Golay complementary array pairs

Abstract

Constructions and nonexistence conditions for multi-dimensional Golay complementary array pairs are reviewed. A construction for a d-dimensional Golay array pair from a (d + 1)-dimensional Golay array pair is given. This is used to explain and expand previously known constructive and nonexistence results in the binary case.

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

“…In this paper we demonstrate the power of the recently-introduced view [11] of a Golay sequence pair as the "projection" of a multi-dimensional Golay array pair. While the paper [11] focused on determining whether at least one Golay array pair of a given size exists, particularly for the binary case, this paper deals with the systematic construction and enumeration of a large set of distinct Golay array pairs of a given size.…”

“…While the paper [11] focused on determining whether at least one Golay array pair of a given size exists, particularly for the binary case, this paper deals with the systematic construction and enumeration of a large set of distinct Golay array pairs of a given size. Although Golay arrays have been previously studied by Lüke [13] and especially Dymond [5], and shown to be of use in coded imaging [15], it appears that for the most part they have been ignored or else regarded as merely another generalisation of a familiar combinatorial object.…”

Quaternary Golay sequence pairs I: even length

“…Stage 2. Take "affine offsets" of the Golay array pairs created in Stage 1, to generate a larger set of Golay array pairs of the same size: Theorem 4 was given for the case (k, ) = (1, 2) in [13]; the form given here follows simply by reordering dimensions.…”

A new source of seed pairs for Golay sequences of length 2m

“…Fiedler, Jedwab and Parker [5], using a viewpoint proposed in [10], showed that all these "standard" Golay sequences can be recovered from a three-stage multi-dimensional construction process, using trivial Golay pairs of length 1 as inputs. Until now, the only known non-standard H -phase Golay sequences of length 2 m were those arising when one or more "cross-over" 4-phase Golay pairs of length 8 [3] are used as inputs to the three-stage construction process [5].…”

“…

In 2007 Jedwab and Parker [10] proposed that the natural viewpoint for a Golay complementary sequence is as a projection of a multi-dimensional Golay array. In 2008 Fiedler, Jedwab and Parker [5] used this viewpoint to show how to construct and enumerate all known 2 h -phase Golay sequences of length 2 m , starting from two sources of Golay seed pairs.

…”