of Elementary Group Theory,, PhD thesis, (1962). Google Scholar
[11]	F. J. MacWilliams, Orthogonal matrices over finite fields,, The American Mathematical Monthly, 76 (1969), 152. doi: 10.2307/2317262. Google Scholar
[12]	C. L. Mallows, V. S. Pless and N. J. A. Sloane, Self-dual codes over GF(3),, SIAM Journal of Applied Mathematics, 31 (1976), 649. doi: 10.1137/0131058. Google Scholar
[13]	M. Marcus and N. Moyls, Linear transformations on algebras of matrices,, Canad. J. Math, 11 (1959), 61. doi: 10.4153/CJM-1959-008-0. Google Scholar
[14]	K. Morrison, Equivalence and Duality for Rank-Metric and Matrix Codes,, PhD thesis, (2012). Google Scholar
[15]	K. Morrison, Equivalence for rank-metric and matrix codes and automorphism groups for Gabidulin codes,, IEEE Transactions on Information Theory, 60 (2014), 7035. doi: 10.1109/TIT.2014.2359198. Google Scholar
[16]	V. S. Pless, On the uniqueness of the Golay codes,, Journal of Combinatorial Theory, 5 (1968), 215. doi: 10.1016/S0021-9800(68)80067-5. Google Scholar
[17]	V. S. Pless, Self-dual codes - Theme and variations,, In S. Boztas and I. Shparlinski, (2001), 13. doi: 10.1007/3-540-45624-4_2. Google Scholar
[18]	E. M. Rains and N. J. A. Sloane, Self-dual codes,, In V. S. Pless and W. C. Huffman, (1998), 177. Google Scholar
[19]	R. M. Roth, Maximum-rank array codes and their application to criss-cross error correction,, IEEE Transactions on Information Theory, (1991), 328. doi: 10.1109/18.75248. Google Scholar
[20]	D. Silva, F. R. Kschischang and R. Kötter, A rank-metric approach to error control in random network coding,, IEEE Transactions on Information Theory, 54 (2008), 3951. doi: 10.1109/TIT.2008.928291. Google Scholar
[21]	D. Taylor, The Geometry of the Classical Groups,, Helderman, (1992). Google Scholar
[22]	C. Vinroot, A note on orthogonal similitudes groups,, Linear and Multilinear Algebra, 54 (2006), 391. doi: 10.1080/03081080500209588. Google Scholar

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References:

[1]	A. Barra and H. Gluesing-Luerssen, MacWilliams extension theorems and local-global property for codes over Frobenius rings,, J. Pure Appl. Algebra, 219* (2015), 703. doi: 10.1016/j.jpaa.2014.04.026. Google Scholar*
[2]	M. Blaum, P. G. Farrell and H. C. A. van Tilborg, Array codes,, In V. Pless and W. C. Huffman, (1998), 1855. Google Scholar
[3]	P. Delsarte, Bilinear forms over a finite field with applications to coding theory,, Journal of Combinatorial Theory, 25 (1978), 226. doi: 10.1016/0097-3165(78)90015-8. Google Scholar
[4]	D. Dummit and R. Foote, Abstract Algebra,, Wiley, (2004). Google Scholar
[5]	D. Grant and M. Varanasi, Duality theory for space-time codes over finite fields,, Advances in Mathematics of Communications, 2 (2008), 35. doi: 10.3934/amc.2008.2.35. Google Scholar
[6]	D. Grant and M. Varanasi, The equivalence of space-time codes and codes defined over finite fields and Galois rings,, Advances in Mathematics of Communications, 2 (2008), 131. doi: 10.3934/amc.2008.2.131. Google Scholar
[7]	L. C. Grove, Classical Groups and Geometrical Algebra,, Graduate Studies in Mathematics, (2002). Google Scholar
[8]	G. Janusz, Parametrization of self-dual codes by orthogonal matrices,, Finite Fields and Their Applications, 13 (2007), 450. doi: 10.1016/j.ffa.2006.05.001. Google Scholar
[9]	R. Kötter and F. R. Kschischang, Coding for errors and erasures in random network coding,, IEEE Transactions on Information Theory, 54 (2008), 3597. doi: 10.1109/TIT.2008.926449. Google Scholar
[10]	F. J. MacWilliams, Combinatorial Problems of Elementary Group Theory,, PhD thesis, (1962). Google Scholar
[11]	F. J. MacWilliams, Orthogonal matrices over finite fields,, The American Mathematical Monthly, 76 (1969), 152. doi: 10.2307/2317262. Google Scholar
[12]	C. L. Mallows, V. S. Pless and N. J. A. Sloane, Self-dual codes over GF(3),, SIAM Journal of Applied Mathematics, 31 (1976), 649. doi: 10.1137/0131058. Google Scholar
[13]	M. Marcus and N. Moyls, Linear transformations on algebras of matrices,, Canad. J. Math, 11 (1959), 61. doi: 10.4153/CJM-1959-008-0. Google Scholar
[14]	K. Morrison, Equivalence and Duality for Rank-Metric and Matrix Codes,, PhD thesis, (2012). Google Scholar
[15]	K. Morrison, Equivalence for rank-metric and matrix codes and automorphism groups for Gabidulin codes,, IEEE Transactions on Information Theory, 60 (2014), 7035. doi: 10.1109/TIT.2014.2359198. Google Scholar
[16]	V. S. Pless, On the uniqueness of the Golay codes,, Journal of Combinatorial Theory, 5 (1968), 215. doi: 10.1016/S0021-9800(68)80067-5. Google Scholar
[17]	V. S. Pless, Self-dual codes - Theme and variations,, In S. Boztas and I. Shparlinski, (2001), 13. doi: 10.1007/3-540-45624-4_2. Google Scholar
[18]	E. M. Rains and N. J. A. Sloane, Self-dual codes,, In V. S. Pless and W. C. Huffman, (1998), 177. Google Scholar
[19]	R. M. Roth, Maximum-rank array codes and their application to criss-cross error correction,, IEEE Transactions on Information Theory, (1991), 328. doi: 10.1109/18.75248. Google Scholar
[20]	D. Silva, F. R. Kschischang and R. Kötter, A rank-metric approach to error control in random network coding,, IEEE Transactions on Information Theory, 54 (2008), 3951. doi: 10.1109/TIT.2008.928291. Google Scholar
[21]	D. Taylor, The Geometry of the Classical Groups,, Helderman, (1992). Google Scholar
[22]	C. Vinroot, A note on orthogonal similitudes groups,, Linear and Multilinear Algebra, 54 (2006), 391. doi: 10.1080/03081080500209588. Google Scholar

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