[Paper Review] Some Notes on Constructions of Binary Sequences with Optimal Autocorrelation

Tongjiang Yan, Guang Gong|arXiv (Cornell University)|Nov 17, 2014

Coding theory and cryptography9 references3 citations

TL;DR

This paper presents generalized constructions of binary sequences with optimal autocorrelation using interleaved methods and almost difference sets, extending known results for sequences with periodic autocorrelation values of {0, -4}, {1, -3}, {2, -2}, and {-1, 3}. The key contribution is a unified framework that derives optimal sequences from known base sequences like Legendre, Sidelnikov, and twin-prime sequences, with explicit autocorrelation functions derived via modular arithmetic and quadratic residue properties.

ABSTRACT

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.

Motivation & Objective

• To develop more general constructions of binary sequences with optimal autocorrelation for applications in CDMA and cryptography.
• To unify and extend existing constructions of sequences with out-of-phase autocorrelation values {0, -4}, {1, -3}, {2, -2}, and {-1, 3}.
• To leverage the interleaved method and properties of almost difference sets to generate new families of sequences with optimal correlation properties.
• To provide explicit autocorrelation functions for newly constructed sequences based on modular arithmetic and quadratic residue sets.

Proposed method

• The interleaved construction method is used, where a sequence w of period 4N is formed by combining four base sequences a_i shifted by elements of a 4-tuple t.
• The autocorrelation of the interleaved sequence w is derived using ideal identities (Ia and Ib) that relate the correlation of w to the correlations of the component sequences a and b.
• The method applies modular arithmetic over Z_{2^n+1} or Z_p for prime p ≡ 1 or 3 mod 4 to classify correlation values based on τ₁ and τ₂, the components of the shift τ.
• Quadratic residue (QR) and non-residue (NQR) sets modulo p are used to determine correlation values for Legendre sequences, especially when p ≡ 1 or 3 mod 4.
• For twin-prime sequences, the method uses the modulus (p+2) to classify shifts and compute correlation values based on congruence conditions.
• The framework systematically evaluates all possible 4-tuples t and base sequence pairs (a, b) to identify those yielding optimal autocorrelation values.

Experimental results

Research questions

• RQ1 Can the interleaved construction method be generalized to produce binary sequences with optimal autocorrelation beyond known families?
• RQ2 What are the necessary and sufficient conditions on the base sequences and shift tuples to achieve optimal autocorrelation values?
• RQ3 How do quadratic residue and non-residue properties modulo p influence the autocorrelation of Legendre-based sequences in the interleaved construction?
• RQ4 Can the correlation behavior of twin-prime sequences be systematically extended using the interleaved method with specific shift patterns?
• RQ5 What role do almost difference sets and the structure of the shift tuple t play in achieving optimal correlation properties?

Key findings

• The paper constructs binary sequences with optimal autocorrelation values {0, -4}, {1, -3}, {2, -2}, and {-1, 3} using interleaved methods and known base sequences.
• For sequences derived from Legendre sequences with p ≡ 3 mod 4, the out-of-phase autocorrelation is -4 when τ₂ = 0, and varies between -4 and 4 based on τ₁ and η modulo p.
• When using Legendre sequences with p ≡ 1 mod 4, the autocorrelation values are -4 for τ₂ = 0, and depend on whether τ₁ + η is a quadratic residue or non-residue modulo p.
• For twin-prime sequences, the autocorrelation is -4 when τ₂ = 0 and τ₁ ≡ 0 mod (p+2), and 4 otherwise, with zero values for τ₂ = 1 and τ₂ = 2.
• The construction yields sequences with optimal autocorrelation for all shift patterns t where H(t) = 1 or 3, and for paired m-sequences or GGMW sequences.
• The framework confirms that the interleaved construction with t = 0111 or t = 0001 produces sequences with {0, -4} or {0, ±4} autocorrelation, extending known results from prior works.