The second and third generalized Hamming weights of Hermitian codes

On Semigroups Generated by Two Consecutive Integers and Improved Hermitian Codes

“…For Hermitian codes this information has appeared previously (see [7,2,9]). We choose to include our own proofs since our methods are new and will be needed later in the analysis of improved codes.…”

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

“…For details, consult [13]. The set {m = aq + b ∈ A(Z, q) | dim C m > dim C m−1 } consists of the quantities in Table 1 and (q 2 + q − 2)q + q − 1, which must be added under a 12 q + q − 1 in the table.…”

“…According to [13], this notion was introduced by Helleseth et al [6]. Later Wei [14] rediscovered the notion and proved the duality between the weight hierarchy {d r (C)} r of C and that of its dual C ⊥ , and the notion became known widely.…”

“…Namely, the theory is not applicable in the two-point case. 1 So we choose another way and restrict our target to the only the second Hamming weight for study the two-point codes, as Munuera and Ramirez did so in an earlier work [13] for the one-point codes on a Hermitian curve. (Actually, they also studied the third one.…”

“…The generalized Hamming weights for the one-point codes on a Hermitian curve were partially determined by Yang et al [16], Munuera [12], and Munuera and Ramirez [13] in the 1990s. Finally Barbero and Munuera described all generalized Hamming weights for an arbitrary one-point code on a Hermitian curve [1].…”