Summary: In a paper from X. Chen, I. S. Reed, T. Helleseth and T. K. Truong [Contemp. Math. 168, 15-22 (1994; Zbl 0812.94018)] (cf. also P. Loustaunau and E. Von York [Appl. Algebra Eng. Commun. Comput. 8, 469-483 (1997; Zbl 0916.94013)]) Gröbner bases are applied as a preprocessing tool in order to devise an algorithm for decoding a cyclic code over \(\text{GF}(q)\) of length \(n\). The Gröbner basis computation of a suitable ideal allows us to produce two finite ordered lists of polynomials over \(\text{GF}(q)\), \[ \{\Gamma_i(X_1,\dots, X_s)\}\quad\text{and}\quad \{G_i(X_1,\dots, X_s, Z)\}; \] upon the receipt of a codeword, one needs to compute the syndromes \(\{S_1,\dots, S_s\}\) and then to compute the maximal value of the index \(i\) such that \(\Gamma_i(S_1,\dots, S_s)= 0\); the error locator polynomial is then \[ \text{gcd}(G_i(S_1,\dots, S_s, Z),Z^n- 1). \] The algorithm proposed in [Chen et al. (loc. cit.)] 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 P. Gianni [Properties of Gröbner bases under specialization, Lect. Notes Comput. Sci. 378, 293-297 (1989)] and M. Kalkbrenner [Solving systems of algebraic equations using Gröbner bases, ibid. 282-292 (1989)]. Using these results we are able to generalize the idea of Chen, Reed, Helleseth and Truong to all cyclic codes.