On Semigroups Generated by Two Consecutive Integers and Improved Hermitian Codes

TL;DR

This paper analyzes semigroups generated by two consecutive integers to optimize Hermitian codes, providing formulas for redundancy and minimum distance, and addressing an open problem in coding theory.

Contribution

It introduces new formulas for the redundancy of Hermitian codes optimized with respect to error correction and geometric genericity, solving an open problem from 1999.

Findings

Formulas for redundancies of optimized Hermitian codes

Order bound on the minimum distance derived

Addresses an open question by Pellikaan and Torres

Abstract

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. The formula for the redundancy of optimal Hermitian codes correcting a given number of errors answers an open question stated by Pellikaan and Torres in 1999.