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Properties of dual codes defined by nondegenerate forms

Year 2017, Volume: 4 Issue: 2 (Special Issue: Noncommutative rings and their applications), 105 - 113, 10.01.2017
https://doi.org/10.13069/jacodesmath.284934

Abstract

Dual codes are defined with respect to non-degenerate sesquilinear or bilinear forms over a finite
Frobenius ring. These dual codes have the properties one expects from a dual code: they satisfy
a double-dual property, they have cardinality complementary to that of the primal code, and they
satisfy the MacWilliams identities for the Hamming weight.

References

  • [1] H. L. Claasen, R. W. Goldbach, A field–like property of finite rings, Indag. Math. (N.S.) 3(1) (1992) 11–26.
  • [2] P. Delsarte, Bounds for unrestricted codes, by linear programming, Philips Res. Rep. 27 (1972) 272–289.
  • [3] M. Hall, A type of algebraic closure, Ann. of Math. 40(2) (1939) 360–369.
  • [4] T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, vol. 189, Springer–Verlag, New York, 1999.
  • [5] G. Nebe, E. M. Rains, N. J. A. Sloane, Self–Dual Codes and Invariant Theory, Algorithms and Computation in Mathematics, vol. 17, Springer–Verlag, Berlin, 2006.
  • [6] J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121(3) (1999) 555–575.
  • [7] J. A. Wood, Foundations of linear codes defined over finite modules: the extension theorem and the MacWilliams identities. Codes over rings, 124–190, Ser. Coding Theory Cryptol., 6, World Sci. Publ., Hackensack, NJ, 2009.
  • [8] J. A. Wood, Anti–isomorphisms, character modules and self–dual codes over non-commutative rings, Int. J. Inf. Coding Theory 1(4) (2010) 429–444.
  • [9] J. A. Wood, Applications of finite Frobenius rings to the foundations of algebraic coding theory. Proceedings of the 44th Symposium on Ring Theory and Representation Theory, 223–245, Symp. Ring Theory Represent. Theory Organ. Comm., Nagoya, 2012.
Year 2017, Volume: 4 Issue: 2 (Special Issue: Noncommutative rings and their applications), 105 - 113, 10.01.2017
https://doi.org/10.13069/jacodesmath.284934

Abstract

References

  • [1] H. L. Claasen, R. W. Goldbach, A field–like property of finite rings, Indag. Math. (N.S.) 3(1) (1992) 11–26.
  • [2] P. Delsarte, Bounds for unrestricted codes, by linear programming, Philips Res. Rep. 27 (1972) 272–289.
  • [3] M. Hall, A type of algebraic closure, Ann. of Math. 40(2) (1939) 360–369.
  • [4] T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, vol. 189, Springer–Verlag, New York, 1999.
  • [5] G. Nebe, E. M. Rains, N. J. A. Sloane, Self–Dual Codes and Invariant Theory, Algorithms and Computation in Mathematics, vol. 17, Springer–Verlag, Berlin, 2006.
  • [6] J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121(3) (1999) 555–575.
  • [7] J. A. Wood, Foundations of linear codes defined over finite modules: the extension theorem and the MacWilliams identities. Codes over rings, 124–190, Ser. Coding Theory Cryptol., 6, World Sci. Publ., Hackensack, NJ, 2009.
  • [8] J. A. Wood, Anti–isomorphisms, character modules and self–dual codes over non-commutative rings, Int. J. Inf. Coding Theory 1(4) (2010) 429–444.
  • [9] J. A. Wood, Applications of finite Frobenius rings to the foundations of algebraic coding theory. Proceedings of the 44th Symposium on Ring Theory and Representation Theory, 223–245, Symp. Ring Theory Represent. Theory Organ. Comm., Nagoya, 2012.
There are 9 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Steve Szabo This is me

Jay A. Wood This is me

Publication Date January 10, 2017
Published in Issue Year 2017 Volume: 4 Issue: 2 (Special Issue: Noncommutative rings and their applications)

Cite

APA Szabo, S., & Wood, J. A. (2017). Properties of dual codes defined by nondegenerate forms. Journal of Algebra Combinatorics Discrete Structures and Applications, 4(2 (Special Issue: Noncommutative rings and their applications), 105-113. https://doi.org/10.13069/jacodesmath.284934
AMA Szabo S, Wood JA. Properties of dual codes defined by nondegenerate forms. Journal of Algebra Combinatorics Discrete Structures and Applications. May 2017;4(2 (Special Issue: Noncommutative rings and their applications):105-113. doi:10.13069/jacodesmath.284934
Chicago Szabo, Steve, and Jay A. Wood. “Properties of Dual Codes Defined by Nondegenerate Forms”. Journal of Algebra Combinatorics Discrete Structures and Applications 4, no. 2 (Special Issue: Noncommutative rings and their applications) (May 2017): 105-13. https://doi.org/10.13069/jacodesmath.284934.
EndNote Szabo S, Wood JA (May 1, 2017) Properties of dual codes defined by nondegenerate forms. Journal of Algebra Combinatorics Discrete Structures and Applications 4 2 (Special Issue: Noncommutative rings and their applications) 105–113.
IEEE S. Szabo and J. A. Wood, “Properties of dual codes defined by nondegenerate forms”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 2 (Special Issue: Noncommutative rings and their applications), pp. 105–113, 2017, doi: 10.13069/jacodesmath.284934.
ISNAD Szabo, Steve - Wood, Jay A. “Properties of Dual Codes Defined by Nondegenerate Forms”. Journal of Algebra Combinatorics Discrete Structures and Applications 4/2 (Special Issue: Noncommutative rings and their applications) (May 2017), 105-113. https://doi.org/10.13069/jacodesmath.284934.
JAMA Szabo S, Wood JA. Properties of dual codes defined by nondegenerate forms. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4:105–113.
MLA Szabo, Steve and Jay A. Wood. “Properties of Dual Codes Defined by Nondegenerate Forms”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 2 (Special Issue: Noncommutative rings and their applications), 2017, pp. 105-13, doi:10.13069/jacodesmath.284934.
Vancouver Szabo S, Wood JA. Properties of dual codes defined by nondegenerate forms. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4(2 (Special Issue: Noncommutative rings and their applications):105-13.