Many hash-based proof systems check that committed words are close to Reed-Solomon codewords by testing one random combination of them. Their soundness analysis uses a proximity gap: if the combination agrees with a codeword on many positions, so do the original words, on the same positions, except for a few ... more >>>

For a linear code $C\subseteq\mathbb{F}_q^n$, we say that $C$ satisfies the \emph{proximity-gaps property} up to distance $\delta_1$ if, for every $\delta_2>\delta_1$ and every $f,g\in\mathbb{F}_q^n$, at least one of which is $\delta_2$-far from $C$ in relative Hamming distance, there are only a small fraction (typically at most $\operatorname{poly}(n)/q$) of \emph{exceptional coefficients} ... more >>>

We provide a deterministic polynomial-time list decoding algorithm for Reed-Solomon codes over prime fields that approaches list decoding capacity on every evaluation set in the low (constant) rate regime.

The line--versus--point test asks the following local-to-global question. Suppose a function $f\colon\mathbb{F}_q^m\to\mathbb{F}_q$ is, on average over a random affine line $L$, correlated with some degree-$d$ polynomial on $L$. Must $f$ then be globally correlated with a single multivariate polynomial of degree at most $d$? Beyond being a natural combinatorial question, ... more >>>

We construct $3$-query relaxed locally decodable codes (RLDCs) with constant alphabet size and length $\tilde{O}(k^2)$ for $k$-bit messages. Combined with the lower bound of $\tilde{\Omega}(k^3)$ of [Alrabiah, Guruswami, Kothari, Manohar, STOC 2023] on the length of locally decodable codes (LDCs) with the same parameters, we obtain a separation between RLDCs ... more >>>

We show that for all $\varepsilon>0$, for sufficiently large prime power $q\in\mathbb{N}$, for all $\delta>0$, it is NP-hard to distinguish whether a $2$-Prover-$1$-Round projection game with alphabet size $q$ has value at least $1-\delta$, or value at most $1/q^{1-\varepsilon}$. This establishes a nearly optimal alphabet-to-soundness tradeoff for $2$-query PCPs ... more >>>