Abstract We introduce greedy weights of matroids, inspired by those for linear codes. We show that a Wei duality holds for two of these types of greedy weights for matroids. Moreover we show that in the cases where the matroids involved are associated to linear codes, our definitions coincide with those for codes. Thus our Wei duality is a generalization of that for linear codes given by Schaathun. In the last part of the paper we show how some important chains of cycles of the matroids appearing, correspond to chains of component maps of minimal resolutions of the independence complex of the corresponding matroids. We also relate properties of these resolutions to chainedness and greedy weights of the matroids, and in many cases codes, that appear.

Abstract We study how some coefficients of two-variable coboundary polynomials can be derived from Betti numbers of Stanley–Reisner rings. We also explain how the connection with these Stanley–Reisner rings forces the coefficients of the two-variable coboundary polynomials and Möbius polynomials to satisfy certain universal equations.

Abstract 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.

We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of C over all field extensions of Fq. Our methods will be a study of the Stanley-Reisner rings of a series of matroids associated to each code C.

We define generalized Hamming weights for almost affine codes. We show that this definition is natural, since we can extend some well-known properties of the generalized Hamming weights for linear codes, to almost affine codes. In addition, we discuss the duality of almost affine codes, and of the smaller class of multilinear codes.

It is shown that each chain of linear codes has an associated demi-matroid, a combinatorial structure that extends the notion of a vector matroid of a linear code. These demi-matroids are proven to determine important properties of the chain, and it is shown that linear code chain duality is represented by demi-matroid duality in a natural and yet surprising way. Profiles are defined for arbitrary demi-matroids and thus for linear code chains, which generalizes previous results in the literature and provides simple and transparent proofs thereof. Finally, applications are given to encryption of messages through wire-tap channels of Type II.

We determine the code parameters of a large class of Goppa codes, constructed from certain curves described by Stichtenoth (see Arch. Math., vol.XXIV, pp.615-631, 1973), in terms of the genus and the Weierstrass gap sequence of a given chosen point on each tune. Examples are codes from Hermitian, Artin-Schreier, and hyperelliptic curves.>