Two Dimensional Deletion Correcting Codes and their Applications

Two dimensional (2D) error correcting codes have been investigated for a long time owing to their numerous applications. Recently, 2D codes correcting row-deletions and column-deletions, also known as criss-cross deletion correcting codes, have been studied as a generalisation of one dimensional deletion correcting codes. In this work, we show that 2D deletion correcting codes are useful to correct errors in racetrack memories. With motivation from both theoretical and practical point of view, we study these 2D codes and aim to improve the previous known results. Our first main result is a construction of an optimal (1,1)-criss-cross deletion correcting code with the redundancy is at most $2n+2\log n+o(\log n)$ bits. Then, we also present a construction of an asymptotic optimal $(t_{r},\ t_{c})$ -criss-cross deletion correcting code with less redundancy than the best known results. Furthermore, since a 2D binary code correcting multiple row-deletions is equivalent to a I1D q-ary code correcting multiple deletions with large $q$, we also improve some previous known results on 1D q-ary code correcting multiple deletions.