Multiple Criss-Cross Deletion-Correcting Codes

Lorenz Welter, Rawad Bitar, Antonia Wachter-Zeh, Eitan Yaakobi

Abstract: This paper investigates the problem of correcting multiple criss-cross deletions in arrays. More precisely, we study the unique recovery of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n\times n$</tex> arrays affected by any combination of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t_{\mathrm{r}}$</tex> row and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t_{\mathrm{c}}$</tex> column deletions such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t_{\mathrm{r}}+t_{\mathrm{c}}=t$</tex> for a given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t$</tex> . 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$tn+t\log n-\log(t!)$</tex> . Then, we present an existential construction of a code capable of correcting t-criss-cross deletions where its redundancy is bounded from above by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$tn+\mathcal{O}(t^{2}\log^{2}n)$</tex> . 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.