On Semigroups Generated by Two Consecutive Integers and Hermitian Codes

Analysis of the Berlekamp-Massey-Sakata algorithm for decoding onepoint 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. Introduction Numerical semigroups have proven to be very useful in the study of one-point algebraic-geometry codes. On one hand the arithmetic of the numerical semigroup associated to the one-point yields a good bound—called the order bound— on minimum distance [5, 9, 11]. On the other hand, a close analysis of the numerical semigroup and the decoding algorithm commonly used for one-point codes shows that significant improvements in rate may be achieved while maintaining a given error correction capability [6]. In this article we discuss the order bound and improvements to the rate for codes constructed from Hermitian curves. Let us briefly recall the definition of one-point algebraic geometry codes and state the notation we will use. Suppose F is a finite field, F/F a function field and P a rational point of F/F. For m ∈ N0 let L(mP ) be the ring of functions in F having poles only at P and of order at most m. Let vP be the valuation of F associated with P and let Λ = {−vP (f) : f ∈ ⋃ m L(mP )}. Λ is a numerical semigroup. That is, a subset of N0, closed under summation, Universitat Autònoma de Barcelona, [email protected] San Diego State University, [email protected]