Linear and polylinear recurring sequences over abelian groups and modulesстатья
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Аннотация:In this paper, the well-known results in the theory of polylinear recurring sequences over fields and their recent generalizations for sequences over modules with commutative rings of coefficients are extended to the class of polylinear sequences over modules with noncommutative rings of coefficients. Possible noncommutativity of the main ring of coefficients requires naturally a study of polylinear recurring sequences over bimodules. To estimate the linear complexity of the considered sequences, we introduce and study polylinear (k-linear) shift registers. A criterion for the theory of polylinear recurring sequences over fields to be adequately generalized in our case is the property of the main bimodule to be quasi-Frobenius with the Artinian (respectively, from the left and the right sides) rings of coefficients (i.e., so-called Artinian duality context).
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 74, Algebra-15, 2000.