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

Katherine Morrison ^1,

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), 703-728. doi: 10.1016/j.jpaa.2014.04.026.
[2]	M. Blaum, P. G. Farrell and H. C. A. van Tilborg, Array codes, In V. Pless and W. C. Huffman, editors, Handbook of Coding Theory, volume 2, pages 1855-1910. Elselvier, 1998.
[3]	P. Delsarte, Bilinear forms over a finite field with applications to coding theory, Journal of Combinatorial Theory, A, 25 (1978), 226-241. doi: 10.1016/0097-3165(78)90015-8.
[4]	D. Dummit and R. Foote, Abstract Algebra, Wiley, 2004.
[5]	D. Grant and M. Varanasi, Duality theory for space-time codes over finite fields, Advances in Mathematics of Communications, 2 (2008), 35-54. doi: 10.3934/amc.2008.2.35.
[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), 131-145. doi: 10.3934/amc.2008.2.131.
[7]	L. C. Grove, Classical Groups and Geometrical Algebra, Graduate Studies in Mathematics, 39. American Mathematical Society, Providence, RI, 2002.
[8]	G. Janusz, Parametrization of self-dual codes by orthogonal matrices, Finite Fields and Their Applications, 13 (2007), 450-491. doi: 10.1016/j.ffa.2006.05.001.
[9]	R. Kötter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Transactions on Information Theory, 54 (2008), 3597-3591. doi: 10.1109/TIT.2008.926449.
[10]	F. J. MacWilliams, Combinatorial Problems of Elementary Group Theory, PhD thesis, Harvard University, Cambridge, Mass, 1962.
[11]	F. J. MacWilliams, Orthogonal matrices over finite fields, The American Mathematical Monthly, 76 (1969), 152-164. doi: 10.2307/2317262.
[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), 649-666. doi: 10.1137/0131058.
[13]	M. Marcus and N. Moyls, Linear transformations on algebras of matrices, Canad. J. Math, 11 (1959), 61-66. doi: 10.4153/CJM-1959-008-0.
[14]	K. Morrison, Equivalence and Duality for Rank-Metric and Matrix Codes, PhD thesis, University of Nebraska-Lincoln, Lincoln, NE, 2012.
[15]	K. Morrison, Equivalence for rank-metric and matrix codes and automorphism groups for Gabidulin codes, IEEE Transactions on Information Theory, 60 (2014), 7035-7046. doi: 10.1109/TIT.2014.2359198.
[16]	V. S. Pless, On the uniqueness of the Golay codes, Journal of Combinatorial Theory, 5 (1968), 215-228. doi: 10.1016/S0021-9800(68)80067-5.
[17]	V. S. Pless, Self-dual codes - Theme and variations, In S. Boztas and I. Shparlinski, editors, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science, pages 13-21. Springer, 2001. doi: 10.1007/3-540-45624-4_2.
[18]	E. M. Rains and N. J. A. Sloane, Self-dual codes, In V. S. Pless and W. C. Huffman, editors, Handbook of Coding Theory, pages 177-294. Elselvier, 1998.
[19]	R. M. Roth, Maximum-rank array codes and their application to criss-cross error correction, IEEE Transactions on Information Theory, 37(2):328-336, Mar 1991. doi: 10.1109/18.75248.
[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), 3951-3967. doi: 10.1109/TIT.2008.928291.
[21]	D. Taylor, The Geometry of the Classical Groups, Helderman, 1992.
[22]	C. Vinroot, A note on orthogonal similitudes groups, Linear and Multilinear Algebra, 54 (2006), 391-396. doi: 10.1080/03081080500209588.

show all references

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), 703-728. doi: 10.1016/j.jpaa.2014.04.026.
[2]	M. Blaum, P. G. Farrell and H. C. A. van Tilborg, Array codes, In V. Pless and W. C. Huffman, editors, Handbook of Coding Theory, volume 2, pages 1855-1910. Elselvier, 1998.
[3]	P. Delsarte, Bilinear forms over a finite field with applications to coding theory, Journal of Combinatorial Theory, A, 25 (1978), 226-241. doi: 10.1016/0097-3165(78)90015-8.
[4]	D. Dummit and R. Foote, Abstract Algebra, Wiley, 2004.
[5]	D. Grant and M. Varanasi, Duality theory for space-time codes over finite fields, Advances in Mathematics of Communications, 2 (2008), 35-54. doi: 10.3934/amc.2008.2.35.
[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), 131-145. doi: 10.3934/amc.2008.2.131.
[7]	L. C. Grove, Classical Groups and Geometrical Algebra, Graduate Studies in Mathematics, 39. American Mathematical Society, Providence, RI, 2002.
[8]	G. Janusz, Parametrization of self-dual codes by orthogonal matrices, Finite Fields and Their Applications, 13 (2007), 450-491. doi: 10.1016/j.ffa.2006.05.001.
[9]	R. Kötter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Transactions on Information Theory, 54 (2008), 3597-3591. doi: 10.1109/TIT.2008.926449.
[10]	F. J. MacWilliams, Combinatorial Problems of Elementary Group Theory, PhD thesis, Harvard University, Cambridge, Mass, 1962.
[11]	F. J. MacWilliams, Orthogonal matrices over finite fields, The American Mathematical Monthly, 76 (1969), 152-164. doi: 10.2307/2317262.
[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), 649-666. doi: 10.1137/0131058.
[13]	M. Marcus and N. Moyls, Linear transformations on algebras of matrices, Canad. J. Math, 11 (1959), 61-66. doi: 10.4153/CJM-1959-008-0.
[14]	K. Morrison, Equivalence and Duality for Rank-Metric and Matrix Codes, PhD thesis, University of Nebraska-Lincoln, Lincoln, NE, 2012.
[15]	K. Morrison, Equivalence for rank-metric and matrix codes and automorphism groups for Gabidulin codes, IEEE Transactions on Information Theory, 60 (2014), 7035-7046. doi: 10.1109/TIT.2014.2359198.
[16]	V. S. Pless, On the uniqueness of the Golay codes, Journal of Combinatorial Theory, 5 (1968), 215-228. doi: 10.1016/S0021-9800(68)80067-5.
[17]	V. S. Pless, Self-dual codes - Theme and variations, In S. Boztas and I. Shparlinski, editors, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science, pages 13-21. Springer, 2001. doi: 10.1007/3-540-45624-4_2.
[18]	E. M. Rains and N. J. A. Sloane, Self-dual codes, In V. S. Pless and W. C. Huffman, editors, Handbook of Coding Theory, pages 177-294. Elselvier, 1998.
[19]	R. M. Roth, Maximum-rank array codes and their application to criss-cross error correction, IEEE Transactions on Information Theory, 37(2):328-336, Mar 1991. doi: 10.1109/18.75248.
[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), 3951-3967. doi: 10.1109/TIT.2008.928291.
[21]	D. Taylor, The Geometry of the Classical Groups, Helderman, 1992.
[22]	C. Vinroot, A note on orthogonal similitudes groups, Linear and Multilinear Algebra, 54 (2006), 391-396. doi: 10.1080/03081080500209588.

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