Multiple Criss-Cross Insertion and Deletion Correcting Codes

This paper investigates the problem of correcting multiple criss-cross insertions and deletions in arrays. More precisely, we study the unique recovery of n× n arrays affected by t-criss-cross deletions defined as any combination of tr row and tc column deletions such that tr + tc = t for a given t. We show an equivalence between correcting t-criss-cross deletions and tcriss-cross insertions and show that a code correcting t-criss-cross insertions/deletions has redundancy at least tn+ t log n− log(t!). Then, we present an existential construction of a t-criss-cross insertion/deletion correcting code with redundancy bounded from above by tn + O(t log n). The main ingredients of the presented code construction are systematic binary t-deletion correcting codes and Gabidulin codes. The first ingredient helps locating the indices of the inserted/deleted rows and columns, thus transforming the insertion/deletion-correction problem into a row/column erasure-correction problem which is then solved using the second ingredient.

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