Lower bounds for the linear complexity of sequences over residue rings
EUROCRYPT '90 Proceedings of the workshop on the theory and application of cryptographic techniques on Advances in cryptology
Discrete Applied Mathematics - Special volume on applied algebra, algebraic algorithms, and error-correcting codes
Algebraic-Geometric Codes
Synthesis of Two-Dimensional Linear Feedback Shift Registers and Groebner Bases
AAECC-5 Proceedings of the 5th International Conference on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
AAECC-6 Proceedings of the 6th International Conference, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Trace-Function on a Galois Ring in Coding Theory
AAECC-12 Proceedings of the 12th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Formal Duality of Linearly Presentable Codes over a Galois Field
AAECC-12 Proceedings of the 12th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
A General Class of Maximal Codes ror Computer Applications
IEEE Transactions on Computers
An upper bound for Weil exponential sums over Galois rings and applications
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Finite multidimensional behaviors
Multidimensional Systems and Signal Processing
On the equivalence of codes over rings and modules
Finite Fields and Their Applications
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We give a short survey of the results obtained in the last several decades that develop the theory of linear codes and polylinear recurrences over finite rings and modules following the well-known results on codes and polylinear recurrences over finite fields. The first direction contains the general results of theory of linear codes, including: the concepts of a reciprocal code and the MacWilliams identity; comparison of linear code properties over fields and over modules; study of weight functions on finite modules, that generalize in some natural way the Hamming weight on a finite field; the ways of representation of codes over fields by linear codes over modules. The second one develops the general theory of polylinear recurrences; describes the algebraic relations between the families of linear recurrent sequences and their periodic properties; studies the ways of obtaining "good" pseudorandom sequences from them. The interaction of these two directions leads to the results on the representation of linear codes by polylinear recurrences and to the constructions of recursive MDS-codes. The common algebraic foundation for the effective development of both directions is the Morita duality theory based on the concept of a quasi-Frobenius module.