Multiple Criss-Cross Deletion-Correcting Codes

02/04/2021 ∙ by Lorenz Welter, et al. ∙ 0

This paper investigates the problem of correcting multiple criss-cross deletions in arrays. More precisely, we study the unique recovery of n × n arrays affected by any combination of t_r row and t_c column deletions such that t_r + t_c = t for a given t. We refer to these type of deletions as t-criss-cross deletions. We show that a code capable of correcting t-criss-cross deletions has redundancy at least tn + t log n - log(t!). Then, we present an existential construction of a code capable of correcting t-criss-cross deletions where its redundancy is bounded from above by tn + 𝒪(t^2 log^2 n). The main ingredients of the presented code are systematic binary t-deletion correcting codes and Gabidulin codes. The first ingredient helps locating the indices of the deleted rows and columns, thus transforming the deletion-correction problem into an erasure-correction problem which is then solved using the second ingredient.

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