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Feng-Rao decoding of primary codes
Año del Documento
Finite Fields and their Applications. Volume 23, pages 35-52 (2013)
We show that the Feng-Rao bound for dual codes and a similar bound by Andersen and Geil 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 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-Miura requires the use of differentials which was not needed in Andersen-Geil. 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.
Revisión por pares
The present work was done while Ryutaroh Matsumoto was visiting Aalborg University as a Velux Visiting Professor supported by the Villum Foundation. The authors gratefully acknowledge this support. The authors also gratefully acknowledge the support from the Danish National Research Foundation and the National Science Foundation of China (Grant No. 11061130539) for the Danish-Chinese Center for Applications of Algebraic Geometry in Coding Theory and Cryptography. Furthermore the authors are thankful for the support from Spanish grant MTM2007-64704, the Spanish MINECO grant No. MTM2012-36917-C03-03, and for the MEXT Grant-in-Aid for Scientific Research (A) No. 23246071.
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