Multiple Criss-Cross Insertion and Deletion Correcting Codes

TL;DR

This paper develops new codes capable of correcting multiple criss-cross insertions and deletions in arrays, providing bounds on redundancy and a construction method using binary deletion-correcting and Gabidulin codes.

Contribution

It introduces an equivalence between criss-cross insertions and deletions and proposes a code construction with bounded redundancy using systematic binary and Gabidulin codes.

Findings

Redundancy lower bound: $tn + t \,\log n - \log(t!)$

Existential code construction with redundancy $tn + \mathcal{O}(t^2 \log^2 n)$

Transformation of insertion/deletion correction to erasure correction using systematic codes

Abstract

This paper investigates the problem of correcting multiple criss-cross insertions and deletions in arrays. More precisely, we study the unique recovery of $n \times n$ arrays affected by $t$-criss-cross deletions defined as any combination of $t_r$ row and $t_c$ column deletions such that $t_r + t_c = t$ for a given $t$. We show an equivalence between correcting $t$-criss-cross deletions and $t$-criss-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 $t$-criss-cross insertion/deletion correcting code with redundancy bounded from above by $tn + \mathcal{O}(t^2 \log^2 n)$. 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.