ECCC-Report TR15-116https://eccc.weizmann.ac.il/report/2015/116Comments and Revisions published for TR15-116en-usMon, 20 May 2019 01:41:15 +0300
Revision 1
| Efficient Low-Redundancy Codes for Correcting Multiple Deletions |
Joshua Brakensiek,
Venkatesan Guruswami,
Samuel Zbarsky
https://eccc.weizmann.ac.il/report/2015/116#revision1We consider the problem of constructing binary codes to recover from $k$-bit deletions with efficient encoding/decoding, for a fixed $k$. The single deletion case is well understood, with the Varshamov-Tenengolts-Levenshtein code from 1965 giving an asymptotically optimal construction with $\approx 2^n/n$ codewords of length $n$, i.e., at most $\log n$ bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than $n^{\Omega(1)}$.
For any fixed $k$, we construct a binary code with $c_k \log n$ redundancy that can be decoded from $k$ deletions in $O_k(n \log^4 n)$ time. The coefficient $c_k$ can be taken to be $O(k^2 \log k)$, which is only quadratically worse than the optimal, non-constructive bound of $O(k)$. We also indicate how to modify this code to allow for a combination of up to $k$ insertions and deletions.
Mon, 20 May 2019 01:41:15 +0300https://eccc.weizmann.ac.il/report/2015/116#revision1
Paper TR15-116
| Efficient Low-Redundancy Codes for Correcting Multiple Deletions |
Joshua Brakensiek,
Venkatesan Guruswami,
Samuel Zbarsky
https://eccc.weizmann.ac.il/report/2015/116We consider the problem of constructing binary codes to recover from $k$-bit deletions with efficient encoding/decoding, for a fixed $k$. The single deletion case is well understood, with the Varshamov-Tenengolts-Levenshtein code from 1965 giving an asymptotically optimal construction with $\approx 2^n/n$ codewords of length $n$, i.e., at most $\log n$ bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than $n^{\Omega(1)}$.
For any fixed $k$, we construct a binary code with $c_k \log n$ redundancy that can be decoded from $k$ deletions in $O_k(n \log^4 n)$ time. The coefficient $c_k$ can be taken to be $O(k^2 \log k)$, which is only quadratically worse than the optimal, non-constructive bound of $O(k)$. We also indicate how to modify this code to allow for a combination of up to $k$ insertions and deletions.
We also note that among linear codes capable of correcting $k$ deletions, the $(k+1)$-fold repetition code is essentially the best possible.Tue, 21 Jul 2015 13:34:48 +0300https://eccc.weizmann.ac.il/report/2015/116