A construction for binary sequence sets with low peak-to-average power ratio

Complementary sequences (CS) have peak-to-average power ratio (PAR) ≤ 2 under the one-dimensional continuous discrete Fourier transform (DFT₁^∞). Davis and Jedwab (see IEEE Trans. Inform. Theory, vol.45, no.7, p.2397-2417, 1999) constructed binary CS (DJ set) for lengths 2ⁿ described by s = 2^-n2/ (-1)^p(x), p(x) = Σ_j=0^L-2x_π(j)x_π(j+1)+c_jx_j+k, c_j, k ∈ Z₂. Hamming distance, D, between sequences in this set satisfies D ≥ 2^n-2. However the rate of the DJ set vanishes for n → ∞, and higher rates are possible for PAR ≤ O(n) and D large. We present such a construction which generalises the DJ set. These codesets have PAR ≤ 2^t under all linear unimodular unitary transforms (LUUTs), including all one and multi-dimensional continuous DFTs, and D ≥ 2^n-d where d is the maximum algebraic degree of the chosen subset of the complete set.