Criss-Cross Deletion Correcting Codes

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

This paper studies the problem of constructing codes correcting deletions in arrays. Under this model, it is assumed that an $n\times n$ array can experience deletions of rows and columns. These deletion errors are referred to as $(t_{\mathrm{r}},t_{\mathrm{c}})$-criss-cross deletions if $t_{\mathrm{r}}$ rows and $t_{\mathrm{c}}$ columns are deleted, while a code correcting these deletion patterns is called a $(t_{\mathrm{r}},t_{\mathrm{c}})$-criss-cross deletion correction code. The definitions for \emph{criss-cross insertions} are similar. Similar to the one-dimensional case, it is first shown that when $t_{\mathrm{r}}=t_{\mathrm{c}}$ the problems of correcting criss-cross deletions and criss-cross insertions are equivalent. Then, we mostly investigate the case of $(1,1)$-criss-cross deletions. An asymptotic upper bound on the cardinality of $(1,1)$-criss-cross deletion correction codes is shown which assures that the asymptotic redundancy is at least $2n-1+2\log n$ bits. Finally, a code construction with an explicit decoding algorithm is presented. The redundancy of the construction is away from the lower bound by at most $2 \log n+8+2\log e$ bits.

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