  In this paper, we study cyclic stabiliser codes over $\mathbb{F}_p$ of length
dividing $p^t+1$ for some positive integer $t$. We call these $t$-Frobenius
codes or just Frobenius codes for short. We give methods to construct them and
show that they have efficient decoding algorithms. An important subclass of
stabiliser codes are the linear stabiliser codes. For linear Frobenius codes we
have stronger results: We completely characterise all linear Frobenius codes.
As a consequence, we show that for every integer $n$ that divides $p^t+1$ for
an odd $t$, there are no linear cyclic codes of length $n$. On the other hand
for even $t$, we give an explicit method to construct all of them. This gives
us a many explicit example of Frobenius codes which include the well studied
Laflamme code. We show that the classical notion of BCH distance can be
generalised to all the Frobenius codes that we construct, including the
non-linear ones, and show that the algorithm of Berlekamp can be generalised to
correct quantum errors within the BCH limit. This gives, for the first time, a
family of codes that are neither CSS nor linear for which efficient decoding
algorithm exits. The explicit examples that we construct are summarised in
Table \ref{tab:explicit-examples-short} and explained in detail in Tables
\ref{tab:explicit-examples-2} (linear case) and \ref{tab:explicit-examples-3}
(non-linear case).
