GENERALIZATION OF THE LEE WEIGHT TO Ζpk

Yıl 2012, Cilt: 2 Sayı: 2, 145 - 153, 01.12.2012

B. Yıldız Z. Ödemiş Özger

Öz

We introduce a new extension of the Lee weight to Zpk and later to Galois rings GR p k , m . The weight we define is a non-homogeneous weight and is different than the one that is generally labeled as “generalized Lee weight”. Unlike the case of generalized Lee weight, we define a distance-preserving Gray map from Zpk , extended Lee distance to F p k−1 p , Hamming distance , thus making our extension practical for coding theory purposes.

Anahtar Kelimeler

Lee weight, extended, Gray map

Kaynakça

• Bhaintwal, M. and Wasan, S. K., (2009), On quasi-cyclic codes overZq, Appl. Algebra Engrg. Comm. Comput., 20, 459-480.
• Blake, I. F., (1972), Codes over Certian Rings, Inf. Control., 20, 396-404.
• Carlet, C., (1998),Z2k-linear codes, IEEE Trans. Inform. Theory, 44, 1543-1547.
• Constantinescu, I. and Heise, T., (1997), A metric for codes over residue class rings of integers, Problemy Peredachi Informatsii, 33, 22-28.
• Greferath, M. and Schmidt, S. E., (1999), Gray Isometries for Şnite chain rings and a nonlinear ternary (36, 312, 15) code, IEEE Trans. Inform. Theory, 45, 2522-2524.
• Hammons, A. R., Kumar, V., Calderbank, A. R., Sloane, N. J. A. and Sol´e, P., (1994), TheZ4-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40, 301-319.
• Huffman, W. C., (1998), Decompositions and extremal Type II codes overZ4, IEEE Trans. Inform. Theory, 44, 800-809.
• Kumar, P. V., Helleseth, T. and Calderbank, A. R., (1995),An upperbound for Weil exponential sums over Galois rings and applications, IEEE Trans. Inform. Theory, 41, 456-468.
• Ling, S. and Blackford, J. T., (2002),Zpk-linear codes, IEEE Trans. Inform. Theory, 48, 2592-2605.
• Ling, S. and Ozbudak, F., (2004), An improvement on the bounds of Weil exponential sums over Galois rings with some applications, IEEE Trans. Inform. Theory, 50, 2529-2539.
• Voloch, J. F. and Walker, J. L., (2003), Homogeneous weights and exponential sums, Finite Fields Appl., 310-321.
• Wasan, S., (1982), On Codes overZm, IEEE Trans. Inform. Theory 28, 117-120.
• Yıldız, B., (2009), A Combinatorial construction of the Gray map over Galois rings, Discrete Mathe- matics, 309(10), 3408-3412.
• Yıldız, B., (2007), Weights modulo peof linear codes over rings, Designs, Codes and Cryptography, 43, 147-165.