The Weight Hierarchy of Hermitian Codes

On the Feng-Rao Bound for Generalized Hamming Weights

“…It is known that (12) gives the actual values of the t generalized Hamming weights of the q-ary Reed-Muller codes (see [7]). It is also known that (12) gives the actual values of the tth generalized Hamming weights of the Hermitian codes (see [2]). For the case of hyperbolic codes (improved q-ary Reed-Muller codes) (13) gives exactly the same estimates as was found in [5].…”

On the generalized Feng-Rao numbers of numerical semigroups generated by intervals

“…The computation of these generalized Feng-Rao distances turns out to be a very hard problem. Actually, very few results are known about this subject, and they are completely scattered in the literature (see for example [1], [9] or [7]). This paper studies the asymptotical behaviour of the generalized Feng-Rao distances, that is, δ r F R (m) for r ≥ 2 and m >> 0.…”

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

“…Finally Barbero and Munuera described all generalized Hamming weights for an arbitrary one-point code on a Hermitian curve [1]. Barbero and Munuera's main tool was the order bound on the generalized Hamming weights obtained by Heijnen and Pellikaan [5], which works only for one-point codes among the geometric Goppa codes.…”

“…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.…”