Hamming correlation properties of the array structure of Sidelnikov sequences

Abstract

In this paper, we investigate the Hamming correlation properties of column sequences from the $(q-1)\times \frac{q^d-1}{q-1}$ array structure of M-ary Sidelnikov sequences of period $q^d-1$ for $M|q-1$ and $d\ge 2$. We prove that the proposed set $\varGamma (d)$ of some column sequences has the maximum non-trivial Hamming correlation upper bounded by the minimum of $\frac{q-1}{M}d-1$ and $\frac{M-1}{M}\left[ (2d-1)\sqrt{q}+1\right] +\frac{q-1}{M}$. When $M=q-1$, we show that $\varGamma (d)$ is optimal with respect to the Singleton bound. The set $\varGamma (d)$ can be extended to a much larger set $\varDelta (d)$ by involving all the constant additions of the members of $\varGamma (d)$, which is also optimal with respect to the Singleton bound when $M=q-1$.

Appendix: Proof of Lemma 1

Recall that $m_l = d$ for any $l \in \varLambda '(d)$ and $m_l$ is the least positive integer which satisfies (5). Then, it is easy to see that

$$\begin{aligned} \left| \varLambda '(d) \right| = \frac{1}{d} \left( \frac{q^d-1}{q-1} - k -1 \right) , \end{aligned}$$

(24)

where

$$\begin{aligned} k = \sum _{\begin{array}{c} r|d \\ r \ne 1\\ r \ne d \end{array}} \left| \left\{ l \mid 1 \le l <\frac{q^d-1}{q-1}, m_l = r \right\} \right| . \end{aligned}$$

(25)

From (5), there are

$$\begin{aligned} \frac{(q^r-1)\gcd \left( \frac{d}{r}, q-1 \right) }{q-1} \end{aligned}$$

column indices from 1 to $\frac{q^d-1}{q-1}-1$, which can be divided by

$$\begin{aligned} \frac{q^d-1}{(q^r-1) \gcd \left( \frac{d}{r}, q-1 \right) }, \end{aligned}$$

where $2 \le r \le d$. So, we have

$$\begin{aligned} k&< \sum _{\begin{array}{c} r|d \\ r \ne d \\ r \ne 1 \end{array}} \frac{(q^r-1) \gcd \left( \frac{d}{r}, q-1 \right) }{q-1}. \end{aligned}$$

(26)

Note that any divisor of d is less than or equal to $\lfloor d/2 \rfloor$ and $\gcd \left( \frac{d}{r}, q-1 \right) \le q-1$. By using these, observe that

$$\begin{aligned} k&< \sum _{r=2}^{\lfloor d/2 \rfloor } \frac{(q^r-1) \gcd \left( \frac{d}{r}, q-1 \right) }{q-1}< \sum _{r=2}^{\lfloor d/2 \rfloor }q^{r} < \frac{q^{\lfloor d/2 \rfloor +1}-1}{q-1} - 1 \end{aligned}$$

(27)

By substituting (27) into (24) finally, we obtain the result. $\square$