Decomposable Subspaces, Linear Sections of Grassmann Varieties, and Higher Weights of Grassmann Codes

Given a homogeneous component of an exterior algebra, we characterize those subspaces in which every nonzero element is decomposable. In geometric terms, this corresponds to characterizing the projective linear subvarieties of the Grassmann variety with its Plucker embedding. When the base field is finite, we consider the more general question of determining the maximum number of points on sections of Grassmannians by linear subvarieties of a fixed (co)dimension. This corresponds to a known open problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties. We recover most of the known results as well as prove some new results. In the process we obtain, and utilize, a simple generalization of the Griesmer-Wei bound for arbitrary linear codes.Comment: 16 page

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 Martinez-Penas and Matsumoto for relative generalized rank weights are derived as a consequence.Comment: 22 pages; with minor revisions in the previous versio

Computation of the aa-invariant of ladder determinantal rings

We solve the problem of effectively computing the $a$-invariant of ladder determinantal rings. In the case of a one-sided ladder, we provide a compact formula, while, for a large family of two-sided ladders, we provide an algorithmic solution.Comment: AmS-LaTeX, 20 pages; minor improvements of presentatio

Arithmetic Progressions in a Unique Factorization Domain

Pillai showed that any sequence of consecutive integers with at most 16 terms possesses one term that is relatively prime to all the others. We give a new proof of a slight generalization of this result to arithmetic progressions of integers and further extend it to arithmetic progressions in unique factorization domains of characteristic zero.Comment: Version 2 (to appear in Acta Arithmetica) with minor typos correcte