Advertisement

Construction and decoding of matrix-product codes from nested codes

  • Fernando Hernando
  • Kristine Lally
  • Diego Ruano
Article

Abstract

We consider matrix-product codes \({[C_1\cdots C_s] \cdot A}\) , where \({C_1, \ldots , C_s}\) are nested linear codes and matrix A has full rank. We compute their minimum distance and provide a decoding algorithm when A is a non-singular by columns matrix. The decoding algorithm decodes up to half of the minimum distance.

Keywords

Linear codes Matrix-product codes Decoding algorithm Minimum distance Quasi-cyclic codes 

Mathematics Subject Classification (2000)

94B05 94B35 94B65 

References

  1. 1.
    Blackmore T., Norton G.H.: Matrix-product codes over \({\mathbb{F}_{q}}\) . Appl. Algebra Eng. Commun. Comput. 12(6), 477–500 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Kasami T.: A Gilbert–Varshamov bound for quasi-cyclic codes of rate 1/2. IEEE Trans. Inf. Theory 20, 679 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Lally K., Fitzpatrick P.: Algebraic structure of quasicyclic codes. Discrete Appl. Math. 111(1–2), 157–175 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ling S., Solé P.: On the algebraic structure of quasi-cyclic codes I: finite fields. IEEE Trans. Inf. Theory IT-47, 2751–2759 (2001)CrossRefGoogle Scholar
  5. 5.
    Macwilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes, ser. North-Holland Mathematical Library, North-Holland (1977)Google Scholar
  6. 6.
    Martínez-Moro E.: A generalization of Niederreiter-Xing’s propagation rule and its commutativity with duality. IEEE Trans. Inf. Theory 50(4), 701–702 (2004)CrossRefGoogle Scholar
  7. 7.
    Niederreiter H., Xing C.: A propagation rule for linear codes. Appl. Algebra Eng. Commun. Comput. 10(6), 425–432 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Özbudak F., Stichtenoth H.: Note on Niederreiter-Xing’s propagation rule for linear codes. Appl. Algebra Eng. Commun. Comput. 13(1), 53–56 (2002)zbMATHCrossRefGoogle Scholar
  9. 9.
    van Asch B.: Matrix-product codes over finite chain rings. Appl. Algebra Eng. Commun. Comput. 19(1), 39–49 (2008)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Fernando Hernando
    • 1
  • Kristine Lally
    • 2
  • Diego Ruano
    • 3
  1. 1.Department of MathematicsUniversity College CorkCorkIreland
  2. 2.Department of Mathematics and StatisticsRMIT UniversityMelbourneAustralia
  3. 3.Department of MathematicsTechnical University of DenmarkKgs. LyngbyDenmark

Personalised recommendations