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

  • Authors:

  • Frank Fiedler;Jonathan Jedwab;Amy Wiebe

  • Affiliations:

  • Mathematics Department, Wesley College, 120 North State Street, Dover, DE 19901, USA;Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby BC, Canada V5A 1S6;Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby BC, Canada V5A 1S6

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2010

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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 non-standard 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.