Explicit and Efficient Constructions of Linear Codes Against Adversarial Insertions and Deletions

  title={Explicit and Efficient Constructions of Linear Codes Against Adversarial Insertions and Deletions},
  author={Roni Con and Amir Shpilka and Itzhak Tamo},
  journal={IEEE Transactions on Information Theory},
In this work, we study linear error-correcting codes against adversarial insertion-deletion (insdel) errors, a topic that has recently gained a lot of attention. We construct linear codes over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula>, for <inline-formula> <tex-math notation="LaTeX">$q= {\mathrm {poly}}(1/\varepsilon)$ </tex-math></inline-formula>, that can efficiently decode from a <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math… 



Efficient Low-Redundancy Codes for Correcting Multiple Deletions

This work considers the problem of constructing binary codes to recover from bit deletions with efficient encoding/decoding, and constructs a binary code with redundancy that can be decoded from <inline-formula> <tex-math notation="LaTeX">$k$ </tex-Math></inline- formula>-fold repetition code.

Explicit Two-Deletion Codes With Redundancy Matching the Existential Bound

An explicit construction of length-based binary codes capable of correcting the deletion of two bits that have size, based on augmenting the classic Varshamov-Tenengolts construction of single deletion codes with additional check equations.

Improved Constructions of Coding Schemes for the Binary Deletion Channel and the Poisson Repeat Channel

This work gives an explicit construction of a family of error correcting codes for the binary deletion channel and for the Poisson repeat channel that have polynomial time encoding and decoding algorithms.

Deletion Codes in the High-Noise and High-Rate Regimes

The first efficient constructions which meet the qualitative goals of correcting a deletion fraction approaching 1 over bounded alphabets, and correcting a constant fraction of bit deletions with rate approach 1 over a fixed alphabet are given.

The zero-rate threshold for adversarial bit-deletions is less than 1/2

It is proved that there exists an absolute constant 6 > 0 such any binary code tolerating (1/2 - δ) N tolerating adversarial deletions must satisfy C ≤ 2<sup>polylog</sup><tex>$N$</tex> and thus have rate asymptotically approaching 0.

Efficient Linear and Affine Codes for Correcting Insertions/Deletions

This paper disprove the existence of binary linear codes of length $n$ and rate just below $1/2$ capable of correcting $\Omega(n)$ insertions and deletions and shows that the $\frac{1}{2}$-rate limitation does not hold for affine codes by giving an explicit affine code of rate $1-\epsilon$ which can efficiently correct a constant fraction of insdel errors.

Synchronization strings: codes for insertions and deletions approaching the Singleton bound

This paper focuses on designing insdel codes, i.e., error correcting block codes (ECCs) for insertion deletion channels, and introduces synchronization strings, which provide a novel way of efficiently dealing with synchronization errors.

A low-complexity algorithm for the construction of algebraic-geometric codes better than the Gilbert-Varshamov bound

This work presents the first low-complexity algorithm for obtaining the generator matrix for AG codes on the curves of GS, and suggests that by concatenating the AG code with short binary block codes, it is possible to obtain binary codes with asymptotic performance close to the G-V bound.

Deterministic Document Exchange Protocols, and Almost Optimal Binary Codes for Edit Errors

This paper gives an efficient deterministic protocol with sketch size O(k log^2 n/k) and obtains the first explicit construction of binary insdel codes that has optimal redundancy for a wide range of error parameters k, and brings the understanding of binaryinsdel codes much closer to that of standard binary error correcting codes.

Synchronization Strings and Codes for Insertions and Deletions—A Survey

The recent progress in designing efficient error-correcting codes over finite alphabets that can correct a constant fraction of worst-case insertions and deletions is surveyed.