9 problems

Let $\mathcal{P}_{d}({\mathbb{F}}^n)$ be the class of polynomials of degree at most $d$ from ${\mathbb{F}}^n$ to ${\mathbb{F}}$. For a function $g:{\mathbb{F}}^n\to{\mathbb{F}}$, l…

Let $RM(r,m)$ denote the Reed–Muller code of order $r$ in $m$ variables, and let its weight spectrum be the set of Hamming weights of its codewords. For a set $S$ of weights, write…

Let $m\geq 3$. For the binary lifted Reed–Muller code ${\mathrm{RM}}_2(1,m)(2|2^2)$, let ${\mathbb{D}}_{3\times 2^{m-2}}({\mathrm{RM}}_2(1,m)(2|2^2))$ denote the design whose block…

Construction 2 weight-spectrum conjecture. The set $S$ contains all weights in the two families

Automorphism-group conjecture. For $q>r>\frac{n(n-1)}{2}$, the automorphism group of $SRM_q[n,r]$ contains a subgroup isomorphic to $M$; equivalently, $SRM_q[n,r]$ is invariant und…

Let $\mathcal{P}$ be a binary-input memoryless symmetric (BMS) channel with output alphabet $\mathcal{Y}$, and let … be its capacity under the uniform input distribution. For a seq…

Let $m\ge 5$ be an integer, let $\alpha$ be a primitive element of ${\mathrm{GF}}(2^m)$, and let $g^*_{2,\alpha}(X)$ denote the generator polynomial of the shortened second-order R…

Gopalan–Klivans–Zuckerman conjecture. For fixed $d$ and finite field $\mathbb{K}$,

Let $d\geq 1$ and let $\varepsilon>0$. For each $n\geq 1$, write $\mathbf{F}_2^n$ for the Boolean vector space, let $f:\mathbf{F}_2^n\to[-1,1]$, let $\|f\|_{U^d(\mathbf{F}_2^n)}$ d…