On Semigroups Generated by Two Consecutive Integers and Improved Hermitian Codes

• Published in: IEEE transactions on information theory Vol. 53; no. 7; pp. 2560 - 2566
• Main Authors: Bras-Amoros, M., O'Sullivan, M.E.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.07.2007; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Algorithms; Analysis; Applied sciences; Arithmetic; Coding, codes; Criteria; Decoding; Design engineering; Design methodology; Error correction codes; Exact sciences and technology; Feng-Rao improved code; Galois fields; Geometry; Hermitian curve; Information systems; Information theory; Information, signal and communications theory; Integer programming; Integers; Mathematical analysis; Mathematical models; Methods; numerical semigroup; Parity check codes; Redundancy; Signal and communications theory; Telecommunications and information theory; Vectors (mathematics); Voting; Voting; numerical semigroup; Redundancy; Hermite interpolation; Berlekamp Massey algorithm; Minimal distance; Numerical method; Feng-Rao improved code; Error correction; Decoding; Parity check; Hermitian curve
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2007.899548

Analysis of the Berlekamp-Massey-Sakata algorithm for decoding one-point codes leads to two methods for improving code rate. One method, due to Feng and Rao, removes parity checks that may be recovered by their majority voting algorithm. The second method is to design the code to correct only those error vectors of a given weight that are also geometrically generic. In this work, formulae are given for the redundancies of Hermitian codes optimized with respect to these criteria as well as the formula for the order bound on the minimum distance. The results proceed from an analysis of numerical semigroups generated by two consecutive integers.