Feng-Rao decoding of primary codes

TL;DR

This paper demonstrates that Feng-Rao bounds for dual and primary codes are interconnected, enabling efficient decoding of a broad class of linear codes, including many multivariate polynomial codes, up to half their minimum distance.

Contribution

It establishes the equivalence of Feng-Rao bounds for dual and primary codes and extends decoding algorithms to new classes of codes without prior efficient decoding methods.

Findings

Feng-Rao decoding applies to primary codes with known well-behaving pairs.

Decoding up to half the minimum distance is possible for many codes.

Connections between different bounds for algebraic geometric codes are clarified.

Abstract

We show that the Feng-Rao bound for dual codes and a similar bound by Andersen and Geil [H.E. Andersen and O. Geil, Evaluation codes from order domain theory, Finite Fields Appl., 14 (2008), pp. 92-123] for primary codes are consequences of each other. This implies that the Feng-Rao decoding algorithm can be applied to decode primary codes up to half their designed minimum distance. The technique applies to any linear code for which information on well-behaving pairs is available. Consequently we are able to decode efficiently a large class of codes for which no non-trivial decoding algorithm was previously known. Among those are important families of multivariate polynomial codes. Matsumoto and Miura in [R. Matsumoto and S. Miura, On the Feng-Rao bound for the L-construction of algebraic geometry codes, IEICE Trans. Fundamentals, E83-A (2000), pp. 926-930] (See also [P. Beelen and T. H{\o}holdt, The decoding of algebraic geometry codes, in Advances in algebraic geometry codes, pp. 49-98]) derived from the Feng-Rao bound a bound for primary one-point algebraic geometric codes and showed how to decode up to what is guaranteed by their bound. The exposition by Matsumoto and Miura requires the use of differentials which was not needed in [Andersen and Geil 2008]. Nevertheless we demonstrate a very strong connection between Matsumoto and Miura's bound and Andersen and Geil's bound when applied to primary one-point algebraic geometric codes.