A Note on Linear Codes over Semigroup Rings

Authors

  • Antonio Aparecido de Andrade
  • Tariq Shah
  • Atlas Khan

DOI:

https://doi.org/10.5540/tema.2011.012.02.0079

Abstract

Abstract. In this paper, we introduced new construction techniques of BCH, alternant, Goppa, Srivastava codes through the semigroup ring B[X; 13Z0] instead of the polynomial ring B[X; Z0], where B is a finite commutative ring with identity, and for these constructions we improve the several results of [1]. After this, we present a decoding principle for BCH, alternant and Goppa codes which is based on modified Berlekamp-Massey algorithm. This algorithm corrects all errors up tothe Hamming weight t r/2, i.e., whose minimum Hamming distance is r + 1.

References

[1] A.A. de Andrade, R. Palazzo Jr, Linear codes over finite rings, TEMA - Tend. Mat. Apl. Comput., 6, No. 2 (2005), 207–217.

[2] T. Shah, A. Khan, A.A. de Andrade, Encoding through generalized polynomial codes, (accepted for publication).

[3] J.C. Interlando, R. Palazzo Jr., M. Elia, On the decoding of Reed-Solomon and BCH codes over integer residue rings, IEEE Trans. Inform. Theory, IT- 43 (1997), 1013–1021.

[4] R. Gilmer, “Commutative Semigroup Rings”, University Chicago Press Chicago and London, 1984.

[5] B.R. McDonlad, “Finite Rings with Identity”, Marcel Dekker, New York, 1974.

[6] H.J. Helgret, Srivastava Codes, IEEE Trans. Inform. Theory, IT-18, No. 2, 1972.

[7] G.D. Forney Jr., On decoding BCH codes, IEEE Trans. Inform. Theory, IT-11 (1965), 549–557.

[8] W.W. Peterson, E.J. Weldon Jr., “Error Correcting Codes”, MIT Press, Cambridge, Mass., 1972.q

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Published

2011-06-01

How to Cite

de Andrade, A. A., Shah, T., & Khan, A. (2011). A Note on Linear Codes over Semigroup Rings. Trends in Computational and Applied Mathematics, 12(2), 79–89. https://doi.org/10.5540/tema.2011.012.02.0079

Issue

Vol. 12 No. 2 (2011)

Section

Original Article

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