Codes over permutations have received significant attention due to their applications in memory devices and powerline transmission systems. In this paper, we construct a family of permutation codes that are capable of correcting a burst of at most $t$ deletions with $\log n+O(\log\log n)$ bits of redundancy where $t$ is a constant, which significantly improves upon the state-of-the-art construction with redundancy $2t\log n$ bits.