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

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

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

Schubert Varieties, Linear Codes and Enumerative Combinatorics

We consider linear error correcting codes associated to higher dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult questions in combinatorics and algebraic geometry. This is illustrated by codes associated to Schubert varieties in Grassmannians, called Schubert codes, which have recently been studied. The basic parameters such as the length, dimension and minimum distance of these codes are known only in special cases. An upper bound for the minimum distance is known and it is conjectured that this bound is achieved. We give explicit formulae for the length and dimension of arbitrary Schubert codes and prove the minimum distance conjecture in the affirmative for codes associated to Schubert divisors.Comment: 12 page