Minimum Distance Decoding of General Algebraic Geometry Codes via Lists

• Published in: IEEE transactions on information theory Vol. 56; no. 9; pp. 4335 - 4340
• Main Authors: Drake, N, Matthews, G L
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.09.2010; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algebra; Algebraic geometry (AG) code; Algorithm design and analysis; Algorithms; Applied sciences; Codes; Coding theory; Coding, codes; Decoding; Error correction codes; Exact sciences and technology; Galois fields; Geometry; Information theory; Information, signal and communications theory; list decoding; Mathematics; minimum distance decoding; multipoint code; Sections; Signal and communications theory; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Transmission and modulation (techniques and equipments); Voting; Algebraic geometry (AG) code; multipoint code; Finite field; Algebraic code; Decoding; Algorithm; Data broadcast; Information dissemination; Multicast; minimum distance decoding; list decoding; Minimal distance; Algebraic geometry
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2010.2054670

Algebraic geometry codes are defined by divisors D and G on a curve over a finite field F. Often, G is supported by a single F-rational point and the resulting code is called a one-point code. Recently, there has been interest in allowing the divisor G to be more general as this can result in superior codes. In particular, one may obtain a code with better parameters by allowing G to be supported by m distinct F-rational points, where m > 1. In this paper, we demonstrate that a multipoint algebraic geometry code C may be embedded in a one-point code C' . Exploiting this fact, we obtain a minimum distance decoding algorithm for the multipoint code C . This is accomplished via list decoding in the one-point code C'.