Related papers: Lower bounds for constant query affine-invariant L…
In this work we explore error-correcting codes derived from the "lifting" of "affine-invariant" codes. Affine-invariant codes are simply linear codes whose coordinates are a vector space over a field and which are invariant under…
Lifted Reed-Solomon codes, a subclass of lifted affine-invariant codes, have been shown to be of high rate while preserving locality properties similar to generalized Reed-Muller codes, which they contain as subcodes. This work introduces a…
Locally Decodable Codes (LDCs) are error-correcting codes $C\colon\Sigma^n\rightarrow \Sigma^m,$ encoding \emph{messages} in $\Sigma^n$ to \emph{codewords} in $\Sigma^m$, with super-fast decoding algorithms. They are important mathematical…
In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist binary LCCs…
A locally correctable code (LCC) is an error correcting code that allows correction of any arbitrary coordinate of a corrupted codeword by querying only a few coordinates. We show that any {\em zero-error} $2$-query locally correctable code…
Ben-Sasson, Goldreich and Sudan showed that a binary error correcting code admitting a $2$-query tester cannot be good, i.e., it cannot have both linear distance and positive rate. The same holds when the alphabet is a finite field…
We give improved lower bounds for binary $3$-query locally correctable codes (3-LCCs) $C \colon \{0,1\}^k \rightarrow \{0,1\}^n$. Specifically, we prove: (1) If $C$ is a linear design 3-LCC, then $n \geq 2^{(1 - o(1))\sqrt{k} }$. A design…
Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affine-invariant property of…
Locally testable codes (LTC) are error-correcting codes that have a local tester which can distinguish valid codewords from words that are "far" from all codewords by probing a given word only at a very few (sublinear, typically constant)…
We prove that a binary linear code of block length $n$ that is locally correctable with $3$ queries against a fraction $\delta > 0$ of adversarial errors must have dimension at most $O_{\delta}(\log^2 n \cdot \log \log n)$. This is almost…
Locally decodable codes (LDCs) are error-correcting codes $C : \Sigma^k \to \Sigma^n$ that admit a local decoding algorithm that recovers each individual bit of the message by querying only a few bits from a noisy codeword. An important…
We prove that for every odd $q\geq 3$, any $q$-query binary, possibly non-linear locally decodable code ($q$-LDC) $E:\{\pm1\}^k \rightarrow \{\pm1\}^n$ must satisfy $k \leq \tilde{O}(n^{1-2/q})$. For even $q$, this bound was established in…
Locally decodable codes (LDC's) are error-correcting codes that allow recovery of individual message indices by accessing only a constant number of codeword indices. For substitution errors, it is evident that LDC's exist -- Hadamard codes…
We prove that the blocklength $n$ of a linear $3$-query locally correctable code (LCC) $\mathcal{L} \colon {\mathbb F}^k \to {\mathbb F}^n$ with distance $\delta$ must be at least $n \geq 2^{\Omega\left(\left(\frac{\delta^2 k}{(|{\mathbb…
A locally recoverable (LRC) code is a code over a finite field $\mathbb{F}_q$ such that any erased coordinate of a codeword can be recovered from a small number of other coordinates in that codeword. We construct LRC codes correcting more…
We present a general framework for constructing high rate error correcting codes that are locally correctable (and hence locally decodable if linear) with a sublinear number of queries, based on lifting codes with respect to functions on…
A code over a finite field is called locally recoverable code (LRC) if every coordinate symbol can be determined by a small number (at most r, this parameter is called locality) of other coordinate symbols. For a linear code with length n,…
We construct the first asymptotically good relaxed locally correctable codes with polylogarithmic query complexity, bringing the upper bound polynomially close to the lower bound of Gur and Lachish (SICOMP 2021). Our result follows from…
We study an approximate version of $q$-query LDCs (Locally Decodable Codes) over the real numbers and prove lower bounds on the encoding length of such codes. A $q$-query $(\alpha,\delta)$-approximate LDC is a set $V$ of $n$ points in…
Lifted Reed-Solomon codes, introduced by Guo, Kopparty and Sudan in 2013, are known as one of the few families of high-rate locally correctable codes. They are built through the evaluation over the affine space of multivariate polynomials…