In this letter, a code-domain nonorthogonal multiple access (NOMA) technique based on an algebraic design is studied. We propose an improved low-density spreading (LDS) sequence design based on projective geometry. In terms of its bit error rate (BER) performance, our proposed improved LDS code set outperforms the existing LDS designs over the frequency-nonselective Rayleigh fading and additive white Gaussian noise (AWGN) channels. We demonstrated that achieving the best BER depends on the minimum distance.

We consider the problem of finding maximal sets of shift-inequivalent decimations of Sidelnikov-Lempel-Cohn-Eastman (SLCE) sequences (as well as the equivalent problem of determining the multiplier groups of the almost difference sets associated with these sequences). We derive a numerical necessary condition for a residue to be a multiplier of an SLCE almost difference set. Using our necessary condition, we show that if $p$ is an odd prime and $S$ is an SLCE almost difference set over $\mathbb{F}_p,$ then the multiplier group of $S$ is trivial. Consequently, for each odd prime $p,$ we obtain a family of $\phi(p-1)$ shift-inequivalent balanced periodic sequences (where $\phi$ is the Euler-Totient function) each having period $p-1$ and nearly perfect autocorrelation.

We consider the problem of determining the cross-correlation values of the sequences in the families comprised of constant multiples of M-ary Sidelnikov sequences over 𝔽_q, where q is a power of an odd prime p. We show that the cross-correlation values of pairs of sequences from such a family can be expressed in terms of certain Jacobi sums. This insight facilitates the computation of the cross-correlation values of these sequence pairs so long as ϕ(M)^ϕ(M)≤ q. We are also able to use our Jacobi sum expression to deduce explicit formulae for the cross-correlation distribution of a family of this type in the special case that there exists an integer x such that p^x ≡ -1 M.

Binary sequences with good autocorrelation properties and large linear complexity are useful in stream cipher cryptography. The Sidelnikov-Lempel-Cohn-Eastman (SLCE) sequences have nearly optimal autocorrelation. However, the problem of determining the linear complexity of the SLCE sequences is still open. It is well known that one can gain insight into the linear complexity of a sequence if one can say something about the divisors of the gcd of a certain pair of polynomials associated with the sequence. Helleseth and Yang (IEEE Trans. Inf. Theory 49(6), 1548–1552 2002), Kyureghyan and Pott (Des. Codes Crypt. 29, 149–164 2003) and Meidl and Winterhof (Des. Codes Crypt. 8, 159–178 2006) were able to obtain some results of this type for the SLCE sequences. Kyureghyan and Pott (Des. Codes Crypt. 29, 149–164 2003) mention that it would be nice to obtain more such results. We derive new divisibility results for the SLCE sequences in this paper. Our approach is to exploit the fact that character values associated with the SLCE sequences can be expressed in terms of a certain type of Jacobi sum. By making use of known evaluations of Gauss and Jacobi sums in the “pure” and “small index” cases, we are able to obtain new insight into the linear complexity of the SLCE sequences.

It is well-known that for each prime power q and for each d ∈ 2N, there exists a circulant weighing matrix of order q d+1−1 q−1 and weight q . We extend this result to show that there exist φ(d+1) 2 inequivalent circulant weighing matrices of order q d+1−1 q−1 and weight q , where φ is the Euler totient function. Further, we obtain a bound on the magnitude of the values taken by the cross-correlation function of any pair of perfect ternary sequences obtained from these matrices.