In this primarily expository
paper, we discuss the connections between two popular and useful
tools in theoretical computer science, namely,
universal hashing and pairwise
independent random variables; and classical combinatorial stuctures
such as error-correcting codes, balanced incomplete block designs,
difference matrices
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We bound the minimum number $w$ of wires needed to compute any (asymptotically good) error-correcting code
$C:\{0,1\}^{\Omega(n)} \to \{0,1\}^n$ with minimum distance $\Omega(n)$,
using unbounded fan-in circuits of depth $d$ with arbitrary gates. Our main results are:

(1) If $d=2$ then $w = \Theta(n ({\log n/ \log \log n})^2)$.

(2) ... more >>>

We establish the first super-polynomial quantum advantage for the tolerant junta testing problem in the adaptive setting. Specifically, we show that within a certain parameter regime, tolerant $k$-junta testing with high precision can be solved using $\mathrm{poly}(k)$ quantum queries, whereas any classical algorithm requires at least $k^{\Omega(\log k)}$ queries.

The ... more >>>

We describe a method that lifts an arbitrary polynomial $f$ with sparsity $s$ to a polynomial that requires a read-once oblivious algebraic program of width $s$ for every variable order. To do so, we introduce a technique for constructing a gadget based on erasure codes over finite fields, where each ... more >>>