  In this paper, we present an infinite dimensional associative diagram algebra
that satisfies the relations of the generalized Temperley--Lieb algebra having
a basis indexed by the fully commutative elements (in the sense of Stembridge)
of the Coxeter group of type affine $C$. Moreover, we provide an explicit
description of a basis for the diagram algebra. In the sequel to this paper, we
show that this diagrammatic representation is faithful. The results of this
paper and its sequel will be used to construct a Jones-type trace on the Hecke
algebra of type affine $C$, allowing us to non-recursively compute leading
coefficients of certain Kazhdan--Lusztig polynomials.
