Michael Viderman

Inspired by recent construction of high-rate locally correctable codes with sublinear query complexity due to

Kopparty, Saraf and Yekhanin (2010) we address the similar question for locally testable codes (LTCs).

In this note we show a construction of high-rate LTCs with sublinear query complexity.

More formally, we show that for ...
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Michael Viderman

Sipser and Spielman (IEEE IT, 1996) showed that any $(c,d)$-regular expander code with expansion parameter $> \frac{3}{4}$ is decodable in \emph{linear time} from a constant fraction of errors. Feldman et al. (IEEE IT, 2007)

proved that expansion parameter $> \frac{2}{3} + \frac{1}{3c}$ is sufficient to correct a constant fraction of ...
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Jonathan Mosheiff, Nicolas Resch, Noga Ron-Zewi, Shashwat Silas, Mary Wootters

We show that Gallager's ensemble of Low-Density Parity Check (LDPC) codes achieve list-decoding capacity. These are the first graph-based codes shown to have this property. Previously, the only codes known to achieve list-decoding capacity were completely random codes, random linear codes, and codes constructed by algebraic (rather than combinatorial) techniques. ... more >>>

Noga Ron-Zewi, Mary Wootters, Gilles Z\'{e}mor

We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let $r > 0$ be any integer. Given an inner code $\cC_0$ of length $d$, and a $d$-regular bipartite expander graph $G$ with $n$ vertices on each side, we give an algorithm to list-decode the expander code $\cC ... more >>>

Oded Goldreich

This text provides a high-level description of the locally testable code constructed by Dinur, Evra, Livne, Lubotzky, and Mozes (ECCC, TR21-151).

In particular, the group theoretic aspects are abstracted as much as possible.

Gil Cohen, Tal Yankovitz

Recently, Kumar and Mon reached a significant milestone by constructing asymptotically good relaxed locally correctable codes (RLCCs) with poly-logarithmic query complexity. Specifically, they constructed $n$-bit RLCCs with $O(\log^{69}n)$ queries. This significant advancement relies on a clever reduction to locally testable codes (LTCs), capitalizing on recent breakthrough works in LTCs.

With ... more >>>