Nearly optimal balanced quaternary sequence pairs of prime period $N\equiv 5\pmod 8$

Abstract

In this paper, for balanced quaternary sequence pairs (BQSPs for short) of period $N\equiv 1\pmod 4$, the optimal cross-correlation is determined and a lower bound on the maximum autocorrelation magnitude is derived. Based on cyclotomic classes of order 4, we obtain two classes of nearly optimal BQSPs of period N, where $N\equiv 5\pmod 8$ is a prime. Those pairs have optimal cross-correlation $\{1,-1\}$ and maximum out-of-phase autocorrelation magnitude no more than $\sqrt{N-4}+4$, whose difference with a lower bound of maximum autocorrelation magnitude is less than 4.

Appendix

In this appendix, we will give the detailed proof of Lemma 6. First we need some knowledge of binary sequences.

A sequence $\mathbf{a }=\{a_k\}_{k=0}^{N-1}$, where $a_k=\{0,1\}$ is called a binary sequence of period N. The set $C=\{0\le k\le N-1:a_i=1\}$ is called the characteristic set of $\mathbf{a}$. If $|C|=N/2$ for even N or $|C|=(N\pm 1)/2$ for odd N, then such $\mathbf{a}$ is called balanced.

Let $\mathbf{a }=\{a_i\}_{i=0}^{N-1}$ and $\mathbf{b }=\{b_i\}_{i=0}^{N-1}$ be two binary sequences of period N. The period cross-correlation function of $\mathbf{a }$ and $\mathbf{b }$ is defined as

$$\begin{aligned} R^{(2)}_{\mathbf{a},\mathbf{b}}(\tau )=\sum \limits _{k=0}^{N-1}(-1)^{a_k-b_{k+\tau }},~0\le \tau \le N-1. \end{aligned}$$

The cross-correlation property of balanced binary sequences was given in [25].

Lemma 9

[25] For any two balanced binary sequences $\mathbf{a}$ and $\mathbf{b}$ of period N, with equal size of characteristic sets,

$$\begin{aligned} R^{(2)}_{\mathbf{a},\mathbf{b}}(\tau )\equiv N\pmod 4. \end{aligned}$$

Any quaternary sequence $\mathbf{u}$ can be transformed into two binary sequences $\mathbf{a}_1$ and $\mathbf{a}_2$ of the same period by the famous Gray mapping $\phi$, denoted by $\phi (\mathbf{u})=(\mathbf{a}_1,\mathbf{a}_2)$, as follows:

$$\begin{aligned} \phi (u_t)=(a_1(t),a_2(t)),~~0\le t<N, \end{aligned}$$

where $\phi$ is defined as:

$$\begin{aligned} \phi (0)=(0,0),~\phi (1)=(0,1),~\phi (2)=(1,1),~\phi (3)=(1,0). \end{aligned}$$

Krone and Sarwate observed the following relation between the correlations of quaternary sequences and binary sequences.

Lemma 10

[9] The cross-correlation function of any two quaternary sequences $\mathbf{u}$ and $\mathbf{v}$ is given by

$$\begin{aligned} R_{\mathbf{u},\mathbf{v}}(\tau )=\frac{1}{2}\left( R^{(2)}_{\mathbf{a}_1,\mathbf{b}_1}(\tau )+R^{(2)}_{\mathbf{a}_2,\mathbf{b}_2}(\tau )\right) +\frac{i}{2}\left( R^{(2)}_{\mathbf{a}_1,\mathbf{b}_2}(\tau )-R^{(2)}_{\mathbf{a}_2,\mathbf{b}_1}(\tau )\right) , \end{aligned}$$

where $\phi (\mathbf{u})=(\mathbf{a}_1,\mathbf{a}_2)$ and $\phi (\mathbf{v})=(\mathbf{b}_1,\mathbf{b}_2)$.

Now we are ready to prove the result of Lemma 6 in Sect. 3.

Proof of Lemma 6

The assumption of $\mathbf{u}, \mathbf{v}$ implies that the resultant four binary sequences are all balanced. Then by Lemmas 9 and 10, we have

$$\begin{aligned} \mathfrak {R}(R_{\mathbf{u},\mathbf{v}}(\tau ))=\frac{1}{2}\left( R^{(2)}_{\mathbf{a}_1,\mathbf{b}_1}(\tau )+R^{(2)}_{\mathbf{a}_2,\mathbf{b}_2}(\tau )\right) \equiv N\pmod 2, \\ \mathfrak {I}(R_{\mathbf{u},\mathbf{v}}(\tau ))=\frac{1}{2}\left( R^{(2)}_{\mathbf{a}_1,\mathbf{b}_2}(\tau )-R^{(2)}_{\mathbf{a}_2,\mathbf{b}_1}(\tau )\right) \equiv 0\pmod 2. \end{aligned}$$

The proof is now completed. $\square$