Codes in the Levenshtein metric that can correct multiple edits (including deletions, insertions, and substitutions) have attracted a lot of attention recently owing to their application in DNA-based storage and racetrack memories. In this work, we present new techniques for designing explicit binary codes of length $n$ that correct $t$ edits with at most $(2t+1)\log n+O_t(\log\log n)$ bits of redundancy. We first propose to use a polynomial hash to construct a systematic code that corrects $t$ edits with $(4t+2)\log n+O_t(\log\log n)$ bits of redundancy. Then we observe that there are many polynomial hashes like this, some are good, some are bad. So, we propose the second technique, called entropy recycling, to use the indices of good hashes to carry information. As a consequence, we obtain a partial-systematic code of length $n$ that corrects $t$ edits with at most $(2t+1)\log n+O_t(\log\log n)$ bits of redundancy. Finally, we provide deterministic polynomial-time encoding and decoding algorithms for our code.

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