In order to complete (and generalize) results of A. Gardiner and C. E. Praeger [Eur. J. Comb. 15, No. 4, 375--381 (1994)] on 4-valent symmetric graphs 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.
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