We study shortest paths in directed graphs whose edge weights are of the form $$wt(e) = a_{e,1} \lambda_1 + a_{e,2} \lambda_2 + a_{e,3} \lambda_3 + \cdots + a_{e,d} \lambda_d + a_{e,d+1}.$$
Here, each $a_{e,i}\in\mathbb{R}$ is a fixed constant for each edge $e$, whereas each $\lambda_i$ is a shared variable ... more >>>

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 >>>