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

Abstract

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. The first source of seed pairs is the trivial Golay pair of length 1, which gives rise to "standard" Golay sequences; the second source is the set of 512 non-standard "cross-over" 4-phase Golay pairs of length 8, which give rise to non-standard 4-phase Golay sequences of length 2 m for each m 4. Beginning with a single length 5 complex-valued Golay sequence pair, we show how to construct a third source of Golay seed pairs (and only the second known non-trivial source), namely a new set of 5184 non-standard 6-phase Golay sequences of length 16 that form 62 208 non-standard ordered Golay pairs. Using the multi-dimensional viewpoint, this new set of Golay seed pairs in turn gives rise to a new infinite family of 6-phase nonstandard Golay sequences of length 2 m for each m 4, and a new infinite family of 12-phase non-standard Golay sequences of length 2 m for each m 8. All currently known H-phase Golay sequences of length 2 m can be constructed from the three sets of seed pairs.

A unified construction of type-Ⅰ even length Z-complementary pairs based on generalized Boolean function

“…In the field of mathematics, the central theoretical questions about the Golay sequence are: for what lengths does a GCP exist [15,16,8]? how to get more GCPs [29,10,11,12], and how many distinct Golay sequences and GCPs of a given length are there [7,24]? In the field of engineering, GCPs have found many applications in wireless communications, which include optimal channel estimation in multiple-input multiple-output (MIMO) frequency-selective channels [27], radar [19], and peak power control in orthogonal frequency division multiplexing (OFDM) [31], etc.…”