Technical Report CS0850

Authors: R.M. Roth and G. Seroussi
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Abstract: We study codes over $GF(q)$ that can correct $t$ channel errors assuming the error {\em 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 {\em 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 Gilbert-Varshamov 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 $t+O(\sqrt{q})$, compared to length $q+1$ which is attainable in the conventional ECC case. We show constructions of optimal $t$-LCCs for $t \in \{1,2,n-2,n-1,n\}$ 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.
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