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Theory of transformation groups of polynomials over \(\mathrm{GF}(2)\) with applications to linear shift register sequences. (English) Zbl 0238.20060
Summary: The group of unimodular transformations on the roots of polynomials over \(\mathrm{GF}(2)\) is considered, and those polynomials with symmetries in the unimodular group are identified. The cross-correlation function between two maximum-length linear shift register sequences of the same degree is shown to be computable as an explicit linear transformation, in matrix form, on either one of the sequences, regarded as a vector. The underlying vector space is the “cyclotomic algebra”, generated by the cyclotomic cosets, or by what Gauss termed the “periods” of the cyclotomic equation. A variety of numerical examples are worked in detail.

94A55 Shift register sequences and sequences over finite alphabets in information and communication theory
11T22 Cyclotomy
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
Full Text: DOI
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