A Framework for the Construction ofGolay Sequences

In 1999, Davis and Jedwab gave an explicit algebraic normal form for $m! \cdot 2^{h(m+2)}$ ordered Golay pairs of length $2^{m}$ over $ {\BBZ }_{2^{h}}$, involving $m!/2 \cdot 2^{h(m+1)}$ Golay sequences. In 2005, Li and Chu unexpectedly found an additional 1024 length 16 quaternary Golay sequences. Fiedler and Jedwab showed in 2006 that these new Golay sequences exist because of a “crossover” of the aperiodic autocorrelation function of certain quaternary length eight sequences belonging to Golay pairs, and that they spawn further new quaternary Golay sequences and pairs of length $2^{m}$ for $m > 4$ under BudiŠin's 1990 iterative construction. The total number of Golay sequences and pairs spawned in this way is counted, and their algebraic normal form is given explicitly. A framework of constructions is derived in which Turyn's 1974 product construction, together with several variations, plays a key role. All previously known Golay sequences and pairs of length $2^{m}$ over $ {\BBZ }_{2^{h}}$ can be obtained directly in explicit algebraic normal form from this framework. Furthermore, additional quaternary Golay sequences and pairs of length $2^{m}$ are produced that cannot be obtained from any other known construction. The framework generalizes readily to lengths that are not a power of $2$ , and to alphabets other than $ {\BBZ }_{2^{h}}$.