Lower bounds for constant query affine-invariant LCCs and LTCs

Affine-invariant codes are codes whose coordinates form a vector space over a finite field and which are invariant under affine transformations of the coordinate space. They form a natural, well-studied class of codes; they include popular codes such as Reed-Muller and Reed-Solomon. A particularly appealing feature of affine-invariant codes is that they seem well-suited to admit local correctors and testers. In this work, we give lower bounds on the length of locally correctable and locally testable affine-invariant codes with constant query complexity. We show that if a code $\mathcal{C} \subset \Sigma^{\mathbb{K}^n}$ is an $r$-query locally correctable code (LCC), where $\mathbb{K}$ is a finite field and $\Sigma$ is a finite alphabet, then the number of codewords in $\mathcal{C}$ is at most $\exp(O_{\mathbb{K}, r, |\Sigma|}(n^{r-1}))$. Also, we show that if $\mathcal{C} \subset \Sigma^{\mathbb{K}^n}$ is an $r$-query locally testable code (LTC), then the number of codewords in $\mathcal{C}$ is at most $\exp(O_{\mathbb{K}, r, |\Sigma|}(n^{r-2}))$. The dependence on $n$ in these bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty and Sudan (ITCS `13) construct affine-invariant codes via lifting that have the same asymptotic tradeoffs. Note that our result holds for non-linear codes, whereas previously, Ben-Sasson and Sudan (RANDOM `11) assumed linearity to derive similar results. Our analysis uses higher-order Fourier analysis. In particular, we show that the codewords corresponding to an affine-invariant LCC/LTC must be far from each other with respect to Gowers norm of an appropriate order. This then allows us to bound the number of codewords, using known decomposition theorems which approximate any bounded function in terms of a finite number of low-degree non-classical polynomials, upto a small error in the Gowers norm

High rate locally-correctable and locally-testable codes with sub-polynomial query complexity

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 and LTCs with block length $n$, constant rate (which can even be taken arbitrarily close to 1), constant relative distance, and query complexity $\exp(\tilde{O}(\sqrt{\log n}))$. Previously such codes were known to exist only with $\Omega(n^{\beta})$ query complexity (for constant $\beta > 0$), and there were several, quite different, constructions known. Our codes are based on a general distance-amplification method of Alon and Luby~\cite{AL96_codes}. We show that this method interacts well with local correctors and testers, and obtain our main results by applying it to suitably constructed LCCs and LTCs in the non-standard regime of \emph{sub-constant relative distance}. Along the way, we also construct LCCs and LTCs over large alphabets, with the same query complexity $\exp(\tilde{O}(\sqrt{\log n}))$, which additionally have the property of approaching the Singleton bound: they have almost the best-possible relationship between their rate and distance. This has the surprising consequence that asking for a large alphabet error-correcting code to further be an LCC or LTC with $\exp(\tilde{O}(\sqrt{\log n}))$ query complexity does not require any sacrifice in terms of rate and distance! Such a result was previously not known for any $o(n)$ query complexity. Our results on LCCs also immediately give locally-decodable codes (LDCs) with the same parameters

List Decoding Tensor Products and Interleaved Codes

We design the first efficient algorithms and prove new combinatorial bounds for list decoding tensor products of codes and interleaved codes. We show that for {\em every} code, the ratio of its list decoding radius to its minimum distance stays unchanged under the tensor product operation (rather than squaring, as one might expect). This gives the first efficient list decoders and new combinatorial bounds for some natural codes including multivariate polynomials where the degree in each variable is bounded. We show that for {\em every} code, its list decoding radius remains unchanged under $m$-wise interleaving for an integer $m$. This generalizes a recent result of Dinur et al \cite{DGKS}, who proved such a result for interleaved Hadamard codes (equivalently, linear transformations). Using the notion of generalized Hamming weights, we give better list size bounds for {\em both} tensoring and interleaving of binary linear codes. By analyzing the weight distribution of these codes, we reduce the task of bounding the list size to bounding the number of close-by low-rank codewords. For decoding linear transformations, using rank-reduction together with other ideas, we obtain list size bounds that are tight over small fields.Comment: 32 page

Combinatorial Construction of Locally Testable Codes

An error correcting code is said to be locally testable if there is a test that checks whether a given string is a codeword, or rather far from the code, by reading only a constant number of symbols of the string. While the best known construction of LTCs by Ben-Sasson and Sudan (STOC 2005) and Dinur (J. ACM 54(3)) achieves very e cient parameters, it relies heavily on algebraic tools and on PCP machinery. In this work we present a new and arguably simpler construction of LTCs that is purely combinatorial, does not rely on PCP machinery and matches the parameters of the best known construction. However, unlike the latter construction, our construction is not entirely explicit

Single-shot decoding of good quantum LDPC codes

Quantum Tanner codes constitute a family of quantum low-density parity-check (LDPC) codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Z\'emor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols.Comment: 35 pages, 3 figure