The extension problem for Lee and Euclidean weights

Year 2017, Volume: 4 Issue: 2 (Special Issue: Noncommutative rings and their applications), 207 - 217, 10.01.2017

Philippe Langevin Jay A. Wood

https://doi.org/10.13069/jacodesmath.284970

Cited By: 2

Abstract

The extension problem is solved for the Lee and Euclidean weights over three families of rings of the form $\Z/N\Z$: $N=2^{\ell + 1}$, $N=3^{\ell + 1}$, or $N=p=2q+1$ with $p$ and $q$ prime. The extension problem is solved for the Euclidean PSK weight over $\Z/N\Z$ for all $N$.

Keywords

Extension problem, Lee weight, Euclidean weight, Egalitarian weight

References

• [1] A. Barra, Equivalence Theorems and the Local-Global Property, ProQuest LLC, PhD thesis University of Kentucky, Ann Arbor, MI, USA, 2012.
• [2] A. Barra, H. Gluesing–Luerssen, MacWilliams extension theorems and the local–global property for codes over Frobenius rings, J. Pure Appl. Algebra 219(4) (2015) 703–728.
• [3] S. Dyshko, P. Langevin, J. A. Wood, Deux analogues au déterminant de Maillet, C. R. Math. Acad. Sci. Paris 354(7) (2016) 649–652.
• [4] F. G. Frobenius, Gesammelte Abhandlungen, Springer–Verlag, Berlin, 1968.
• [5] M. Greferath, T. Honold, C. Mc Fadden, J. A.Wood, J. Zumbrägel, MacWilliams’ extension theorem for bi-invariant weights over finite principal ideal rings, J. Combin. Theory Ser. A 125 (2014) 177–193.
• [6] M. Greferath, C. Mc Fadden, J. Zumbrägel, Characteristics of invariant weights related to code equivalence over rings, Des. Codes Cryptogr. 66(1) (2013) 145–156.
• [7] M. Greferath, S. E. Schmidt, Finite–ring combinatorics and MacWilliams’ equivalence theorem, J. Combin. Theory Ser. A 92(1) (2000) 17–28.
• [8] W. Heise, T. Honold, Homogeneous and egalitarian weights on finite rings, Proceedings of the Seventh International Workshop on Algebraic and Combinatorial Coding Theory (ACCT-2000), Bansko, Bulgaria, 183–188, 2000.
• [9] T. Honold, Characterization of finite Frobenius rings, Arch. Math. 76(6) (2001) 406–415.
• [10] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, second ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990.
• [11] J. A. Wood, Extension theorems for linear codes over finite rings, Applied algebra, algebraic algorithms and error–correcting codes (Toulouse, 1997), Lecture Notes in Comput. Sci. 1255 (1997) 329–340.
• [12] J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121(3) (1999) 555–575.
• [13] J. A. Wood, Weight functions and the extension theorem for linear codes over finite rings, Finite fields: theory, applications, and algorithms (Waterloo, ON, 1997), Contemp. Math. 225 (1999) 231–243.
• [14] J. A.Wood, Factoring the semigroup determinant of a finite chain ring, Coding Theory, Cryptography and Related Areas (2000) 249–264.
• [15] J. A. Wood, The structure of linear codes of constant weight, Trans. Amer. Math. Soc. 354(3) (2002) 1007–1026.
• [16] 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.
• [17] 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.
• [18] J. A. Wood, Relative one-weight linear codes, Des. Codes Cryptogr. 72(2) (2014) 331–344.