t-Deletion-1-Insertion-Burst Correcting Codes
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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-1-insertion-burst ((t,1)-burst for short) proposed by Schoeny et. al, which deletes t consecutive symbols and inserts an arbitrary symbol at the same coordinate. We provide a sphere-packing upper bound on the size of binary codes that can correct (t,1)-burst errors, showing that the redundancy of such codes is at least log n+t-1. An explicit construction of a binary (t,1)-burst correcting code with redundancy log n+(t-2)loglog n+O(1) is given. In particular, we construct a binary (3,1)-burst correcting code with redundancy at most log n+9, which is optimal up to a constant.
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