Barker sequences and difference sets
Let \(A=(a_ 1,\dots,a_ v)\) be a binary sequence (i.e. \(a_ i=\mp 1)\) such that the sums \(\gamma=\sum^ v_{i=1}a_ ia_{i+j}\) \((j=1,\dots,v-1)\) reading the indices modulo \(v\) do not depend on \(j\). Associating to \(A\) the set \(D=\{i\mid a_ i=-1\}\) we have a difference set on \(\mathbb{Z}_ v\), obtaining a bijection between such sequences \(A\) and cyclic difference sets. If \(|\gamma|\leq 1\), the sequence \(A\) is called a periodic Barker sequence. A nontrivial Barker sequence is a periodic Barker sequence, but it is not true that each periodic Barker sequence is a Barker sequence. On this line, and widely using the multiplier theorems (also as tools to see a difference set as union of orbits of a group of multipliers) the authors give a interesting survey (both on Barker sequences and difference sets) with many new proofs and results, mainly on existence questions. The case \(\gamma=0\) is equivalent to circular Hadamard sequences, but the authors prefer to discuss Turyn results on the possible values of \(v=4N^ 2\), \(N\geq 55\). For \(\gamma=1\) many nonexistence results are given in a table. The case \(\gamma=-1\) is equivalent to the study of Hadamard difference sets of order \(v=4n-1\) and an explicit complete classification up to \(n=100\) is given, of course using previous discussions and machine enumerations. Also usual (aperiodic) Barker sequence are studied: if \(v\) is an even integer having a prime factor \(p\equiv 3\pmod 4\), there is no Barker sequence of length \(v\), and so there is no nontrivial Barker sequence of even length less than 1,898,884.
scientific article; zbMATH DE number 221268