Enumeration of Linear Transformation Shift Registers

We consider the problem of counting the number of linear transformation shift registers (TSRs) of a given order over a finite field. We derive explicit formulae for the number of irreducible TSRs of order two. An interesting connection between TSRs and self-reciprocal polynomials is outlined. We use this connection and our results on TSRs to deduce a theorem of Carlitz on the number of self-reciprocal irreducible monic polynomials of a given degree over a finite field.Comment: 16 page

Efficient linear feedback shift registers with maximal period

We introduce and analyze an efficient family of linear feedback shift registers (LFSR's) with maximal period. This family is word-oriented and is suitable for implementation in software, thus provides a solution to a recent challenge posed in FSE '94. The classical theory of LFSR's is extended to provide efficient algorithms for generation of irreducible and primitive LFSR's of this new type

Cyclic Quantum Error-Correcting Codes and Quantum Shift Registers

We transfer the concept of linear feed-back shift registers to quantum circuits. It is shown how to use these quantum linear shift registers for encoding and decoding cyclic quantum error-correcting codes.Comment: 18 pages, 15 figures, submitted to Proc. R. Soc.