We prove that any linear $3$-query locally correctable code $C \colon \mathbb{F}_2^k \to \mathbb{F}_2^n$ with constant correction radius must have length $n \geq 2^{\Omega(\sqrt{k})}$. This improves on the prior best lower bound of $2^{\Omega(\sqrt{k/\log k})}$, due to [AG24], and shows that the folklore construction of linear $3$-LCCs from Reed--Muller codes, ... more >>>

Symmetry of Information (SoI) [ZL70] is a fundamental theorem of classical Kolmogorov complexity whose status for time-bounded computation is closely related to central open problems in complexity theory and cryptography. In a recent development, [HMO26] showed that SoI fails for $rKt$ complexity, namely in the setting of randomized computation. By ... more >>>

The sign rank of a ${\pm 1}$-valued matrix $A$ is the minimum rank of a real matrix that agrees with $A$ entrywise in sign. For every sufficiently large $d$, we give an explicit matrix $A$ with $2^{\Theta(d^2)}$ rows and columns and sign rank $\Theta(d)$ such that $A \land A$ has ... more >>>