  In order to complete (and generalize) results of Gardiner and Praeger on
4-valent symmetric graphs (European J. Combin, 15 (1994)) we apply the method
of lifting automorphisms in the context of elementary-abelian covering
projections. In particular, the vertex- and edge-transitive graphs whose
quotient by a normal $p$-elementary abelian group of automorphisms, for $p$ an
odd prime, is a cycle, are described in terms of cyclic and negacyclic codes.
Specifically, the symmetry properties of such graphs are derived from certain
properties of the generating polynomials of cyclic and negacyclic codes, that
is, from divisors of $x^n \pm 1 \in {\mathbb Z}_p[x]$. As an application, a
short and unified description of resolved and unresolved cases of Gardiner and
Praeger are given.
