Exploring the missing link among d-separable, d¯-separable and d-disjunct matrices

Abstractd-Disjunct matrices, d¯-separable matrices and d-separable matrices are well studied in various problems including group testing, coding, extremal set theory and, recently, DNA sequencing. The implications from the first two matrices to the last one are well documented. This paper gives an implication of the other direction for the first time

2-cancellative hypergraphs and codes

A family of sets F (and the corresponding family of 0-1 vectors) is called t-cancellative if for all distict t+2 members A_1,... A_t and B,C from F the union of A_1,..., A_t and B differs from the union of A_1, ..., A_t and C. Let c(n,t) be the size of the largest t-cancellative family on n elements, and let c_k(n,t) denote the largest k-uniform family. We significantly improve the previous upper bounds, e.g., we show c(n,2) n_0). Using an algebraic construction we show that the order of magnitude of c_{2k}(n,2) is n^k for each k (when n goes to infinity).Comment: 20 page

Improving Uniquely Decodable Codes in Binary Adder Channels

We present a general method to modify existing uniquely decodable codes in the $T$-user binary adder channel. If at least one of the original constituent codes does not have average weight exactly half of the dimension, then our method produces a new set of constituent codes in a higher dimension, with a strictly higher rate. Using our method we improve the highest known rate for the $T$-user binary adder channel for all $T \geq 2$. This information theory problem is equivalent to co-Sidon problems initiated by Lindstr{\"o}m in the 1960s, and also the multi-set union-free problem. Our results improve the known lower bounds in these settings as well.Comment: 8 page

A Better Bound for Locally Thin Set Families

AbstractA family of subsets of an n-set is 4-locally thin if for every quadruple of its members the ground set has at least one element contained in exactly 1 of them. We show that such a family has at most 20.4561n members. This improves on our previous results with Noga Alon. The new proof is based on a more careful analysis of the self-similarity of the graph associated with such set families by the graph entropy bounding technique