Location-correcting codes

We study codes over GF (q) that can correct t channel errors assuming the error values are known. This is a counterpart to the well-known problem of erasure correction, where error values are found assuming the locations are known. The correction capabilities of these so called t-location correcting codes (t-LCCs) are characterized by a new metric, the decomposability distance, which plays a role analogous to that of the Hamming metric in conventional error-correcting codes (ECCs). Based on the new metric, we prove bounds on the parameters of t-LCCs that are counterparts to the classical Singleton, sphere packing and GilbertVarshamov bounds for ECCs. In particular, we show examples of perfect LCCs, and we study optimal (MDS-like) LCCs that attain the Singleton-type bound on the redundancy. We show that these optimal codes are generally much shorter than their erasure (or conventional ECC) analogues: The length n of any t-LCC that attains the Singleton-type bound for t > 1 is bounded from above by t + O( √ q), compared to length q+1 which is attainable in the conventional ECC case. We show constructions of optimal t-LCCs for t ∈ {1, 2, n−2, n−1, n} that attain the asymptotic length upper bounds, and constructions for other values of t that are optimal, yet their lengths fall short of the upper bounds. The resulting asymptotic gap remains an open research problem. All the constructions presented can be efficiently decoded.