The structure of one weight linear and cyclic codes over Z_{2}^r x (Z_{2} + uZ_{2})^s

Ismail Aydogdu

Abstract


Inspired by the Z2Z4-additive codes, linear codes over Z_{2}^r x  (Z_{2} + uZ_{2})^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear codes over Z_{2}^r x  (Z_{2} + uZ_{2})^s have some advantages compared to Z2Z4-additive codes. A code is called constant weight(one weight) if all the codewords have the same weight. It is well known that constant weight or one weight codes have many important applications. In this paper, we study the structure of one weight Z2Z2[u]-linear and cyclic codes. We classify these type of one weight codes and also give some illustrative examples.

 

 

 

 

 

 

 

 


Keywords


One weight codes; Z2Z2[u]-linear codes; duality.

Full Text:

PDF

References


Hammons, A. R., Kumar, V., Calderbank, A. R., Sloane, N.J.A. and Sol´e, P. (1994). The Z4-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory, 40, 301-319.

Calderbank, A.R. and Sloane, N.J.A. (1995). Modular and p-adic cyclic codes. Designs, Codes and Cryptog- raphy, 6, 21-35.

Greferath, M. and Schmidt, S. E. (1999). Gray isometries for finite chain rings. IEEE Trans. Info. Theory, 45(7), 2522-2524.

Honold, T. and Landjev, I. (1998). Linear codes over finite chain rings. In Optimal Codes and Related Topics, 116-126, Sozopol, Bulgaria.

Borges, J., Fern´andez-C´ordoba, C., Pujol, J., Rif`a, J. and Villanueva, M. (2010). Z2Z4-linear codes: Generator Matrices and Duality. Designs, Codes and Cryptography, 54(2), 167-179.

Aydogdu, I. and Siap, I. (2013). The structure of Z2Z2s−Additive codes: bounds on the minimum distance. Applied Mathematics and Information Sciences(AMIS), 7(6), 2271-2278.

Aydogdu, I. and Siap, I. (2015). On prZps -additive codes. Linear and Multilinear Algebra, , 63(10), 2089-2102.

Abualrub, T., Siap, I. and Aydin, N. (2014). Z2Z4-additive cyclic codes. IEEE Trans. Inf. Theory, 60(3), 1508-1514.

Dougherty, S.T., Liu, H. and Yu, L. (2016). One Weight 24 additive codes. Applicable Algebra in Engineering, Communication and Computing, 27, 123-138.

Carlet, C. (2000). One-weight 4-linear codes, In: Buchmann, J., Høholdt, T., Stichtenoth, H., Tapia Recillas,H. (eds.) Coding Theory, Cryptography and Related Areas. 57-72. Springer, Berlin.

Wood, J.A.(2002) The structure of linear codes of constant weight. Trans. Am. Math. Soc. 354, 1007-1026.

Skachek, V. and Schouhamer Immink, K.A. (2014). Constant weight codes: An approach based on Knuth’s balancing method. IEEE Journal on Selected Areas in Communications, 32(5), 909-918.

Telatar, I.E. and Gallager, R.G (1990). Zero error decision feedback capacity of discrete memoryless channels. in BILCON-90: Proceedings of 1990 Bilkent International Conference on New Trends in Communication, Control and Signal Processing, E. Arikan, Ed. Elsevier, 228-233.

Dyachkov, A.G. (1984). Random constant composition codes for multiple access channels. Problems Control Inform. Theory/Problemy Upravlen. Teor. Inform., 13(6), 357-369.

Ericson, T. and Zinoviev, V. (1995). Spherical codes generated by binary partitions of symmetric point sets. IEEE Trans. Inform. Theory, 41(1), 107-129.

King, O.D. (2003). Bounds for DNA codes with constant GC-content. Electron. J. Combin., 10(1), Research Paper 33, (electronic).

Milenkovic, O. and Kashyap, N. (2006). On the design of codes for DNA computing. Ser. Lecture Notes in Computer Science, vol. 3969. Berlin: Springer-Verlag, 100-119.

Colbourn, C. J., Kløve, T. and Ling, A. C. H. (2004). Permutation arrays for powerline communication and mutually orthogonal Latin squares. IEEE Trans. Inform. Theory, 50(6), 1289-1291.

Abualrub, T. and Siap, I. (2007). Cyclic codes over the rings Z2 + uZ2 and Z2 + uZ2 + u2Z2. Designs Codes and Cryptography, 42(3), 273-287.

Al-Ashker, M. and Hamoudeh, M. (2011). Cyclic codes over Z2 + uZ2 + u2Z2 + · · · + uk−1Z2. Turk. J. Math., 35, 37-749.

Dinh, H. Q. (2010). Constacyclic codes of length ps over Fpm + uFpm. Journal of Algebra, 324, 940-950.

Bonisoli, A. (1984). Every equidistant linear code is a sequence of dual Ham- ming codes. Ars Combin., 18, 181-186.

Aydogdu, I., Abualrub, T. and Siap, I. (2015). On Z2Z2[u]−additive codes. International Journal of Computer Mathematics, 92(9), 1806-1814.

Aydogdu, I., Abualrub, T. and Siap, I. (2017). Cyclic and constacyclic codes. IEEE Trans. Inf. Theory, 63(8), 4883-4893.

Van Lint, J.H. (1992). Introduction to Coding Theory. Springer-Verlag, New York.

Grassl, M., Code tables: Bounds on the parameters of various types of codes. Online database. Available at http://www.codetables.de/




DOI: http://dx.doi.org/10.11121/ijocta.01.2018.00512

Refbacks

  • There are currently no refbacks.


Copyright (c) 2017 ismail aydogdu

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

footer_771

   ithe_170     crossref_284         ind_131_43_x_117_117  Scopus  EBSCO_Host    ULAKBIM   PROQUEST   ZBMATH more...