A construction of binary Golay sequence pairs from odd-length Barker sequences

Binary Golay sequence pairs exist for lengths 2, 10 and 26 and, by Turyn’s product construction, for all lengths of the form 21026 where a, b, c are non-negative integers. Computer search has shown that all inequivalent binary Golay sequence pairs of length less than 100 can be constructed from five “seed” pairs, of length 2, 10, 10, 20 and 26. We give the first complete explanation of the origin of the length 26 binary Golay seed pair, involving a Barker sequence of length 13 and a related Barker sequence of length 11. This is the special case m = 1 of a general construction for a length 16m+ 10 binary Golay pair from a related pair of Barker sequences of length 8m+ 5 and 8m+ 3, for integer m ≥ 0. In the case m = 0, we obtain an alternative explanation of the origin of one of the length 10 binary Golay seed pairs. The construction cannot produce binary Golay sequence pairs for m > 1, having length greater than 26, because there are no Barker sequences of odd length greater than 13.