A direct construction of complete complementary code with zero correlation zone property for prime-power length

Abstract

In this paper, we propose a direct construction of a novel type of code set which has combined properties of complete complementary code (CCC) and zero correlation zone (ZCZ) sequence set and which we call complete complementary-ZCZ (CC-ZCZ) code set. The code set is constructed using multivariate functions. The proposed construction also provides Golay-ZCZ codes of prime-power lengths. The proposed Golay-ZCZ codes are optimal and asymptotically optimal for binary and non-binary cases, respectively, by Tang-Fan-Matsufuzi bound. Furthermore, the proposed direct construction provides novel ZCZ sequences of length $\varvec{p^k}$, where $\varvec{p}$ is a prime number and $\varvec{k}$ is an integer $\varvec{\ge 2}$. We establish a relationship between the proposed CC-ZCZ code set and the first-order generalized Reed-Muller (GRM) code, and prove that both have the same Hamming distance. We also count the number of CC-ZCZ code sets in first-order GRM codes. The column sequence peak-to-mean envelope power ratio (PMEPR) of the proposed CC-ZCZ code set is derived and compared with existing works. From the proposed construction, the Golay-ZCZ code and ZCZ sequences are also derived and compared with the existing works. The proposed construction generalizes many of the existing works.

Appendix

1.1 A: Proof of Theorem 1

We state and prove a lemma which is used to prove the Theorem 1.

Lemma 3

Let $(i_1,i_2,\hdots ,i_m)$ and $(j_1,j_2,\hdots ,j_m)$ be p-ary representation of non-negative integers i and j respectively. Let $f:\mathbb {Z}_p^m\rightarrow \mathbb {Z}_\lambda$ be a function as defined in (15). Now, from (16), we can write sequence $a_{\mathbf {u_k}}^{\mathbf {v_k}}$ as

$$\begin{aligned} a_{\mathbf {u_k}}^{\mathbf {v_k}}=f+\frac{\lambda }{p}\sum _{\beta =1}^{k}{x_{\pi _{\beta }{(1)}}u_{\beta }}+\frac{\lambda }{p}\sum _{\beta =1}^{k}{x_{\pi _{\beta }{(n_\beta )}}v_{\beta }}. \end{aligned}$$

(A1)

Further, suppose we have $i_{\pi _\beta (1)}=j_{\pi _\beta (1)}~\forall ~\beta =1,2,\hdots ,k$. For a certain $\beta '\le k$, let us assume t be the smallest integer such that $i_{\pi _{\beta '}(t)}\ne j_{\pi _{\beta '}(t)}$. Let us define $i^\eta$ to be an integer whose vector representation with base p is

$$\begin{aligned} (i_1,i_2,\hdots ,i_{\pi _{\beta '}(t-1)}-\eta ,\hdots ,i_m), \end{aligned}$$

(A2)

which differs from that of i only at the position $\pi _{\beta '}(t-1)$ and $\eta =1,2,\hdots ,(p-1)$. Similarly, we define $j^\eta$ such that its vector representation with base p is

$$\begin{aligned} (j_1,j_2,\hdots ,j_{\pi _{\beta '}(t-1)}-\eta ,\hdots ,j_m). \end{aligned}$$

(A3)

Then we have

$$\begin{aligned} \sum _{\eta =1}^{p-1}{\omega _\lambda ^{(a_{\mathbf {u_k}}^{\mathbf {v_k}})_{j^\eta }-(a_{\mathbf {u_k}}^{\mathbf {v_k}})_{i^\eta }}}+\omega _\lambda ^{(a_{\mathbf {u_k}}^{\mathbf {v_k}})_{j}-(a_{\mathbf {u_k}}^{\mathbf {v_k}})_{i}}=0. \end{aligned}$$

(A4)

Proof

For notational convenience, we write $a_{\mathbf {u_k}}^{\mathbf {v_k}}=\textbf{c}$. Now using (15) and (A1), we can write

$$\begin{aligned} \textbf{c}_{i^\eta }-\textbf{c}_{i}=&f_{i^\eta }-f_{i} \nonumber \\ =&\frac{\lambda }{p}[i_{\pi _{\beta '}(t-2)}i_{\pi _{\beta '}(t-1)}^\eta +i_{\pi _{\beta '}(t-1)}^\eta i_{\pi _{\beta '}(t)}-i_{\pi _{\beta '}(t-2)}i_{\pi _{\beta '}(t-1)}^\eta -i_{\pi _{\beta '}(t-1)}i_{\pi _{\beta '}(t)}] \nonumber \\&+g_{\pi _{\beta '}(t-1)}[i_{\pi _{\beta '}(t-1)}^\eta -i_{\pi _{\beta '}(t-1)}] \nonumber \\ =&[-\eta \frac{\lambda }{p} i_{\pi _{\beta '}(t)}-\eta \frac{\lambda }{p} i_{\pi _{\beta '}(t-2)}-\eta g_{\pi _{\beta '}(t-1)}] \nonumber \\ =&-\eta [\frac{\lambda }{p}i_{\pi _{\beta '}(t)}+\frac{\lambda }{p}i_{\pi _{\beta '}(t-2)}+g_{\pi _{\beta '}(t-1)}]. \end{aligned}$$

(A5)

Similarly,

$$\begin{aligned} \textbf{c}_{j^\eta }-\textbf{c}_{j}=f_{j^\eta }-f_{j}=-\eta [\frac{\lambda }{p}j_{\pi _{\beta '}(t)}+\frac{\lambda }{p}j_{\pi _{\beta '}(t-2)}+g_{\pi _{\beta '}(t-1)}]. \end{aligned}$$

(A6)

From (A5) and (A6), we have

$$\begin{aligned} (\textbf{c}_{j^\eta }-\textbf{c}_{i^\eta })-(\textbf{c}_{j}-\textbf{c}_{i})=\eta \frac{\lambda }{p}(i_{\pi _{\beta '}(t)}-j_{\pi _{\beta '}(t)}). \end{aligned}$$

(A7)

Now, taking sum of $\omega _\lambda ^{(\textbf{c}_{j^\eta }-\textbf{c}_{i^\eta })-(\textbf{c}_{j}-\textbf{c}_{i})}$ over $\eta$, we have

$$\begin{aligned} \sum _{\eta =1}^{p-1}{\omega _\lambda ^{\textbf{c}_{j^\eta }-\textbf{c}_{i^\eta }-(\textbf{c}_{j}-\textbf{c}_{i})}=\sum _{\eta =1}^{p-1}{\omega _\lambda ^{\eta \frac{\lambda }{p}(i_{\pi _{\beta '}(t)}-j_{\pi _{\beta '}(t)})}}}=\sum _{\eta =1}^{p-1}{\omega _p^{\eta (i_{\pi _{\beta '}(t)}-j_{\pi _{\beta '}(t)})}}. \end{aligned}$$

(A8)

It is given that $i_{\pi _{\beta '}(t)}\ne j_{\pi _{\beta '}(t)}$, implies that $i_{\pi _{\beta '}(t)}-j_{\pi _{\beta '}(t)}\ne 0$ and hence RHS of (A8) is the sum of roots of polynomial $z^p-1=0$ except the root $z=1$. Hence

$$\begin{aligned} \sum _{\eta =1}^{p-1}{\omega _p^{\eta (i_{\pi _{\beta '}(t)}-j_{\pi _{\beta '}(t)})}}=-1 \end{aligned}$$

(A9)

Therefore, from (A8) and (A9), we have

$$\begin{aligned} \sum _{\eta =1}^{p-1}{\omega _\lambda ^{\textbf{c}_{j^\eta }-\textbf{c}_{i^\eta }-(\textbf{c}_{j}-\textbf{c}_{i})}}=-1, \end{aligned}$$

(A10)

which further implies that

$$\begin{aligned} \sum _{\eta =1}^{p-1}{\omega _\lambda ^{\textbf{c}_{j^\eta }-\textbf{c}_{i^\eta }}+\omega _\lambda ^{(\textbf{c}_{j}-\textbf{c}_{i})}}=0. \end{aligned}$$

(A11)

$\square$

Proof of Theorem

1 We need to show that for a fixed $\mathbf {u_k}$, $\mathbf {C_{u_k}}$ is a $(p^k,(p-1)p^{\pi _1(2)-1},p^m)$-ZCZ. Except the ZCZ width all the parameters are directly inherited from Lemma 2. So, we only need to show that the ZCZ width is $(p-1)p^{\pi _1(2)-1}$. Let $\textbf{c}\in \mathbf {C_{u_k}}$, by (17), $\textbf{c}=\psi (a_{\mathbf {u_k}}^{\mathbf {v_{k_1}}})$ for some $0\le \mathbf {v_{k_1}}\le p^k-1$. First, we find the PACF of $\textbf{c}$ and show that for $0<\tau \le (p-1)p^{\pi _1(2)-1}$,

$$\begin{aligned} \mathcal {P}(\textbf{c})(\tau )=\sum _{i=0}^{L-1}{\omega _\lambda ^{\textbf{c}_{(i+\tau )mod~L}-\textbf{c}_i}}=0, \end{aligned}$$

(A12)

where L is the length of the sequence, i.e., $L=p^m$. For any integer i, let us denote another integer $j=(i+\tau )mod~L$. Then we consider two cases and demonstrate that for each pair (i, j) there exist other $(p-1)$ pairs $(i^\eta ,j^\eta ),~\eta =1,2,\hdots ,p-1$ such that

$$\begin{aligned} \sum _{\eta =1}^{p-1}{\omega _\lambda ^{\textbf{c}_{j^\eta }-\textbf{c}_{i^\eta }}+\omega _\lambda ^{(\textbf{c}_{j}-\textbf{c}_{i})}}=0, \end{aligned}$$

(A13)

in each case.

Case 1 ($i_{\pi _{1}(2)}=j_{\pi _{1}(2)}$) In this case, we have $i_{\pi _{\beta }(1)}=j_{\pi _{\beta }(1)},~\forall \beta =1,2,\hdots ,k$. On the contrary, suppose this is not true. Then, assume that $\bar{\beta }$ is the largest integer such that $i_{\pi _{\bar{\beta }}(1)}\ne j_{\pi _{\bar{\beta }}(1)}$. For ease of presentation, let $d=\pi _{\bar{\beta }}(1)$, now if $j_d>i_d$, we have

$$\begin{aligned} \tau =j-i=&\sum _{s=1}^{d}{(j_s-i_s)p^{s-1}} \nonumber \\ =&(j_d-i_d)p^{d-1}+\sum _{s=1,s\ne \pi _{1}(2)}^{d-1}{(j_s-i_s)p^{s-1}}\nonumber \\ \ge&(j_d-i_d)p^{d-1}-(p-1)\sum _{s=1}^{d-1}{p^{s-1}}+(p-1)p^{\pi _{1}(2)-1}\nonumber \\ =&(j_d-i_d)p^{d-1}-(p-1)\Big [\frac{p^{d-1}-1}{p-1}\Big ]+(p-1)p^{\pi _{1}(2)-1}\nonumber \\ =&(j_d-i_d-1)p^{d-1}+1+(p-1)p^{\pi _{1}(2)-1}>(p-1)p^{\pi _{1}(2)-1}. \end{aligned}$$

(A14)

Hence (A14) implies that $\tau >(p-1)p^{\pi _{1}(2)-1}$ which is a contradiction. Similarly, if $j_d<i_d$, then

$$\begin{aligned} \tau&=j-i+p^m=\sum _{s=1}^{d}{(j_s-i_s)p^{s-1}}+p^m\nonumber \\&=(j_d-i_d)p^{m-\bar{\beta }}+p^m+\sum _{s=1,s\ne \pi _{1}(2)}^{d-1}{(j_s-i_s)p^{s-1}}\nonumber \\&=(j_d-i_d+p^{\bar{\beta }})p^{m-\bar{\beta }}+\sum _{s=1,s\ne \pi _{1}(2)}^{d-1}{(j_s-i_s)p^{s-1}}\nonumber \\&\ge (j_d-i_d+p^{\bar{\beta }})p^{m-\bar{\beta }}-(p-1)\sum _{s=1}^{d-1}{p^{s-1}}+(p-1)p^{\pi _{1}(2)-1}\nonumber \\ =&(j_d-i_d+p^{\bar{\beta }}-1)p^{m-\bar{\beta }}+1+(p-1)p^{\pi _{1}(2)-1}>(p-1)p^{\pi _{1}(2)-1}. \end{aligned}$$

(A15)

Again we got a contradiction. Hence $i_{\pi _{\beta }(1)}=j_{\pi _{\beta }(1)}~\forall \beta =1,2,\hdots ,k$. Now without loss of generality, we assume that there exists a positive integer $\beta '\le k$ such that $i_{\pi _{\beta }(r)}=j_{\pi _{\beta }(r)},~\forall \beta =1,2,\hdots ,\beta '-1$ and $r=1,2,\hdots ,n_{\beta }$. Assume t be the smallest integer with $i_{\pi _{\beta '}(t)}\ne j_{\pi _{\beta '}(t)}$. Now, let us define $i^\eta$ and $j^\eta$ same as in (A2) and (A3) respectively. Then it can easily be obtained that $j^\eta =(i^\eta +\tau )mod~L$ and hence using Lemma 3, we get (A13). Case 2 ($i_{\pi _{1}(2)}\ne j_{\pi _{1}(2)}$) In this case, let $i^\eta$ and $j^\eta$ are modified from i and j by changing only last bit of i and j as $i_{m}^\eta =i_{m}-\eta$ and $j_{m}^\eta =j_{m}-\eta$. Then

$$\begin{aligned} \textbf{c}_{i^\eta }-\textbf{c}_{i}&=f_{i^\eta }-f_{i}+\frac{\lambda }{p}(i_m-\eta -i_m)u_1\nonumber \\&=\frac{\lambda }{p}\left[ i_m^\eta i_{\pi _1(2)}-i_mi_{\pi _1(2)}\right] +g_{m}(i_m-\eta -i_m)+\frac{\lambda }{p}(i_m-\eta -i_m)u_1\nonumber \\&=-\eta \left[ \frac{\lambda }{p}i_{\pi _1(2)}+g_m+\frac{\lambda }{p}u_1\right] . \end{aligned}$$

(A16)

Similarly,

$$\begin{aligned} \textbf{c}_{j^\eta }-\textbf{c}_{j}=-\eta \left[ \frac{\lambda }{p}j_{\pi _1(2)}+g_m+\frac{\lambda }{p}u_1\right] . \end{aligned}$$

(A17)

By subtracting (A17) from (A16), we get

$$\begin{aligned} (\textbf{c}_{j^\eta }-\textbf{c}_{i^\eta })-(\textbf{c}_{j}-\textbf{c}_{i})=\eta \frac{\lambda }{p}(i_{\pi _{1}(2)}-j_{\pi _{1}(2)}). \end{aligned}$$

(A18)

Now following the same steps as in (A7), (A8), (A9), and (A10), we get

$$\begin{aligned} \sum _{\eta =1}^{p-1}{\omega _\lambda ^{\textbf{c}_{j^\eta }-\textbf{c}_{i^\eta }}+\omega _\lambda ^{(\textbf{c}_{j}-\textbf{c}_{i})}}=0. \end{aligned}$$

(A19)

Till now we have proved that for $0<\tau \le (p-1)p^{\pi _1(2)-1}$, the value of $\mathcal {P}(\textbf{c})(\tau )=0$. Now in the rest of the proof, we will prove that for any $0\le \tau \le (p-1)p^{\pi _1(2)-1}$, the PCCF of any two different sequences in $\mathbf {C_{u_k}}$ is zero. For that let $0\le \varvec{\gamma _k},\varvec{\delta _k}\le p^k-1$ such that $\psi (a_{\mathbf {u_k}}^{\varvec{\gamma _k}}),\psi (a_{\mathbf {u_k}}^{\varvec{\delta _k}})\in \mathbf {C_{u_k}}$. Again for notational convenience, we denote $\psi (a_{\mathbf {u_k}}^{\varvec{\gamma _k}})$ and $\psi (a_{\mathbf {u_k}}^{\varvec{\delta _k}})$ by $\textbf{b}$ and $\textbf{c}$ respectively. Then, we need to prove that for $0\le \tau \le (p-1)p^{\pi _1(2)-1}$,

$$\begin{aligned} \mathcal {P}(\textbf{c},\textbf{b})(\tau )=\sum _{i=0}^{L-1}{\omega _\lambda ^{\textbf{c}_{(i+\tau )mod~L}-\textbf{b}_i}}=0. \end{aligned}$$

(A20)

Let $j=(i+\tau )mod~L$. Following similar arguments as in Case 1 and Case 2, for any pair (i, j), we can find other pairs $(i^\eta ,j^\eta ),~\eta =1,2,\hdots ,k-1$ such that

$$\begin{aligned} \sum _{\eta =1}^{p-1}{\omega _\lambda ^{\textbf{c}_{j^\eta }-\textbf{b}_{i^\eta }}+\omega _\lambda ^{(\textbf{c}_{j}-\textbf{b}_{i})}}=0, \end{aligned}$$

(A21)

for $\tau \ne 0$, where $j^\eta =(i^\eta +\tau )mod~L$. Therefore, we can obtain that (A20) holds for $\tau \ne 0$. Now, it remains to prove that,

$$\begin{aligned} \mathcal {P}(\textbf{c},\textbf{b})(0)=\sum _{i=0}^{L-1}{\omega _\lambda ^{\textbf{c}_{i}-\textbf{b}_i}}=0. \end{aligned}$$

(A22)

Taking (A1) into consideration, let $\textbf{c}-\textbf{b}=\frac{\lambda }{p}.\textbf{d}$. Then $\textbf{d}$ is a non-zero codeword in $GRM_p(m, 1)$. Now let $\textbf{d}=(d_1,d_2,\hdots ,d_{p^m})$ and hence $\textbf{d}$ can be written as linear combination of $\theta (x_1),\theta (x_2),$ $\hdots ,\theta (x_m)$ as $\textbf{d}=c_1\cdot \theta (x_1)+c_2\cdot \theta (x_2)+\cdots +c_m\cdot \theta (x_m)$, where $c_i\in \mathbb {Z}_p,~1\le i\le m$. For each i, $\theta (x_i)$ contains each element of the set $\{0,1,\hdots ,p-1\}$, $p^{m-1}$ times. Hence $\textbf{d}$ will also contains each element of the set $\{0,1,\hdots ,p-1\}$, $p^{m-1}$ times. Now (A22) can be written as

$$\begin{aligned} \mathcal {P}(\textbf{c},\textbf{b})(0)=\sum _{i=0}^{L-1}{\omega _\lambda ^{\frac{\lambda }{p}\textbf{d}_{i}}}=\sum _{i=0}^{L-1}{\omega _\lambda ^{\frac{\lambda }{p}\textbf{d}_{i}}}=\sum _{i=0}^{L-1}{\omega _p^{\textbf{d}_{i}}}=0. \end{aligned}$$

(A23)

Hence the Theorem 1 is proved.$\square$