A Remark on the Problem of Nonnegative k-Subset Sums

Aydinian H, Blinovsky VM. A Remark on the Problem of Nonnegative k-Subset Sums. Problems Of Information Transmission. 2012;48(4):347-351.Given a set of n real numbers with a nonnegative sum, consider the family of all its k-element subsets with nonnegative sums. How small can the size of this family be? We show that this problem is closely related to a problem raised by Ahlswede and Khachatrian in [1]. The latter, in a special case, is nothing else but the problem of determining a minimal number c(n)(k) such that any k-uniform hypergraph on n vertices having c(n)(k) + 1 edges has a perfect fractional matching. We show that results obtained in [1] can be applied for the former problem. Moreover, we conjecture that these problems have in general the same solution

List Decoding from Erasures: Bounds and Code Constructions

We consider the problem of list decoding from erasures. We establish lower and upper bounds on the rate of a (binary linear) code that can be list decoded with list size L when up to a fraction p of its symbols are adversarially erased. Our results show that in the limit of large L, the rate of such a code approaches the \capacity" (1 p) of the erasure channel. Such nicely list decodable codes are then used as inner codes in a suitable concatenation scheme to give a uniformly constructive family of asymptotically good binary linear codes of that can be eciently list decoded using lists of size O(1=") from up to a fraction (1 ") of erasures, for arbitrary " > 0. This improves previous results from [19] in this vein, which achieved a rate of log(1="))