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 \lceil \tau n \rceil -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 \lceil \tau n \rceil -grain-correcting codes of length n on the interval \tau \in (0,{1}/{8}] and meets the lower bound of {1}/{2} for \tau \geq 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 on the minimum redundancy of $\infty $ -grain-detecting codes of length $n$ is presented. [ABSTRACT FROM PUBLISHER]