A polymatroid approach to generalized weights of rank metric codes

We consider the notion of a (q, m)-polymatroid, due to Shiromoto, and the more general notion of (q, m)-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality for generalized Hamming weights of linear codes. The corresponding results of Ravagnani for Delsarte rank metric codes, and Martínez-Peñas and Matsumoto for relative generalized rank weights are derived as a consequence.

On the generalized Hamming weights of certain Reed–Muller-type codes

There is a nice combinatorial formula of P. Beelen and M. Datta for the r-th generalized Hamming weight of an a ne cartesian code. Using this combinatorial formula we give an easy to evaluate formula to compute the r-th generalized Hamming weight for a family of a ne cartesian codes. If 𝕏 is a set of projective points over a finite field we determine the basic parameters and the generalized Hamming weights of the Veronese type codes on 𝕏 and their dual codes in terms of the basic parameters and the generalized Hamming weights of the corresponding projective Reed–Muller-type codes on 𝕏 and their dual codes.

Ideals of Numerical Semigroups and Error-Correcting Codes

Several results relating additive ideals of numerical semigroups and algebraic-geometrycodes are presented. In particular, we deal with the set of non-redundant parity-checks, the codelength, the generalized Hamming weights, and the isometry-dual sequences of algebraic-geometrycodes from the perspective of the related Weierstrass semigroups. These results are related tocryptographic problems such as the wire-tap channel, t-resilient functions, list-decoding, networkcoding, and ramp secret sharing schemes.