Multiple Criss-Cross Deletion-Correcting Codes

This paper investigates the problem of correcting multiple criss-cross deletions in arrays. More precisely, we study the unique recovery of n\times n arrays affected by any combination of t_{\mathrm{r}} row and t_{\mathrm{c}} column deletions such that t_{\mathrm{r}}+t_{\mathrm{c}}=t for a given t. We refer to these type of deletions as t-criss-cross deletions. We show that the asymptotic redundancy of a code correcting t-criss-cross deletions is 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+\mathcal{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.