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GENERALIZATION OF THE LEE WEIGHT TO Ζpk

Year 2012, Volume 2, Issue 2, 145 - 153, 01.12.2012

Abstract

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.

References

  • 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.

Year 2012, Volume 2, Issue 2, 145 - 153, 01.12.2012

Abstract

References

  • 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.

Details

Primary Language English
Journal Section Research Article
Authors

B. YILDIZ This is me
Department of Mathematics, Fatih University, 34500, Istanbul, Turkey


Z. ÖDEMİŞ ÖZGER This is me
Department of Mathematics, Fatih University, 34500, Istanbul, Turkey

Publication Date December 1, 2012
Published in Issue Year 2012, Volume 2, Issue 2

Cite

Bibtex @ { twmsjaem761707, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2012}, volume = {2}, number = {2}, pages = {145 - 153}, title = {GENERALIZATION OF THE LEE WEIGHT TO Ζpk}, key = {cite}, author = {Yıldız, B. and Ödemiş Özger, Z.} }