On the Existence of MDS Codes Over Small Fields With Constrained Generator Matrices

We study the existence over small fields of Maximum Distance Separable (MDS) codes with generator matrices having specified supports (i.e. having specified locations of zero entries). This problem unifies and simplifies the problems posed in recent works of Yan and Sprintson (NetCod'13) on weakly secure cooperative data exchange, of Halbawi et al. (arxiv'13) on distributed Reed-Solomon codes for simple multiple access networks, and of Dau et al. (ISIT'13) on MDS codes with balanced and sparse generator matrices. We conjecture that there exist such $[n,k]_q$ MDS codes as long as $q \geq n + k - 1$, if the specified supports of the generator matrices satisfy the so-called MDS condition, which can be verified in polynomial time. We propose a combinatorial approach to tackle the conjecture, and prove that the conjecture holds for a special case when the sets of zero coordinates of rows of the generator matrix share with each other (pairwise) at most one common element. Based on our numerical result, the conjecture is also verified for all $k \leq 7$. Our approach is based on a novel generalization of the well-known Hall's marriage theorem, which allows (overlapping) multiple representatives instead of a single representative for each subset.Comment: 8 page

Orthogonal Bases of Invariants in Tensor Models

Representation theory provides a suitable framework to count and classify invariants in tensor models. We show that there are two natural ways of counting invariants, one for arbitrary rank of the gauge group and a second, which is only valid for large N. We construct bases of invariant operators based on the counting, and compute correlators of their elements. The basis associated with finite N diagonalizes the two-point function of the theory and it is analogous to the restricted Schur basis used in matrix models. We comment on future lines of investigation.Comment: Two overlapping but independent results are merged to a joint work. 16 pages, 1 tabl

Multiple Flag Varieties of Finite Type

We classify all products of flag varieties with finitely many orbits under the diagonal action of the general linear group. We also classify the orbits in each case and construct explicit representatives. This generalizes the classical Schubert decompostion, which states that the GL(n)-orbits on a product of two flag varieties correspond to permutations. Our main tool is the theory of quiver representations.Comment: 18pp. to appear in Adv. Mat

Efficient and Equitable Airport Slot Allocation

This paper studies slot allocation at congested airports in Europe. First, I discuss the inefficiencies of the current regulation, introduced as part of the liberalisation process of the air transport market. Then, I consider three marked based methods which are suitable to achieve a more efficient allocation of slots to airlines: congestion pricing, auctions and secondary trading. These methods are examined in terms of their ability to improve efficiency and in terms of their implications on the distribution of slots’ scarcity rents. Special attention is drawn to complementarities between slots. Finally, I propose to auction slots periodically, allowing secondary trading well before the first auction takes place. By selling slots before the first auction incumbents can be partially compensated for the subsequent withdrawal of their slots.