On Construction and Generalization of Algebraic Geometry Codes
Ryutaroh Matsumoto, Shinji Miura · 2007
The construction, estimation of minimum distance, and decoding algorithms of algebraic geometry codes can be explained without using advanced mathematics by the notion of weight domains. We clarify the relation between algebraic geometry codes and linear codes from weight domains. Then we review a systematic construction which yields all weight domains. 1 Introduction Algebraic geometry codes were defined in an algebraic geometric way, and many facts about them, in particular estimation of minimum distance and decoding algorithms, are also stated and proved algebraic geometrically. So it had been difficult to understand algebraic geometry codes without the theory of algebraic curves. Recently Hholdt et al. [HvLP97, HvLP98] observed that definition, estimation of minimum distance, and decoding algorithms of algebraic geometry codes could be explained using only the notion of a weight function, which is essentially a discrete valuation, and made understanding of algebraic geometry codes...