A Matching Vector ($\mathbf{MV}$) family modulo a positive integer $m\ge 2$ is a
pair of ordered lists $U=(u_1,\ldots,u_K)$ and $V=(v_1,\ldots,v_K)$ with
$u_i,v_j\in \Z_m^n$ such that $\langle u_i,v_i\rangle=0 \pmod m$ for every
$i\in[K]$, while $\langle u_i,v_j\rangle\ne 0 \pmod m$ for every $i\ne j$. It
is called $r$-restricted if the set of ... more >>>

We derive the four principal asymptotic rate-distance tradeoffs for binary codes---Plotkin, Elias--Bassalygo, and the two McEliece--Rodemich--Rumsey--Welch (MRRW) bounds---from one theorem, the ``pretty good criterion.'' If the bit error rate under the pretty good measurement (PGM)---the quantum analog of posterior sampling---of a binary-input output-symmetric classical--quantum (cq) channel lies below $\delta$, then ... more >>>

We prove lower bounds for $k$-OV, $k$-XOR, and $k$-SUM in nonuniform AC$^0$, tracking how the circuit-size exponent scales with $k$ and using no running-time hypothesis. Our framework gives depth-zero projections from colored subgraph isomorphism to the three targets at dimension, row count, or bit width $O(k\log n)$, without increasing depth ... more >>>