A multi-dimensional approach to the construction and enumeration of Golay complementary sequences

We argue that a Golay complementary sequence is naturally viewed as a projection of a multi-dimensional Golay array. We present a three-stage process for constructing and enumerating Golay array and sequence pairs:

1.

construct suitable Golay array pairs from lower-dimensional Golay array pairs;

2.

apply transformations to these Golay array pairs to generate a larger set of Golay array pairs; and

3.

take projections of the resulting Golay array pairs to lower dimensions.

This process greatly simplifies previous approaches, by separating the construction of Golay arrays from the enumeration of all possible projections of these arrays to lower dimensions.We use this process to construct and enumerate all h²-phase Golay sequences of length m² obtainable under any known method, including all 4-phase Golay sequences obtainable from the length 16 examples given in 2005 by Li and Chu [Y. Li, W.B. Chu, More Golay sequences, IEEE Trans. Inform. Theory 51 (2005) 1141-1145].