Binary $t_1$-Deletion-$t_2$-Insertion-Burst Correcting Codes and Codes Correcting a Burst of Deletions

We first give a construction of binary $t1$-deletion-$t2$-insertion-burst correcting codes with redundancy at most $\log(n)+(t1-t2-1)\log\log(n)+O(1)$, where $t1\ge 2t2$. Then we give an improved construction of binary codes capable of correcting a burst of $4$ non-consecutive deletions, whose redundancy is reduced from $7\log(n)+2\log\log(n)+O(1)$ to $4\log(n)+6\log\log(n)+O(1)$. Lastly, by connecting non-binary $b$-burst-deletion correcting codes with binary $2b$-deletion-$b$-insertion-burst correcting codes, we give a new construction of non-binary $b$-burst-deletion correcting codes with redundancy at most $\log(n)+(b-1)\log\log(n)+O(1)$. This construction is different from previous results.