t-Deletion-s-Insertion-Burst Correcting Codes
Abstract
Motivated by applications in DNA-based storage and communication systems, we study deletion and insertion errors simultaneously in a burst. In particular, we study a type of error named t-deletion-s-insertion-burst ((t,s)-burst for short) which is a generalization of the (2,1)-burst error proposed by Schoeny et. al. Such an error deletes t consecutive symbols and inserts an arbitrary sequence of length s at the same coordinate. We provide a sphere-packing upper bound on the size of binary codes that can correct a (t,s)-burst error, showing that the redundancy of such codes is at least n+t-1. For t≥ 2s, an explicit construction of binary (t,s)-burst correcting codes with redundancy n+(t-s-1) n+O(1) is given. In particular, we construct a binary (3,1)-burst correcting code with redundancy at most n+9, which is optimal up to a constant.