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

We first give a construction of binary $t_1$-deletion-$t_2$-insertion-burst correcting codes with redundancy at most $\log (n)+(t_1-t_2-1)\log \log (n)+O(1)$, where $t_1\ge 2t_2$. 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.