A Barker sequence is a finite sequence a o , ..., a n -1 , each term ±1, for which every sum Σ i a i a i +k with 0 < k < n is either 0, 1, or -- 1. It is widely conjectured that no Barker sequences of length n > 13 exist, and this conjecture has been verified for the case when n is odd. We show that in this case the problem can in fact be reduced to a question of irreducibility for a certain family of univariate polynomials: No Barker sequence of length 2 m + 1 exists if a particular integer polynomial of degree 4 m is irreducible over Q. A proof of irreducibility for this family would thus provide a short, alternative proof that long Barker sequences of odd length do not exist. However, we also prove that the polynomials in question are always reducible modulo p , for every prime p .
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Irreducible polynomials and barker sequences
Borwein, Peter; Kaltofen, Erich; Mossinghoff, Michael J.
Follow ACM Communications in Computer Algebra
, Volume 41 (4)
Association for Computing Machinery – Dec 1, 2007
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