  Let $\mathbb{F}_p$ denote the finite field of order $p$, where $p$ is an odd
prime. We study certain quantum negacyclic codes over $\mathbb{F}_p$ which we
call $t$-Frobenius negacyclic codes. We obtain a criterion for constructing
such codes from certain subspaces of $\mathbb{F}_p^n\times\mathbb{F}_p^n$ in
terms of generating pairs of ideals of cyclotomic rings and we completely
classify all linear $t$-Frobenius negacyclic codes. Further, the notion of BCH
distance is extended for these codes and several new codes are identified.
