A pair of sequences is called a Golay complementary pair (GCP) if their aperiodic auto-correlation sums are zero for all out-of-phase time shifts. Existing known binary GCPs only have even-lengths in the form of 2 α 10 β 26 γ (where α, β, γ are non-negative integers). To fill the gap left by the odd-lengths, we investigate the optimal odd-length binary pairs which display the closest correlation property to that of GCPs. Our criteria of "closeness" is that each pair has the maximum possible zerocorrelation zone (ZCZ) width and minimum possible out-of-zone aperiodic auto-correlation sums. Such optimal pairs are called optimal odd-length binary Z-complementary pairs (OB-ZCP) in this paper. We show that each optimal OB-ZCP has maximum ZCZ width of (N + 1)/2, and minimum out-of-zone aperiodic sum magnitude of 2, where N denotes the sequence length (odd). Systematic constructions of such optimal OP-ZCPs are proposed by insertion and deletion of certain binary GCPs, which settle the 2011 Li-Fan-Tang-Tu open problem positively. The proposed optimal OB-ZCPs may serve as a replacement for GCPs in many engineering applications where odd sequence lengths are preferred. In addition, they give rise to a new family of base-two almost difference families (ADF) which are useful in studying partially balanced incomplete block design (BIBD).
Abstract-To support interference-free quasi-synchronous code-division multiple-access (QS-CDMA) communication with low spectral density profile in a cognitive radio (CR) network, it is desirable to design a set of CDMA spreading sequences with zero-correlation zone (ZCZ) property. However, traditional ZCZ sequences (which assume the availability of the entire spectral band) cannot be used because their orthogonality will be destroyed by the spectrum hole constraint in a CR channel. To date, analytical construction of ZCZ CR sequences remains open. Taking advantage of the Kronecker sequence property, a novel family of sequences (called "quasi-ZCZ" CR sequences) which displays zero cross-correlation and near-zero auto-correlation zone property under arbitrary spectrum hole constraint is presented in this paper. Furthermore, a novel algorithm is proposed to jointly optimize the peak-to-average power ratio (PAPR) and the periodic auto-correlations of the proposed quasi-ZCZ CR sequences. Simulations show that they give rise to single-user biterror-rate performance in CR-CDMA systems which outperform traditional non-contiguous multicarrier CDMA and transform domain communication systems; they also lead to CR-CDMA systems which are more resilient than non-contiguous OFDM systems to spectrum sensing mismatch, due to the wideband spreading.
Levenshtein improved the Welch bound on aperiodic correlation by weighting the cyclic shifts of the sequences over complex roots-of-unity. Although many works have been concerned on meeting the Welch bound with equality, no such effort has been reported for the Levenshtein bound. We show that the Levenshtein bound with equality is met if and only if the non-trivial aperiodic correlations have identical amplitude for all time-shifts, and the sequences form a novel class of complementary set whose aperiodic correlation is defined as the conventional aperiodic correlation modulated by a simplex weighting vector.
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