Title :
Vector bundles and codes on the Hermitian curve
Author_Institution :
Dept. of Math., Bloomsburg Univ., PA
fDate :
6/1/2005 12:00:00 AM
Abstract :
The construction of algebraic-geometry (AG) codes can be seen as a distinctly geometric process, and yet decoding procedures tend to rely on algebraic ideas that have no direct geometric interpretation. Recently, however, Trygve Johnsen observed that decoding can be viewed in abstract terms of a class of vector bundles on the underlying curve. The present paper describes these objects at a concrete computational level for the Hermitian codes COmega(D,mPinfin) defined over Fq 2 (q a power of 2). The construction of explicit representations of the vector bundles by transition matrices involves finding functions on the curve that satisfy a certain property in their power series expansions around Pinfin, computing the image of the corresponding global sections under Serre duality, and finding a suitable open cover of the curve. The cover enables any rational point to be expressed as a line bundle by a simple kind of transition function. A special case is considered in which these functions can be realized as ratios of linear forms
Keywords :
Hermitian matrices; algebraic geometric codes; decoding; Hermitian code; Serre duality; abstract term; algebraic-geometry code; construction code; decoding procedure; geometric process; power series expansion; transition matrix; underlying curve; vector bundle; Codes; Computer science; Concrete; Decoding; Galois fields; Linear systems; Mathematics; Statistics; Vectors; Voting; Algebraic-geometry (AG) codes; Hermitian curve; Serre duality; transition matrix; vector bundles;
Journal_Title :
Information Theory, IEEE Transactions on
DOI :
10.1109/TIT.2005.847730