A Lower Bound on List Size for List Decoding

A q-ary error-correcting code C ⊆ {1, 2,..., q} n is said to be list decodable to radius ρ with list size L if every Hamming ball of radius ρ contains at most L codewords of C. We prove that in order for a q-ary code to be list-decodable up to radius (1 − 1/q)(1 − ε)n, we must have L = Ω(1/ε 2). Specifically, we prove that there exists a constant cq> 0 and a function fq such that for small enough ε> 0, if C is list-decodable to radius (1 − 1/q)(1 − ε)n with list size cq/ε 2, then C has at most fq(ε) codewords, independent of n. This result is asymptotically tight (treating q as a constant), since such codes with an exponential (in n) number of codewords are known for list size L = O(1/ε 2). A result similar to ours is implicit in Blinovsky [Bli1] for the binary (q = 2) case. Our proof is simpler and works for all alphabet sizes, and provides more intuition for why the lower bound arises.

On the number of step functions with restrictions

Ahlswede R, Blinovsky V. On the number of step functions with restrictions. THEORY OF PROBABILITY AND ITS APPLICATIONS. 2005;50(4):537-560.We obtain an asymptotic formula for the number of scaled step functions with restrictions on the length and height of steps (shapes of Young diagrams) of a given area in the neighborhood of a given curve. This allows us to find the asymptotics of the whole number of such functions and find the limit shape - the curve of concentration of the step functions

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="))