1 Bounds and Constructions for Granular Media Coding

Abstract

Abstract—Bounds on the rates of grain-correcting codes are presented. The lower bounds are Gilbert–Varshamov-like ones, whereas the upper bounds improve on the previously known result by Mazumdar et al. Constructions of t-grain-correcting codes of length n for certain values of n and t are discussed. Finally, an infinite family of codes of rate approaching 1 that can detect an arbitrary number of grain errors is shown to exist. Index Terms—convex optimization, Gilbert–Varshamov bound, grain-correcting codes, granular media, lower bounds, magnetic recording, Markov chain, upper bounds. I