American Institute of Mathematical Sciences

Advanced
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

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

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.  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

[1]

Steven T. Dougherty, Cristina Fernández-Córdoba, Roger Ten-Valls, Bahattin Yildiz. Quaternary group ring codes: Ranks, kernels and self-dual codes. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020023
[2]

Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011
[3]

Steven T. Dougherty, Joe Gildea, Abidin Kaya, Bahattin Yildiz. New self-dual and formally self-dual codes from group ring constructions. Advances in Mathematics of Communications, 2020, 14 (1) : 11-22. doi: 10.3934/amc.2020002
[4]

Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031
[5]

Steven T. Dougherty, Joe Gildea, Adrian Korban, Abidin Kaya. Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020037
[6]

Amita Sahni, Poonam Trama Sehgal. Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (4) : 437-447. doi: 10.3934/amc.2015.9.437
[7]

Suat Karadeniz, Bahattin Yildiz. Double-circulant and bordered-double-circulant constructions for self-dual codes over $R_2$. Advances in Mathematics of Communications, 2012, 6 (2) : 193-202. doi: 10.3934/amc.2012.6.193
[8]

Masaaki Harada, Akihiro Munemasa. Classification of self-dual codes of length 36. Advances in Mathematics of Communications, 2012, 6 (2) : 229-235. doi: 10.3934/amc.2012.6.229
[9]

Stefka Bouyuklieva, Anton Malevich, Wolfgang Willems. On the performance of binary extremal self-dual codes. Advances in Mathematics of Communications, 2011, 5 (2) : 267-274. doi: 10.3934/amc.2011.5.267
[10]

Nikolay Yankov, Damyan Anev, Müberra Gürel. Self-dual codes with an automorphism of order 13. Advances in Mathematics of Communications, 2017, 11 (3) : 635-645. doi: 10.3934/amc.2017047
[11]

T. Aaron Gulliver, Masaaki Harada, Hiroki Miyabayashi. Double circulant and quasi-twisted self-dual codes over $\mathbb F_5$ and $\mathbb F_7$. Advances in Mathematics of Communications, 2007, 1 (2) : 223-238. doi: 10.3934/amc.2007.1.223
[12]

Crnković Dean, Vedrana Mikulić Crnković, Bernardo G. Rodrigues. On self-orthogonal designs and codes related to Held's simple group. Advances in Mathematics of Communications, 2018, 12 (3) : 607-628. doi: 10.3934/amc.2018036
[13]

Masaaki Harada, Akihiro Munemasa. On the covering radii of extremal doubly even self-dual codes. Advances in Mathematics of Communications, 2007, 1 (2) : 251-256. doi: 10.3934/amc.2007.1.251
[14]

Stefka Bouyuklieva, Iliya Bouyukliev. Classification of the extremal formally self-dual even codes of length 30. Advances in Mathematics of Communications, 2010, 4 (3) : 433-439. doi: 10.3934/amc.2010.4.433
[15]

Hyun Jin Kim, Heisook Lee, June Bok Lee, Yoonjin Lee. Construction of self-dual codes with an automorphism of order $p$. Advances in Mathematics of Communications, 2011, 5 (1) : 23-36. doi: 10.3934/amc.2011.5.23
[16]

Bram van Asch, Frans Martens. Lee weight enumerators of self-dual codes and theta functions. Advances in Mathematics of Communications, 2008, 2 (4) : 393-402. doi: 10.3934/amc.2008.2.393
[17]

Bram van Asch, Frans Martens. A note on the minimum Lee distance of certain self-dual modular codes. Advances in Mathematics of Communications, 2012, 6 (1) : 65-68. doi: 10.3934/amc.2012.6.65
[18]

Masaaki Harada, Katsushi Waki. New extremal formally self-dual even codes of length 30. Advances in Mathematics of Communications, 2009, 3 (4) : 311-316. doi: 10.3934/amc.2009.3.311
[19]

Suat Karadeniz, Bahattin Yildiz. New extremal binary self-dual codes of length $68$ from $R_2$-lifts of binary self-dual codes. Advances in Mathematics of Communications, 2013, 7 (2) : 219-229. doi: 10.3934/amc.2013.7.219
[20]

Steven T. Dougherty, Cristina Fernández-Córdoba. Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes. Advances in Mathematics of Communications, 2011, 5 (4) : 571-588. doi: 10.3934/amc.2011.5.571

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences