We prove new lower bounds for computing some functions $f:\{0,1\}^{n}\to\{0,1\}$ in $E^{NP}$ by polynomials modulo $2$, constant-depth circuits with parity gates ($AC^{0}[\oplus]$), and related classes. Results include:

(1) $\Omega(n/\log^{2}n)$ lower bounds probabilistic degree. This is optimal up to a factor $O(\log^{2}n)$. The previous best lower bound was $\Omega(\sqrt{n})$ proved in ... more >>>

A tree code is an edge-coloring of the complete infinite binary tree such that every two nodes of equal depth have a fraction--bounded away from $0$--of mismatched colors between the corresponding paths to their least common ancestor. Tree codes were introduced in a seminal work by Schulman (STOC 1993) and ... more >>>

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 >>>