Criss-Cross Insertion and Deletion Correcting Codes

This paper studies the problem of constructing codes correcting deletions in arrays. Under this model, it is assumed that an n × n array can experience deletions of rows and columns. These deletion errors are referred to as (tr, {tc)-criss-cross deletions if tr rows and tc columns are deleted, while a code correcting these deletion patterns is called a (tr, tc)-criss-cross deletion correction code. The definitions for criss-cross insertions are similar. It is first shown that when tr=tc the problems of correcting criss-cross deletions and criss-cross insertions are equivalent. The focus of this paper lies on the case of (1, 1)-criss-cross deletions. A non-asymptotic upper bound on the cardinality of (1, 1)-criss-cross deletion correction codes is shown which assures that the redundancy is at least 2n-3+2 log n bits. A code construction with an existential encoding and an explicit decoding algorithm is presented. The redundancy of the construction is at most 2n+4 log n + 7 +2 log e. A construction with explicit encoder and decoder is presented. The explicit encoder adds an extra 5 log n + 5 bits of redundancy to the construction.