(2,1)-separating systems beyond the probabilistic bound
Abstract
Building on previous results of Xing, we give new lower bounds on the rate of intersecting codes over large alphabets. The proof is constructive, and uses algebraic geometry, although nothing beyond the basic theory of linear systems on curves. Then, using these new bounds within a concatenation argument, we construct binary (2,1)-separating systems of asymptotic rate exceeding the one given by the probabilistic method, which was the best lower bound available up to now. This answers (negatively) the question of whether this probabilistic bound was exact, which has remained open for more than 30 years. (By the way, we also give a formulation of the separation property in terms of metric convexity, which may be an inspirational source for new research problems.)
Keywords
Cite
@article{arxiv.1010.5764,
title = {(2,1)-separating systems beyond the probabilistic bound},
author = {Hugues Randriambololona},
journal= {arXiv preprint arXiv:1010.5764},
year = {2012}
}
Comments
Version 7 is a shortened version, so that numbering should match with the journal version (to appear soon). Material on convexity and separation in discrete and continuous spaces has been removed. Readers interested in this material should consult version 6 instead