This paper presents a method for the synthesis of a one-dimensional linear hybrid cellula from a given irreducible polynomial. A detailed description of the algorithm is given, toget the theoretical background. It is shown that two CA exist for each irreducible polynomial, open CA existence conjecture. An in-depth example of the synthesis is presented, along benchmarks and an operation count. The algorithm solves the previously open problem o all practical applications Index Terms Inspec Controlled Indexing cellular automata finite state machines polynomials Non-controlled Indexing CA existence conjecture irreducible polynomial linear finite state machines o dimensional linear hybrid cellular automata operation count timing benchmark Author Keywords Not Available References 1 P. H. Bardell, "Analysis of cellular automata used as pseudorandom pattern generato Test Conf., pp. 762-767, 1990.

Many applications call for exhaustive lists of strings subject to various constraints, such as inequivalence under group actions. A k-ary necklace is an Ž . equivalence class of k-ary strings under rotation the cyclic group . A k-ary unlabeled necklace is an equivalence class of k-ary strings under rotation and permutation of alphabet symbols. We present new, fast, simple, recursive algo-Ž . rithms for generating i.e., listing all necklaces and binary unlabeled necklaces. These algorithms have optimal running times in the sense that their running times are proportional to the number of necklaces produced. The algorithm for generating necklaces can be used as the basis for efficiently generating many other equivalence classes of strings under rotation and has been applied to generating bracelets, fixed density necklaces, and chord diagrams. As another application, we describe the implementation of a fast algorithm for listing all degree n irreducible Ž . and primitive polynomials over GF 2 . ᮊ

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