New Bounds and Constructions for Granular Media Coding

Improved lower and upper bounds on the size and the rate of grain-correcting codes are presented. The lower bound is Gilbert-Varshamov-like combined with a construction by Gabrys et al., and it improves on the previously best known lower bounds on the asymptotic rate of ⌈τn⌉-grain-correcting codes of length n on the interval [0, 0.0668]. One of the two newly presented upper bounds improves on the best known upper bounds on the asymptotic rate of ⌈τn⌉-grain-correcting codes of length n on the interval τ ∈ (0, 1/8] and meets the lower bound of 1/2 for τ ≥ 1/8. Moreover, in a nonasymptotic regime, both upper bounds improve on the previously best known results on the largest size of t-grain-correcting codes of length n, for certain values of n and t. Constructions of 1-grain-correcting codes based on a partitioning technique are presented for lengths up to 18. Finally, a lower bound of 1/2 log₂ n on the minimum redundancy of ∞-grain-detecting codes of length n is presented.