An improvement of the Feng-Rao bound for primary codes (Q2350925)

The Feng-Rao bound, originally proposed for decoding the so-called one-point algebraic-geometry (AG) codes [\textit{G.-L. Feng} and \textit{T. R. N. Rao}, IEEE Trans. Inf. Theory 39, No. 1, 37--45 (1993; Zbl 0765.94021)], provides a bound for the minimum distance of a linear error-correcting code. This paper presents an improvement of that bound, with particular attention to the case of affine variety codes, for which the new bound is particularly suited. In a previous paper [Finite Fields Appl. 30, 33--48 (2014; Zbl 1343.94104)] the authors already studied an improvement of the Feng-Rao bound for dual codes. The paper begins (Section 2) studying the new bound for primary affine variety codes, recalling the two order domains conditions (C1) and (C2) for affine varieties [the first author, Advances in algebraic geometry codes. Hackensack, NJ: World Scientific, 153--180 (2008; Zbl 1178.94244)]. The authors argue that the proposed improvement ``can produce good estimates in the case that the order domain condition (C1) is satisfied but (C2) is not''. Section 3 looks for curves in such a situation, introducing the \textit{generalized \(C_{ab}\) polynomials}. Sections 4 and 5 deepen in the study of codes defined by optimal \(C_{ab}\) polynomials. Section 6 shows the application of the new method to generalized Hamming weights. Section 7 reformulates the new bound (up to now considered for affine variety codes) for any linear code and finally Sections 8 and 9 study the connections of the bound in this paper and the bound for dual codes above mentioned. The paper outlines as an open problem the question of the existence of ``examples where our new method improves on the Feng-Rao bound for one-point algebraic geometric codes (the case where both order domain conditions are satisfied).''