Algebraic Structures of Constacyclic Codes Over Finite Chain Rings and Power Series Rings

  • M. Beygi
  • S. NamaziEmail author
  • H. Sharif
Research Paper
Part of the following topical collections:
  1. Mathematics


Let R be a chain ring with the maximal ideal \(\langle \gamma \rangle\). In this paper, we shall study \((r_1+r_2\gamma )\)-constacyclic codes of arbitrary length over R, where \(r_1, r_2\) are units in R. We shall obtain the generators of these codes and their duals. Moreover, a minimal spanning set (and so a generator matrix) for a constacyclic code is obtained. We shall determine the minimum Hamming distance of a constacyclic code over the chain ring R. At last, we shall study some constacyclic codes over formal power series rings. The minimal spanning set for these codes are also established.


Linear code Constacyclic code Finite chain ring Formal power series ring Minimum Hamming distance 

Mathematics Subject Classification

94B05 94A55 94B15 58F15 58F17 


  1. Bakshi TGK, Raka M (2012) A class of constacyclic codes over a finite field. Finite Field Appl 18:362–377MathSciNetCrossRefzbMATHGoogle Scholar
  2. Batoul A, Guenda K (2016) Some constacyclic codes over finite chain rings. Adv Math Commun 10:683–694MathSciNetCrossRefzbMATHGoogle Scholar
  3. Cao Y (2013) On constacyclic codes over finite chain rings. Finite Field Appl 24:124–135MathSciNetCrossRefzbMATHGoogle Scholar
  4. Chen B, Fan Y, Lin L, Liu H (2012) Constacyclic codes over finite fields. Finite Field Appl 18:1217–1231MathSciNetCrossRefzbMATHGoogle Scholar
  5. Dinh HQ (2010) Constacyclic codes of length \(p^{s}\) over \(F_{p^{m}}+uF_{p^{m}}\). J Algebra 324:940–950MathSciNetCrossRefzbMATHGoogle Scholar
  6. Dinh HQ, López-Permouth SR (2004) Cyclic and negacyclic codes over finite chain rings. IEEE Trans Inf Theory 50:1728–1744MathSciNetCrossRefzbMATHGoogle Scholar
  7. Dinh HQ, Nguyen HDT (2012) On some classes of constacyclic codes over polynomial residue rings. Adv Math Commun 6(2):175–191MathSciNetCrossRefzbMATHGoogle Scholar
  8. Dinh HQ, Nguyen HDT, Sriboonchitta S (2017) Repeated-root constacyclic codes of prime power lengths over finite chain rings. Finite Fields Appl 43:22–41MathSciNetCrossRefzbMATHGoogle Scholar
  9. Dougherty ST, Liu H (2013) Cyclic codes over formal power series rings. Acta Math Sci 31:331–343MathSciNetCrossRefzbMATHGoogle Scholar
  10. Dougherty ST, Liu H, Park YH (2011) Lifted codes over finite chain rings. Math J Okayama Univ 53:39–53MathSciNetzbMATHGoogle Scholar
  11. Kai X, Zhu S, Li P (2010) \((1+\lambda u)\)-Constacyclic codes over \(\frac{F_{p}[u]}{<u^{m}>}\). J Franklin Inst 347:751–762MathSciNetCrossRefzbMATHGoogle Scholar
  12. Liu H (2013) On the characterization of cyclic codes over two classes of rings. Acta Math Sci 33:413–422MathSciNetCrossRefzbMATHGoogle Scholar
  13. Matsumura Commutative (1990) Ring theory. Cambridge University Press, CambridgeGoogle Scholar
  14. McDonald BR (1974) Finite rings with identity. Marcel Dekker, New YorkzbMATHGoogle Scholar
  15. Norton GH, Sǎlǎgean A (2000a) On the Hamming distance of linear codes over a finite chain ring. IEEE Trans Inf Theory 46(3):1060–1067MathSciNetCrossRefzbMATHGoogle Scholar
  16. Norton GH, Sǎlǎgean A (2000b) On the structure of linear and cyclic codes over a finite chain ring. Appl Algebra Eng Commun Comput 10:489–506MathSciNetCrossRefzbMATHGoogle Scholar
  17. Prange E (1957) Cyclic error-correcting codes in two symbols. Technical report TN-57-103. Air Force Cambridge Research Center, CambridgeGoogle Scholar
  18. Rani S (2019) Structure of repeated-root constacyclic codes of length \(8l^mp^n\). Asian-Eur J Math 12(1):1950050-1–1950050-17MathSciNetGoogle Scholar
  19. Roman S (1974) Coding and information theory. Marcel Dekker, New YorkGoogle Scholar
  20. Sharma A, Rani S (2016) Repeated-root constacyclic codes of length \(4l^mp^n\). Finite Fields Appl 40:163–200MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Shiraz University 2019

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesShiraz UniversityShirazIran

Personalised recommendations