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The Chen-Reed-Helleseth-Truong Decoding Algorithm and the Gianni-Kalkbrenner Gröbner Shape Theorem

  • Massimo Caboara
  • Teo Mora

Abstract.

 In a paper from Chen, Reed, Helleseth and Truong, [10] (cf. also Loustaunau and York [10]) Gröbner bases are applied as a preprocessing tool in order to devise an algorithm for decoding a cyclic code over GF(q) of length n. The Gröbner basis computation of a suitable ideal allows us to produce two finite ordered lists of polynomials over GF(q),

$$$$

upon the receipt of a codeword, one needs to compute the syndromes \(\) and then to compute the maximal value of the index i s.t.\(\) the error locator polynomial is then

$$$$

The algorithm proposed in [4] needs the assumption that the computed Gröbner basis associated to a cyclic code has a particular structure; this assumption is not satisfied by every cyclic code. However the structure of the Gröbner basis of a 0-dimensional ideal has been deeply analyzed by Gianni [7] and Kalkbrenner [8]. Using these results we were able to generalize the idea of Chen, Reed, Helleseth and Truong to all cyclic codes.

Keywords

Basis Computation Cyclic Code Decode Algorithm Preprocessing Tool 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Massimo Caboara
    • 1
  • Teo Mora
    • 2
  1. 1.DIMA, Univ. Genova, Viale Dodecaneso 35, 16146 Genova, Italy (e-mail: caboara@dima.unige.it)IT
  2. 2.DISI, Univ. Genova, Viale Dodecaneso 35, 16146 Genova, Italy (e-mail: theomora@dima.unige.it)IT

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