Properties of dual codes defined by nondegenerate forms

Öz

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.

Anahtar Kelimeler

• Frobenius ring,Sesquilinear form,Bilinear form,Dual code,Generating character,MacWilliams identities

Kaynakça

1. [1] H. L. Claasen, R. W. Goldbach, A field–like property of finite rings, Indag. Math. (N.S.) 3(1) (1992) 11–26.
2. [2] P. Delsarte, Bounds for unrestricted codes, by linear programming, Philips Res. Rep. 27 (1972) 272–289.
3. [3] M. Hall, A type of algebraic closure, Ann. of Math. 40(2) (1939) 360–369.
4. [4] T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, vol. 189, Springer–Verlag, New York, 1999.
5. [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. [6] J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121(3) (1999) 555–575.
7. [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. [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. [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.