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
$n \times n$
arrays affected by
${t}$
-criss-cross deletions
defined as 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 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
${t} n + {t}\log n - \log ({t}!)$
. Then, we present an existential construction of a
${t}$
-criss-cross insertion/deletion correcting code with redundancy bounded from above by
${t} n + \mathcal {O}({t}^{2} \log ^{2} n)$
. 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.