• SARI, MUSTAFA (Department of Mathematics Yildiz Technical University) ;
  • SIAP, IRFAN (Department of Mathematics Yildiz Technical University) ;
  • SIAP, VEDAT (Department of Mathematical Engineering Yildiz Technical University)
  • Received : 2014.10.14
  • Published : 2015.11.30


This paper determines the structures of one-homogeneous weight codes over finite chain rings and studies the algebraic properties of these codes. We present explicit constructions of one-homogeneous weight codes over finite chain rings. By taking advantage of the distance-preserving Gray map defined in [7] from the finite chain ring to its residue field, we obtain a family of optimal one-Hamming weight codes over the residue field. Further, we propose a generalized method that also includes the examples of optimal codes obtained by Shi et al. in [17].


linear codes;constant weight codes;gray map;optimal codes


  1. E. Agrell, A. Vardy, and K. Zeger, Upper bounds for constant-weight codes, IEEE Trans. Inform. Theory 46 (2000), no. 7, 2373-2395.
  2. A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin. 18 (1984), 181-186.
  3. A. E. Brouwer, J. B. Shearer, N. J. A. Sloane, andW. D. Smith, A new table of constant weight codes, IEEE Trans. Inform. Theory 36 (1990), no. 6, 1334-1380.
  4. C. Carlet, One-weight $\mathbb{Z}_4$-linear codes, In J. Buchmann, T. Hoholdt, H. Stichtenoth, and H. Tapia-Recillas, editors, Coding, Cryptography and Related Areas, 57-72, Springer, 2000.
  5. I. Constantinescu and W. Heise, A metric for codes over residue class rings, Probl. Inf. Transm. 33 (1997), no. 3, 22-28.
  6. M. Greferath and E. S. Schmidt, Gray isometries for finite chain rings and a nonlinear ternary (36, $3^{12}$, 15) code, IEEE Trans. Inform. Theory 45 (1999), no. 7, 2522-2524.
  7. S. Jitman and P. Udomkavanich, The gray image of codes over finite chain rings, Int. J. Contemp. Math. Sci. 5 (2010), no. 9-12, 449-458.
  8. P. J. Kuekes, W. Robinett, R. M. Roth, G. Seroussi, G. S. Snider, and R. S. Williams, Resistor-logic demultiplexers for nano electronics based on constant-weight codes, Nanotechnol. 17 (2006), no. 4, 1052-1061.
  9. S. Ling and C. Xing, Coding Theory: A First Course, Cambridge University Press, UK, 2004.
  10. J. van Lint and L. Tolhuizen, On perfect ternary constant-weight codes, Des. Codes Cryptogr. 18 (1999), no. 1-3, 231-234.
  11. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North- Holland, Amsterdam, The Netherlands, 1977.
  12. J. N. J. Moon, L. A. Hughes, and D. H. Smith, Assignment of frequency lists in frequency hopping networks, IEEE Trans. Veh. Technol. 54 (2005), no. 3, 1147-1159.
  13. G. H. Norton and A. Salagean, On the structure of linear and cyclic codes over a finite chain ring, Appl. Algebra Engrg. Comm. Comput. 10 (2000), no. 6, 489-506.
  14. W. W. Peterson and E. J. Jr. Weldon, Error-Correcting Codes, The MIT Press, USA, 1972.
  15. E. M. Rains and N. J. A. Sloane, Table of constant-weight binary codes, [Online]. Avail- able: njas/codes/Andw/
  16. M. Shi, Optimal p-ary codes from one-weight linear codes over $\mathbb{Z}_{p^}m$, Chin. J. Electron. 22 (2013), no. 4, 799-802.
  17. M. Shi, S. Zhu, and S. Yang, A class of optimal p-ary codes from one-weight codes over $\mathbb{F}_p[u]/(u^m)$, J. Franklin Inst. 350 (2013), no. 5, 929-937.
  18. D. M. Smith and R. Montemanni, Bounds for constant-weight binary codes with n > 28, [Online]. Available: roberto/Andw29/
  19. J. A. Wood, The structure of linear codes of constant weight, Trans. Amer. Math. Soc. 354 (2002), no. 3, 1007-1026.

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