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

Abstract

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 2 h -phase Golay sequences of length 2 m 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].

Golay complementary array pairs

“…A further paper [12] considers the explicit construction, structure, and enumeration of Golay array pairs of a given size s 1 × · · · × s r , especially when r k=1 s k is a power of 2.…”

“…A further question, considered in [12], is how many different Golay array pairs of a given size exist.…”

A generalized Boolean function generator for complementary sequences

“…To prove the complementarity for a general permutation we have to use the results from [17] and view the sequence as a projection of a multidimensional array.…”

Close Encounters with Boolean Functions of Three Different Kinds

“…We then extend this problem to the design of bipolar arrays with good aperiodic properties and show how, for 'perfect' arrays, their aperiodic properties can be carried over to related sequences, where the sequences are obtained from the arrays by recursive joining of dimensions [22,23,13] We also show how to interpret this relationship in the Fourier domain.…”

“…A construction exists for complementary sequences, as proposed by Golay [14,15], and Shapiro-Rudin [31], and later generalised by Turyn [30]. Pairs of Boolean functions constructed via an array form of the Golay-Turyn [24,23,13] construction have optimised type-I properties. We call such a pair a type-I pair.…”