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Abstract: We show that the Feng–Rao bound for dual codes and a similar bound by Andersen and Geil (2008) [1] 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 (2000) [30] (see also Beelen and Høholdt, 2008 [3]) 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 in Matsumoto and Miura (2000) [30] requires the use of differentials which was not needed in Andersen and Geil (2008) [1]. Nevertheless we demonstrate a very strong connection between Matsumoto and Miuraʼs bound and Andersen and Geilʼs bound when applied to primary one-point algebraic geometric codes. [Copyright &y& Elsevier]

We consider weighted Reed-Muller codes over point ensemble S × · · · × S where S needs not be of the same size as S. For m = 2 we determine optimal weights and analyze in detail what is the impact of the ratio | S|/| S| on the minimum distance. In conclusion the weighted Reed-Muller code construction is much better than its reputation. For a class of affine variety codes that contains the weighted Reed-Muller codes we then present two list decoding algorithms. With a small modification one of these algorithms is able to correct up to 31 errors of the [49,11,28] Joyner code. [ABSTRACT FROM AUTHOR]

We generalize the unique decoding algorithm for one-point AG codes over the Miura–Kamiya C a b curves proposed by Lee et al. (2012) to general one-point AG codes, without any assumption. We also extend their unique decoding algorithm to list decoding, modify it so that it can be used with the Feng–Rao improved code construction, prove equality between its error correcting capability and half the minimum distance lower bound by Andersen and Geil (2008) that has not been done in the original proposal except for one-point Hermitian codes, remove the unnecessary computational steps so that it can run faster, and analyze its computational complexity in terms of multiplications and divisions in the finite field. As a unique decoding algorithm, the proposed one is empirically and theoretically as fast as the BMS algorithm for one-point Hermitian codes. As a list decoding algorithm, extensive experiments suggest that it can be much faster for many moderate size/usual inputs than the algorithm by Beelen and Brander (2010) . It should be noted that as a list decoding algorithm the proposed method seems to have exponential worst-case computational complexity while the previous proposals ( Beelen and Brander, 2010; Guruswami and Sudan, 1999 ) have polynomial ones, and that the proposed method is expected to be slower than the previous proposals for very large/special inputs. [ABSTRACT FROM AUTHOR]

We generalize the list decoding algorithm for Hermitian codes proposed by Lee and OʼSullivan (2009) based on Gröbner bases to general one-point AG codes, under an assumption weaker than one used by Beelen and Brander (2010). Our generalization enables us to apply the fast algorithm to compute a Gröbner basis of a module proposed by Lee and OʼSullivan (2009), which was not possible in another generalization by Lax (2012). [ABSTRACT FROM AUTHOR]