Golay-Davis-Jedwab Complementary Sequences and Rudin-Shapiro Constructions

A Golay Complementary Sequence (CS) has a Peak-to-AveragePower-Ratio (PAPR) ≤ 2.0 for its one-dimensional continuous Discrete Fourier Transform (DFT) spectrum. Davis and Jedwab showed that all known length 2 CS, (GDJ CS), originate from certain quadratic cosets of Reed-Muller (1,m). These can be generated using the RudinShapiro construction. This paper shows that GDJ CS have a PAPR ≤ 2.0 under all 2m×2m unitary transforms whose rows are unimodular linear (Linear Unimodular Unitary Transforms (LUUTs)), including oneand multi-dimensional generalised DFTs. In this context we define Constahadamard Transforms (CHTs) and show how all LUUTs can be formed from tensor combinations of CHTs. We also propose tensor cosets of GDJ sequences arising from Rudin-Shapiro extensions of near-complementary pairs, thereby generating many more infinite sequence families with tight low PAPR bounds under LUUTs. We then show that GDJ CS have a PAPR ≤ 2m−⌊m2 ⌋ under all 2 × 2 unitary transforms whose rows are linear (Linear Unitary Transforms (LUTs)). Finally we present a radix-2 tensor decomposition of any 2 × 2 LUT. M.G.Parker is with the Code Theory Group, Inst. for Informatikk, Hoyteknologisenteret i Bergen, University of Bergen, Bergen 5020, Norway. E-mail: matthew@ii.uib.no. Web: http://www.ii.uib.no/∼matthew/MattWeb.html C.Tellambura is with the School of Computer Science and Software Engineering, Monash University, Clayton, Victoria 3168, Australia. E-mail: chintha@dgs.monash.edu.au. Phone/Fax: +61 3 9905 3196/5146