Some Notes on Constructions of Binary Sequences with Optimal Autocorrelation

Constructions of binary sequences with low autocorrelation are considered in the paper. Based on recent progresses about this topic, several more general constructions of binary sequences with optimal autocorrelations and other low autocorrelations are presented.

💡 Research Summary

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The paper investigates binary sequences whose autocorrelation functions are as low as theoretically possible. After recalling the definitions of periodic autocorrelation and cross‑correlation, the authors collect elementary shift properties (Lemma 1) and express the autocorrelation of a binary sequence in terms of its support set (Equation 2). They then list the minimal out‑of‑phase autocorrelation values (perfect autocorrelation) and the next‑best values (optimal autocorrelation) in Tables 1 and 2, citing known families such as Legendre, Sidelnikov, and twin‑prime sequences.

The central contribution is a set of generalized interleaved constructions. Construction A builds a sequence by interleaving a zero column of length K with T‑1 column sequences that are cyclic shifts of each other; this is a generalized GMW construction. Construction B modifies A by replacing the zero column with an all‑one column. Lemmas 3 and 4 give the cross‑correlation and autocorrelation of the sequences produced by A and B, especially when T = 2ⁿ + 1 and K = 2ⁿ − 1, showing that the out‑of‑phase values are either –1 or –4, depending on the shift.

Next, the paper treats Legendre sequences (both the first‑type and second‑type) and twin‑prime sequences. Lemmas 5‑8 provide their exact autocorrelation and cross‑correlation values for primes p ≡ 1 (mod 4) and for twin‑prime pairs (p, p + 2). These results reveal a characteristic pattern: for Legendre sequences the autocorrelation takes the three values p, 1, –3, while for twin‑prime sequences the out‑of‑phase autocorrelation is –1 except at multiples of p + 2 where it becomes 3 or 1.

The authors then introduce the interleaving operator “⊗” that merges two sequences a and b into a single sequence of double length by alternating their symbols. Lemmas 8‑12 derive the autocorrelation of such merged sequences, showing that when the two component sequences are related by a shift or complement, the merged sequence can have zero autocorrelation for all odd lags. This observation leads to Definition 1, a four‑column interleaved sequence
v(t) = (a, b, L_{N+1/2}( ¯a ), L_{N+1/2}(b)),
where L_m denotes a cyclic shift by m. Theorem 1 gives a complete description of the autocorrelation of v(t) in terms of the autocorrelations of a and b, with the remarkable property that all odd‑lag values are zero.

Theorem 2 (N ≡ 3 (mod 4)) and Theorem 3 (N ≡ 1 (mod 4)) characterize when the sequence u(t) = a ⊗ b has optimal autocorrelation {4N, ±4, 0}. The condition is that both a and b must themselves have ideal autocorrelation (for N ≡ 3) or optimal autocorrelation (for N ≡ 1). Concrete examples are provided (e.g., a = 01000, b = 10000).

Theorem 4 extends the construction to the generalized GMW sequences from Construction A and its modified version B, showing that after appropriate cyclic shifts L_{η₁} and L_{η₂} the interleaved sequence still exhibits a small set of autocorrelation values, namely {±4, 0, ±8}. Theorem 5 does the same for Legendre sequences, while Theorem 6 treats twin‑prime sequences, each time giving explicit formulas for the autocorrelation values depending on the shift parameters and the underlying prime modulus.

Finally, Definition 2 and Theorem 7 introduce another four‑column interleaved structure
w(t) = (a, L_η( ¯a ), b, L_η(b)),
and prove that its autocorrelation takes the values 2R_a(τ₁)+2R_b(τ₁) for even lags, zero for certain odd lags, and controlled linear combinations of shifted autocorrelations for the remaining odd lags. This flexibility allows designers to suppress specific lag values, which is valuable in CDMA and stream‑cipher applications where low cross‑interference is crucial.

Overall, the paper provides a unified framework that generalizes several known optimal‑autocorrelation constructions, supplies rigorous proofs for all correlation properties, and demonstrates how to generate large families of binary sequences with prescribed autocorrelation spectra by simple choices of base sequences and shift parameters. The results are directly applicable to spread‑spectrum communications, cryptographic pseudo‑random generators, and radar waveform design, where sequences with low out‑of‑phase autocorrelation improve performance and security.