7 problems

Let $q=p^{\ell}$ be a prime power, let $m$ be a positive integer, and let $N$ divide $(q^m-1)/(q-1)$. Let $C_0^{(N,q^m)}$ be the subgroup of $\mathbb{F}_{q^m}^*$ of index $N$, and…

Cyclotomic-class PDS classification conjecture. All partial difference sets formed by a single cyclotomic class take one of three forms: the subfield case, the semi-primitive case,…

Cyclotomic product formula conjecture. For these vectors and matrix,

Let $q=p^s$ be a prime power, let $Q=q^m$, and let $N$ be a divisor of $Q-1$. Write $N_1=(Q-1)/N$ and let $A$ be the circulant matrix defined in equation from the relevant Gaussian…

Let $q=p^{\ell}$ be an odd prime power, let $N\geq 2$ be an even divisor of $q-1$, and let $C_0^{(N,q)}$ be the subgroup of $\mathbb{F}_q^*$ of index $N$. A subset…

Let $q=p^{\ell}$ be an odd prime power, let $N\geq 2$ be an even divisor of $q-1$, and let $C_0^{(N,q)}$ be the subgroup of $\mathbb{F}_q^*$ of index $N$. A subset…

Cyclotomic difference-set classification conjecture. $H_{q,m}$ is a nontrivial difference set in $\mathbb{F}_q$ if and only if one of the following holds: (a) $m=2$ and…