On Semigroups Generated by Two Consecutive Integers and Improved Hermitian Codes

Abstract

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.

Unique Decoding of Plane AG Codes via Interpolation

“…Finally it is clear thatB generates the moduleĨ. From (12), (13), and (14), we see thatB is a Gröbner basis ofĨ with respect to > s−1 , by the criterion in Proposition 1.…”

“…x + · · · α 3 x 2 + · · · α 7 x 3 + · · · x 4 + · · · 2x 6 + · · · x 8 + · · · From the result f i g i c i w i f 0 g 1 2 0 f 1 g 2 2 0 f 2 g 0 0 2 we set w 12 = 0, and the Gröbner basis with respect to > 12 for v (12) = v (13) − ev(0 · x 3 y) is y 2 z yz z y 2 y 1 g 0 1 α 3 x + · · · α 7 x 2 + · · · α 3 x 4 + · · · α 7 x 6 + · · · α 3 x 8 + · · · g 1…”

Ideals of Numerical Semigroups and Error-Correcting Codes

“…It is fundamental in the computation of bounds for the minimum distance of algebraic-geometry codes based on a single point as well as in the optimization of the redundancy of those codes. Its properties and applications can be seen in [7][8][9][10][11][12][13][14] and in the survey [1]. As a curiosity, it was proved in [7,15] that the set of elements of a numerical semigroup is determined by its ν sequence.…”

“…On one side, we can use inequality (9) and inequality (11), and obtain E r ≥ ( − 1)(n −1 + 1). On the other side, we can use inequality (9) and inequality (12), and then inequality (6), as follows:…”

Improved Two-Point Codes on Hermitian Curves

“…By adding two independent words from C(62)\C(60) to C(59) we obtain a [64, 56,5] code. The parameters of improved one-point Hermitian codes are given in closed form in [9].…”