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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]

In 1985, Gabidulin introduced the rank metric in coding theory over finite fields, and used this kind of codes in a McEliece cryptosystem, six years later. In this paper, we consider rank metric codes over Galois rings. We propose a suitable metric for codes over such rings, and show its main properties. With this metric, we define Gabidulin codes over Galois rings, propose an efficient decoding algorithm for them, and hint their cryptographic application. [ABSTRACT FROM AUTHOR]

Reed-Solomon codes have gained a lot of interest due to its encoding simplicity, well structuredness and list-decoding capability [6] in the classical setting. This interest also translates to other metric setting, including the insertion and deletion (insdel for short) setting which is used to model synchronization errors caused by positional information loss in communication systems. Such interest is supported by the construction of a deletion correcting algorithm of insdel Reed-Solomon code in [22] which is based on the Guruswami-Sudan decoding algorithm [6]. Nevertheless, there have been few studies [3] on the insdel error-correcting capability of Reed-Solomon codes. In this paper, we discuss a criterion for a 2-dimensional insdel Reed-Solomon codes to have optimal asymptotic error-correcting capabilities, which are up to their respective lengths. Then we provide explicit constructions of 2-dimensional insdel Reed-Solomon codes that satisfy the established criteria. The family of such constructed codes can then be shown to extend the family of codes with asymptotic error-correcting capability reaching their respective lengths provided in [3, Theorem 2] which provide larger error-correcting capability compared to those defined in [25]. [ABSTRACT FROM AUTHOR]