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

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.