The Two-Point Codes on a Hermitian Curve with the Designed Minimum Distance

Abstract

This is the second part of the series of papers devoted to the determination of the minimum distance of two-point codes on a Hermitian curve. We study the case where the minimum distance agrees with the designed one. In order to construct a function which gives a codeword with the designed minimum distance, we use functions arising from conics in the projective plane.

The second generalized Hamming weight for two-point codes on a Hermitian curve

“…We prove Theorems 1-6 step by step from the next section, and arrange the sections from the viewpoint of the technique or the tool for proving of each statement, because the proof involves many materials such as Munuera's results on generalized Hamming weights on a Goppa code, Munuera-Ramirez' results on C m , fine geometry of a Hermitian curve progressed by [8], Coppens' results on base-point-free pencils on a curve, and the method of RDF. 2 This construction of the article may confuse the reader.…”

“…As for the choice of constants the ϕ i 's and the κ i 's, consult [8,Sect. 3] and/or [9, the proof of Lemma 3].…”

“…In the suite of articles [7][8][9][10], we found the value of minimum weight for all two-point codes on a Hermitian curve. The target of this paper is to determine the second generalized Hamming weight for these codes.…”

“…3]). We follow the notation and the terminology of the series of papers [7][8][9][10], some of which is not in common use. We denote by A(Z, q) the array of integers with the infinite length of column .…”

“…These labels also correspond to the labels used in [7][8][9][10]; however, the subscript 2 on a label means that there is a small difference between the region that appears here from the region that appeared in our previous works. Moreover, the superscript * denotes that the case appears in a boundary of the indicated area in the table.…”