Generalized Barker sequences

Abstract

A generalized Barker sequence is a finite sequence <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{a_{r}\}</tex> of complex numbers having absolute value <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</tex> , and possessing a correlation function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C(\tau)</tex> satisfying the constraint <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|C(\tau)| \leq 1, \tau \neq 0</tex> . Classes of transformations leaving <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|C(\tau)|</tex> invariant are exhibited. Constructions for generalized Barker sequences of various lengths and alphabet sizes are given. Sextic Barker sequences are investigated and examples are given for all lengths through thirteen. No theoretical limit to the length of sextic sequences has been found.