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Elementary fuzzy Cellular Automata (CA) are known as continuous counterpartof elementary CA, which are 2-state CA, via the polynomial representation oflocal rules. In this paper, we first develop a new fuzzification methodologyfor $q$-state CA. It is based on the vector representation of $q$-state CA,that is, the $q$-states are assigned to the standard basis vectors of the$q$-dimensional real space and the local rule can be expressed by a tuple of$q$ polynomials. Then, the $q$-state vector-valued fuzzy CA are defined byexpanding the set of the states to the convex hull of the standard basisvectors in the $q$-dimensional real space. The vector representation of statesenables us to enumerate the number-conserving rules of 3-state vector-valuedfuzzy CA in a systematic way.

$q$-VFCA: $q$-state Vector-valued Fuzzy Cellular Automata