Advances in Mathematics of Communications (AMC)

Duality for some families of correction capability optimized evaluation codes

Pages: 15 - 33, Volume 2, Issue 1, February 2008      doi:10.3934/amc.2008.2.15

       Abstract        Full Text (244.6K)       Related Articles

Maria Bras-Amorós - Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Avinguda Països Catalans, 26, 43007 Tarragona, Catalonia, Spain (email)
Michael E. O'Sullivan - Department of Mathematics and Statistics, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182-7720, United States (email)

Abstract: Improvements to code dimension of evaluation codes, while maintaining a fixed decoding radius, were discovered by Feng and Rao, 1995, and nicely described in terms of an order function by Høholdt, van Lint, Pellikaan, 1998. In an earlier work, 2006, we considered a different improvement, based on the observation that the decoding algorithm corrects an error vector based not so much on the weight of the vector but rather the ''footprint'' of the error locations. In both cases one can find minimal sets of parity checks defining the codes by means of the order function. In this paper we show that these minimal sets have a very useful closure property. For several important families of codes that we consider, this property allows us to construct a generating matrix for the code that has properties amenable to encoding. The generating matrix can be constructed by evaluating monomials in a set which also has the closure property.

Keywords:  Algebraic-geometry code, Reed-Muller code, Hermitian code, toric code, dual code.
Mathematics Subject Classification:  94B27.

Received: July 2007;      Revised: November 2007;      Available Online: January 2008.