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Title	The Chen-Reed-Helleseth-Truong Decoding Algorithm and the Gianni-Kalkbrenner Shape Theorem
Author(s)	Massimo Caboara, Ferdinando Mora
Type	Article in Journal
Abstract	In a paper from Chen, Reed, Helleseth and Truong, (cf. also Lousteanaou and York ) Gr\"obner 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 \Gr\ basis computation of a suitable ideal allows us to produce two finite ordered lists of polynomials over $GF(q)$, $$ \{\Gamma_i(X_1,\ldots,X_s)\}\text{\ and\ }\{G_i(X_1,\ldots,X_s, Z)\}; $$ upon the receipt of a codeword, one needs to compute the syndromes $\{{\sf s}_1,\ldots,{\sf s}_s\}$ and then to compute the maximal value of the index $i$ s.t. $\Gamma_i({\sf s}_1,\ldots,{\sf s}_s) = 0$; the error locator polynomial is then $$ \gcd(G_i({\sf s}_1,\ldots,{\sf s}_s, Z), Z^n-1). $$ The algorithm proposed in Chen, Reed Helleseth and Truong needs the assumption that the computed Gr\"obner basis associated to a cyclic code has a particular structure; this assumption is not satisfied by every cyclic code. However the structure of the Gr\"obner basis of a $0$-dimensional ideal has been deeply analyzed by Gianni and Kalkbrenner. Using these results we were able to generalize the idea of Chen, Reed, Helleseth and Truong to all cyclic codes.
Length	22
File
Language	English
Journal	Applicable Algebra in Engineering, Communications and Computing
Volume	13
Number	3
Pages	209-232
Publisher	Springer
Year	2002
Edition	0
Translation	No
Refereed	Yes