A Note on Linear Codes over Semigroup Rings

Antonio Aparecido de Andrade, Tariq Shah, Atlas Khan


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.


[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

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

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Trends in Computational and Applied Mathematics

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