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This paper presents a method for the synthesis of a one-dimensional linear hybrid cellular automaton (CA) from a given irreducible polynomial. A detailed description of the algorithm is given, together with an outline of the theoretical background. It is shown that two CA exist for each irreducible polynomial, solving the previously open CA existence conjecture. An in-depth example of the synthesis is presented, along with timing benchmarks and an operation count. The algorithm solves the previously open problem of synthesizing CA for all practical applications