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

Abstract

The aim of this article is the determination of the second generalized Hamming weight of any two-point code on a Hermitian curve of degree q + 1. The determination involves results of Coppens on base-point-free pencils on a plane curve. To avoid nonessential trouble, we assume that q > 4.

Higher Hamming weights for locally recoverable codes on algebraic curves

“…For these reasons, the GHWs (and their extended version, the relative generalized Hamming weights [21,19]) play a central role in coding theory. In particular, generalized and relative generalized Hamming weights are studied for Reed-Muller codes [10,23] and for codes constructed by using an algebraic curve [6] as Goppa codes [24,38], Hermitian codes [12,25] and Castle codes [27].…”

Generalized Hamming weights of three classes of linear codes

“…In this paper we investigate the generalized Hamming weights of three classes of linear codes and determine them partly for some cases. The generalized Hamming weights of many families of codes have been determined, such as Hamming codes [28], Reed-Muller codes [13], BCH codes [11,2,27], cyclic codes [9,18], trace codes [24], binary Kasami codes [15], Melas and dual Melas codes [26], AG codes [1,29,21,17,16], etc. The readers are referred to [25] for a survey.…”

Bounds for generalized Hamming weights of general AG codes

“…It seems that these bounds may give simpler proofs for the results like in Homma and Kim [23] in efforts of determining higher generalized Hamming weights of multi-point AG codes.…”

“…, m r )|: m 1 < m 2 < · · · < m r ∈Λ ≤0 }. As is well known [21,22], δ 1 is the true minimum distance of the code, and δ 2 is the true second generalized Hamming weight of the code according to Homma and Kim [23]. By Munuera's arithmetic interpretation [24], we verified that δ 3 is also the true third generalized Hamming weight, which is realized by the linear space spanned by the three codewords ev(μ 1 ) = (0, 0, 0, 0, 0, α 7 , α, 2, 0, α 7 , 0, α 6 , 1, α 3 , 0, 1, 0, 0, α 2 , 0, 0, α 7 , 0, 2, α 6 , α), ev(μ 2 ) = (0, 0, 0, 0, 0, α 7 , 2, α 5 , 0, α, 0, 2, α, α 2 , 0, 1, 0, 0, 1, 0, 0, α 3 , 0, α 6 , α 3 , 2), ev(μ 3 ) = (0, 0, 0, 0, 0, α 3 , 1, α, 0, 1, 0, α 2 , α 7 , 1, 0, α, 0, 0, α 3 , 0, 0, 1, 0, 1, α 5 , α 6 ),…”