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Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields
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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 |
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] | |
[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] | |
[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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