Six constructions of asymptotically optimal codebooks via the character sums

Abstract

In this paper, using additive characters of finite field, we find a codebook which is equivalent to the measurement matrix in Mohades et al. (IEEE Signal Process Lett 21(7):839–843, 2014). The advantage of our construction is that it can be generalized naturally to construct the other five classes of codebooks using additive and multiplicative characters of finite field. We determine the maximum cross-correlation amplitude of these codebooks by the properties of characters and character sums. We prove that all the codebooks we constructed are asymptotically optimal with respect to the Welch bound. The parameters of these codebooks are new.

1 Introduction

An (N, K) codebook $\mathcal {C}=\{\mathbf {c}_0,\mathbf {c}_1, ...,\mathbf {c}_{N-1}\}$ is a set of N unit-norm complex vectors $\mathbf {c}_i \in \mathbb {C}^K$ over an alphabet A, where $i=0, 1, \ldots , N-1$. The size of A is called the alphabet size of $\mathcal {C}$. As a performance measure of a codebook in practical applications, the maximum cross-correlation magnitude of an (N, K) codebook $\mathcal {C}$ is defined by

$$\begin{aligned} I_{max}(\mathcal {C})=\underset{0\le i\ne j \le N-1}{\max }|\mathbf {c}_i\mathbf {c}_j^H|, \end{aligned}$$

where $\mathbf {c}_j^H$ denotes the conjugate transpose of the complex vector $\mathbf {c}_j$. To evaluate an (N, K) codebook $\mathcal {C}$, it is important to find the minimum achievable $I_{max}(\mathcal {C})$ or its lower bound. The Welch bound [27] provides a well-known lower bound on $I_{max}(\mathcal {C})$,

$$\begin{aligned} I_{max}(\mathcal {C})\ge I_W=\sqrt{\frac{N-K}{(N-1)K}}. \end{aligned}$$

The equality holds if and only if for all pairs of (i, j) with $i\ne j$

$$\begin{aligned} |\mathbf {c}_i\mathbf {c}_j^H|=\sqrt{\frac{N-K}{(N-1)K}}. \end{aligned}$$

A codebook $\mathcal {C}$ achieving the Welch bound equality is called a maximum-Welch-bound-equality (MWBE) codebook [24] or an equiangular tight frame [14]. MWBE codebooks are employed in various applications including code-division multiple-access(CDMA) communication systems [20], communications [24], combinatorial designs [3, 4, 29], packing [2], compressed sensing [1], coding theory [5] and quantum computing [23]. To our knowledge, only the following MWBE codebooks are presented as follows:

• (N, N) orthogonal MWBE codebooks for any $N>1$ [24, 29];

• $(N,N-1)$ MWBE codebooks for $N>1$ based on discrete Fourier transformation matrices [24, 29] or m-sequences [24];

• (N, K) MWBE codebooks from conference matrices [2, 25], where $N=2K=2^{d+1}$ for a positive integer d or $N=2K=p^d+1$ for an odd prime p and a positive integer d;

• (N, K) MWBE codebooks based on $(N, K, \lambda )$ difference sets in cyclic groups [29] and abelian groups [3, 4];

• (N, K) MWBE codebooks from $(2, k, \nu )$-Steiner systems [7];

• (N, K) MWBE codebooks depended on graph theory and finite geometries [6, 8, 9, 22].

The construction of an MWBE codebook is known to be very hard in general, and the known classes of MWBE codebooks only exist for very restrictive N and K. Many researches have been done instead to construct asymptotically optimal codebooks, i.e., codebook $\mathcal {C}$ whose $I_{max}(\mathcal {C})$ asymptotically achieves the Welch bound. In [24], Sarwate gave some asymptotically optimal codebooks from codes and signal sets. As an extension of the optimal codebooks based on difference sets, various types of asymptotically optimal codebooks based on almost difference sets, relative difference sets and cyclotomic classes were proposed, see [3, 13, 31,32,33]. Asymptotically optimal codebooks constructed from binary row selection sequences were presented in [12, 30]. In [10, 11, 17,18,19], some asymptotically optimal codebooks were constructed via Jacobi sums and hyper Eisenstein sum.

In [21], the authors combined a Reed–Solomon generator matrix with itself by the tensor product and employed this generated matrix to construct a complex measurement matrix. They proved that this matrix is asymptotically optimal according to the Welch bound. In this paper, we find a codebook which is equivalent to the measurement matrix in [21]. The codebook is actually the first construction in Sect. 3, using additive characters of finite field. The advantage of our construction is that it can be generalized naturally to construct the other five classes of codebooks using additive and multiplicative characters of finite field. We determine the maximum cross-correlation amplitude of these codebooks by the properties of characters and character sums. All of these codebooks we constructed are asymptotically optimal according to the Welch bound. As a comparison, in Table 1, we list the parameters of some known classes of asymptotically optimal codebooks and those of the new ones.

This paper is organized as follows. In Sect. 2, we recall some notations and basic results which will be needed in our discussion. In Sect. 3, we present our six constructions of asymptotically optimal codebooks. In Sect. 4, we derive another family of codebooks, which are also asymptotically optimal. In Sect. 5, we conclude this paper.

Table 1 The parameters of codebooks asymptotically meeting the Welch bound

2 Preliminaries

In this section, we introduce some basic results on characters and character sums over finite fields, which will play important roles in the constructions of codebooks.

In this paper, we set q be a power of a prime p, and $\mathbb {F}_q$ be a finite field with q elements. For a set E, $\#E$ denotes the cardinality of E.

2.1 Characters over finite fields

Let $\mathbb {F}_q$ be a finite field. In this subsection, we recall the definitions of the additive and multiplicative characters of $\mathbb {F}_q$.

For each $a\in \mathbb {F}_q$, an additive character of $\mathbb {F}_q$ is defined by the function $\chi _a(x)=\zeta _p^{{\text {Tr}}_{q/p}(ax)}$, where $\zeta _p$ is a primitive p-th root of complex unity and ${\text {Tr}}_{q/p}(\cdot )$ is the trace function from $\mathbb {F}_q$ to $\mathbb {F}_p$. By the definition, $\chi _a(x)=\chi _1(ax)$. When $a=0$, we call $\chi _0$ the trivial additive character of $\mathbb {F}_q$. When $a=1$, we call $\chi _1$ the canonical additive character of $\mathbb {F}_q$. Let $\widehat{\mathbb {F}_q}$ be the set of all additive characters of $\mathbb {F}_q$. The orthogonal relation of additive characters (see [16]) is given by

$$\begin{aligned} \sum _{x\in \mathbb {F}_q}\chi _a(x)=\left\{ \begin{array}{ll} q,&{} \hbox {if } a=0,\\ 0,&{} \hbox {otherwise.} \end{array} \right. \end{aligned}$$

As in [16], the multiplicative characters of $\mathbb {F}_q$ is defined as follows. For $j=0,1,...,q-2$, the functions $\varphi _j$ defined by

$$\begin{aligned} \varphi _j(\alpha ^i)=\zeta _{q-1}^{ij}, \end{aligned}$$

are all the multiplicative characters of $\mathbb {F}_q$, where $\alpha$ is a primitive element of $\mathbb {F}_q^*$, and $0\le i\le q-2$. If $j=0$, we have $\varphi _0(x)=1$ for any $x\in \mathbb {F}_q^*$, $\varphi _0$ is called the trivial multiplicative character of $\mathbb {F}_q$. Let $\widehat{\mathbb {F}_q^*}$ be the set of all the multiplicative characters of $\mathbb {F}_q^*$.

Let $\varphi$ be a multiplicative character of $\mathbb {F}_q$. The orthogonal relation of multiplicative characters (see [16]) is given by

$$\begin{aligned} \sum _{x\in \mathbb {F}_q^*}\varphi (x)=\left\{ \begin{array}{ll} q-1,&{} \hbox {if } \varphi =\varphi _0,\\ 0,&{} \hbox {otherwise.} \end{array} \right. \end{aligned}$$

2.2 Character sums over finite fields

2.2.1 Gauss sum

Let $\varphi$ be a multiplicative character of $\mathbb {F}_q$ and $\chi$ an additive character of $\mathbb {F}_q$. Then the Gauss sum over $\mathbb {F}_q$ is given by

$$\begin{aligned} G(\varphi ,\chi )=\sum _{x\in \mathbb {F}_q^*}\varphi (x){\chi }(x). \end{aligned}$$

For simplicity, we write $G(\varphi ,\chi _1)$ over $\mathbb {F}_q$ simply as $g(\varphi )$. It is easy to see the absolute value of $G(\varphi ,\chi )$ is at most $q-1$, but is much smaller in general. The following lemma shows all the cases.

Lemma 2.1

[16, Theorem 5.11] Let $\varphi$ be a multiplicative character and $\chi$ an additive character of $\mathbb {F}_q$. Then the Gauss sum $G(\varphi ,\chi )$ over $\mathbb {F}_q$ satisfies

$$\begin{aligned} G(\varphi ,\chi )=\left\{ \begin{array}{ll} q-1, &{} \hbox {if } \varphi =\varphi _0, \chi =\chi _0, \\ -1,&{} \hbox {if } \varphi =\varphi _0, \chi \ne \chi _0,\\ 0,&{} \hbox {if } \varphi \ne \varphi _0, \chi =\chi _0. \end{array} \right. \end{aligned}$$

For $\varphi \ne \varphi _0$ and $\chi \ne \chi _0$, we have $\left| G(\varphi ,\chi )\right| =\sqrt{q}$.

Lemma 2.2

[16] Gauss sums for the finite field $\mathbb {F}_q$ satisfy the following property:

$G(\varphi ,\chi _{ab})=\overline{\varphi }(a)G(\varphi ,\chi _{b})$ for $a\in \mathbb {F}_q^*$, $b\in \mathbb {F}_q$, where $\overline{\varphi }$ denotes the complex conjugate of $\varphi$.

2.2.2 Jacobi sum

The definition of a multiplicative character $\varphi$ can be extended as follows.

$$\begin{aligned} \varphi (0)=\left\{ \begin{array}{ll} 1, &{} \hbox {if } \varphi =\varphi _0,\\ 0,&{} \hbox {if } \varphi \ne \varphi _0. \end{array} \right. \end{aligned}$$

Let $\varphi _1$ and $\varphi _2$ be multiplicative characters of $\mathbb {F}_q$. The sum

$$\begin{aligned} J(\varphi _1,\varphi _2)=\sum _{c_1+c_2=1, c_1,c_2\in \mathbb {F}_q}\varphi _1(c_1)\varphi _2(c_2) \end{aligned}$$

is called a Jacobi sum in $\mathbb {F}_q$.

The values of Jacobi sums are given as follows.

Lemma 2.3

[16, Theorem 5.19, Theorem 5.20] For the values of Jacobi sums, we have the following results.

• (1) If $\varphi _1$ and $\varphi _2$ are trivial, then $J(\varphi _1,\varphi _2)=q$.

• (2) If one of the $\varphi _1$ and $\varphi _2$ is trivial, the other is nontrivial, $J(\varphi _1,\varphi _2)=0$.

• (3) If $\varphi _1$ and $\varphi _2$ are both nontrivial and $\varphi _1\varphi _2$ is nontrivial, then $|J(\varphi _1,\varphi _2)|=\sqrt{q}$.

• (4) If $\varphi _1$ and $\varphi _2$ are both nontrivial and $\varphi _1\varphi _2$ is trivial, then $|J(\varphi _1,\varphi _2)|=1$.

2.3 A general construction of codebooks

Let D be a set and $K=\#D$. Let E be a set of some functions which satisfy

$$\begin{aligned} f:D\rightarrow S,\ \ {\hbox {where S is the unit circle on the complex plane.}} \end{aligned}$$

A general construction of codebooks is stated as follows in the complex plane,

$$\begin{aligned} \mathcal {C}(D;E)=\left\{ \mathbf {c}_f:=\frac{1}{\sqrt{K}}(f(x))_{x\in D}, f\in E\right\} . \end{aligned}$$

3 Six constructions of codebooks

In [21], the authors combined a Reed–Solomon generator matrix with itself by the tensor product and employed this generated matrix to construct a complex measurement matrix. They proved that this matrix is asymptotically optimal according to the Welch bound. In this paper, we find a codebook which is equivalent to the measurement matrix in [21]. The codebook is actually the first construction in Sect. 3, which has been obtained in [26] recently. The advantage of the first construction is that it can be generalized naturally to construct the other five classes of asymptotically optimal codebooks using additive and multiplicative characters of finite field.

3.1 The first construction of codebooks

Let

$$\begin{aligned} D_1:=\{(x,y,z)\in \mathbb {F}_{q}\times \mathbb {F}_q\times \mathbb {F}_q: z=xy\}. \end{aligned}$$

Then $\#D_1=q^2$.

The codebook $\mathcal {C}_1$ is constructed as

$$\begin{aligned} \mathcal {C}_1:=\left\{ \frac{1}{q}(\chi _a(x)\chi _b(y)\chi _c(z))_{(x,y,z)\in D_1}:a,b,c\in \mathbb {F}_q\right\} . \end{aligned}$$

We can derive the following Theorem.

Theorem 3.1

([26, Theorem 1]) $\mathcal {C}_1$ is a codebook with $N_1=q^3$, $K_1=q^2$ and $I_{max}(\mathcal {C}_1)=\frac{1}{q}$.

Theorem 3.2

([26, Remark 1]) Let $I_{W1}$ be the Welch bound equality, for the given $N_1$, $K_1$ in the current section. We have

$$\begin{aligned} \lim _{{q}\rightarrow \infty }\frac{I_{max}(\mathcal {C}_1)}{I_{W1}}=1, \end{aligned}$$

then the codebooks we proposed are asymptotically optimal.

When codebooks employed in practical applications, only those with only a few correlation values are interesting. The distribution of correlation magnitudes of $\mathcal {C}_1$ is given as follows.

Corollary 3.3

([26, Theorem 1])

$$\begin{aligned} |\mathbf {c}\mathbf {c}'^H|=\left\{ \begin{array}{ll} 1,&{} q^3\hbox { times},\\ \frac{1}{q},&{} q^6-q^5\hbox { times},\\ 0,&{} q^5-q^3\hbox { times}. \end{array} \right. \end{aligned}$$

3.2 The second construction of codebooks

Let

$$\begin{aligned} D_2:=\{(x,y,z)\in \mathbb {F}_{q}^*\times \mathbb {F}_q\times \mathbb {F}_q: z=xy\}. \end{aligned}$$

Then $\#D_2=(q-1)q$.

The codebook $\mathcal {C}_2$ is constructed as

$$\begin{aligned} \mathcal {C}_2:=\left\{ \frac{1}{\sqrt{(q-1)q}}(\varphi (x)\chi _b(y)\chi _c(z))_{(x,y,z)\in D_2}:\varphi \in \widehat{\mathbb {F}_q^*},b,c\in \mathbb {F}_q\right\} . \end{aligned}$$

We can derive the following Theorem.

Theorem 3.4

$\mathcal {C}_2$ is a codebook with $N_2=(q-1)q^2$, $K_2=(q-1)q$ and $I_{max}(\mathcal {C}_2)=\frac{1}{q-1}$.

Proof

By the definition of $\mathcal {C}_2$, it contains $(q-1)q^2$ codewords of length $(q-1)q$. Then it is easy to see $N_2=(q^2-1)q$ and $K_2=(q-1)q$. Let $\mathbf {c}$ and $\mathbf {c}'$ be any different codewords in $\mathcal {C}_2$, $\mathbf {c}=\frac{1}{\sqrt{K_2}}(\varphi _j(x)\chi _{b_1}(y)\chi _{c_1}(z))_{(x,y,z)\in D_2}$ and $\mathbf {c}'=\frac{1}{\sqrt{K_2}}(\varphi _k(x)\chi _{b_2}(y)\chi _{c_2}(z))_{(x,y,z)\in D_2}$, where $\varphi _j,\varphi _k\in \widehat{\mathbb {F}_q^*},b_1,b_2,c_1,c_2\in \mathbb {F}_q$. Then the correlation of $\mathbf {c}$ and $\mathbf {c}'$ is as follows.

$$\begin{aligned}&K_2\mathbf {c}\mathbf {c}'^H\\&\quad =\sum _{(x,y,z)\in D_2}\varphi _j(x)\chi _{b_1}(y)\chi _{c_1}(z)\overline{\varphi _k(x)\chi _{b_2}(y)\chi _{c_2}(z)}\\&\quad =\sum _{x\in \mathbb {F}_q^*, y\in \mathbb {F}_q}\varphi _j\overline{\varphi _k}(x)\chi ((b_1-b_2)y+(c_1-c_2)xy)\\&\quad =\sum _{x\in \mathbb {F}_q^*, y\in \mathbb {F}_q}\varphi (x)\chi (by+cxy) \ \ (\hbox {where } \varphi =\varphi _j\overline{\varphi _k}, b=b_1-b_2 \hbox { and } c=c_1-c_2)\\&\quad =\sum _{x\in \mathbb {F}_q^*}\varphi (x)\sum _{y\in \mathbb {F}_q}\chi ((b+cx)y)\\&\quad =\sum _{x\in \mathbb {F}_q^*,b+cx=0}\varphi (x)q. \end{aligned}$$

The last equation holds by the orthogonal relation of additive characters.

When $b\ne 0$ and $c\ne 0$, we have

$$\begin{aligned} K_2\mathbf {c}\mathbf {c}'^H=q\varphi \left( -\frac{b}{c}\right) . \end{aligned}$$

When $b\ne 0$, $c=0$, or $b=0, c\ne 0$, it is easy to see $\mathbf {c}\mathbf {c}'^H=0$.

When $b=0$ and $c=0$, since $\mathbf {c}\ne \mathbf {c}'$, $\varphi$ is nontrivial. We have

$$\begin{aligned} K_2\mathbf {c}\mathbf {c}'^H=\sum _{x\in \mathbb {F}_q^*}\varphi (x)q=0, \end{aligned}$$

by the orthogonal relation of multiplicative characters.

Therefore, we have

$$\begin{aligned} I_{max}(\mathcal {C}_2)=max\{|\mathbf {c}\mathbf {c}'^H|:\mathbf {c},\mathbf {c}'\in \mathcal {C}, and\ \mathbf {c}\ne \mathbf {c}'\}=\frac{q}{K_2}=\frac{1}{q-1} . \end{aligned}$$

$\square$

Using Theorem 3.4, we can derive the ratio of $I_{max}(\mathcal {C}_2)$ of the proposed codebooks to that of the MWBE codebooks and show the asymptotic optimality of the proposed codebooks as in the following theorem.

Theorem 3.5

Let $I_{W2}$ be the Welch bound equality, for the given $N_2$, $K_2$ in the current section. We have

$$\begin{aligned} \lim _{{q}\rightarrow \infty }\frac{I_{max}(\mathcal {C}_2)}{I_{W2}}=1, \end{aligned}$$

then the codebooks we proposed are asymptotically optimal.

Proof

Note that $N_2=(q-1)q^2$ and $K_2=(q-1)q$. Then the corresponding Welch bound is

$$\begin{aligned} I_{W2}=\sqrt{\frac{N_2-K_2}{(N_2-1)K_2}}=\sqrt{\frac{(q-1)q^2-(q-1)q}{((q-1)q^2-1)(q-1)q}}=\sqrt{\frac{q-1}{q^3-q^2-1}}, \end{aligned}$$

we have

$$\begin{aligned} \frac{I_{max}(\mathcal {C}_2)}{I_{W2}}=\sqrt{\frac{q^3-q^2-1}{(q-1)^3}}. \end{aligned}$$

It is obvious that $\lim _{q\rightarrow +\infty }\frac{I_{max}(\mathcal {C}_2)}{I_{W2}}=1$. The codebook $\mathcal {C}_2$ asymptotically meets the Welch bound. This completes the proof. $\square$

In Table 2, we provide some explicit values of the parameters of the codebooks we proposed for some given q, and corresponding numerical data of the Welch bound for comparison. The numerical results show that the codebooks $\mathcal {C}_2$ asymptotically meet the Welch bound.

Table 2 Parameters of the $(N_2,K_2)$ codebook of Section III

The distribution of correlation magnitudes of $\mathcal {C}_2$ is given as follows.

Corollary 3.6

$$\begin{aligned} |\mathbf {c}\mathbf {c}'^H|=\left\{ \begin{array}{ll} 1,&{} (q-1)q^2\hbox { times},\\ \frac{1}{q-1},&{} q-1)^4q^2\hbox { times},\\ 0,&{} q-1)q^3(2q-3)\hbox { times}. \end{array} \right. \end{aligned}$$

Example 1

Let $q=p=3$. Then

$$\begin{aligned} D_2=\{(1,0,0), (1,1,1), (1,2,2), (2,0,0), (2,1,2), (2,2,1)\} \end{aligned}$$

and $K_2=\#D_2=6$. Let $\zeta =\zeta _3$. Thus, the set $\mathcal {C}_2$ consists of the following 18 codewords of length 6:

$$\begin{aligned} \mathbf {c}_0= & {} \frac{1}{\sqrt{6}}(1,1,1,1,1,1),\ \ \ \mathbf {c}_1=\frac{1}{\sqrt{6}}(1,\zeta ,\zeta ^2,1,\zeta ^2,\zeta ),\ \ \ \mathbf {c}_2=\frac{1}{\sqrt{6}}(1,\zeta ^2,\zeta ,1,\zeta ,\zeta ^2),\\ \mathbf {c}_3= & {} \frac{1}{\sqrt{6}}(1,\zeta ,\zeta ^2,1,\zeta ,\zeta ^2),\ \ \ \mathbf {c}_4=\frac{1}{\sqrt{6}}(1,\zeta ^2,\zeta ,1,1,1),\ \ \ \mathbf {c}_5=\frac{1}{\sqrt{6}}(1,1,1,1,\zeta ^2,\zeta ),\\ \mathbf {c}_6= & {} \frac{1}{\sqrt{6}}(1,\zeta ^2,\zeta ,1,\zeta ^2,\zeta ),\ \ \ \mathbf {c}_7=\frac{1}{\sqrt{6}}(1,1,1,1,\zeta ,\zeta ^2),\ \ \ \mathbf {c}_8=\frac{1}{\sqrt{6}}(1,\zeta ,\zeta ^2,1,1,1),\\ \mathbf {c}_9= & {} \frac{1}{\sqrt{6}}(1,1,1,-1,-1,-1),\ \ \ \mathbf {c}_{10}=\frac{1}{\sqrt{6}}(1,\zeta ,\zeta ^2,-1,-\zeta ^2,-\zeta ),\\ \mathbf {c}_{11}= & {} \frac{1}{\sqrt{6}}(1,\zeta ^2,\zeta ,-1,-\zeta ,-\zeta ^2),\\ \mathbf {c}_{12}= & {} \frac{1}{\sqrt{6}}(1,\zeta ,\zeta ^2,-1,-\zeta ,-\zeta ^2),\ \ \ \mathbf {c}_{13}=\frac{1}{\sqrt{6}}(1,\zeta ^2,\zeta ,-1,-1,-1),\\ \mathbf {c}_{14}= & {} \frac{1}{\sqrt{6}}(1,1,1,-1,-\zeta ^2,-\zeta ),\\ \mathbf {c}_{15}= & {} \frac{1}{\sqrt{6}}(1,\zeta ^2,\zeta ,-1,-\zeta ^2,-\zeta ),\ \ \ \mathbf {c}_{16}=\frac{1}{\sqrt{6}}(1,1,1,-1,-\zeta ,-\zeta ^2),\\ \mathbf {c}_{17}= & {} \frac{1}{\sqrt{6}}(1,\zeta ,\zeta ^2,-1,-1,-1). \end{aligned}$$

It is easy to verify that this codebook is a (18, 6) codebook with $I_{max}=\frac{1}{2}$. This is consistent with the conclusion of Theorem 3.5.

3.3 The third construction of codebooks

Let

$$\begin{aligned} D_3:=\{(x,y,z)\in \mathbb {F}_{q}^*\times \mathbb {F}_q^*\times \mathbb {F}_q^*: z=xy\}. \end{aligned}$$

Then $\#D_3=(q-1)^2$.

The codebook $\mathcal {C}_3$ is constructed as

$$\begin{aligned} \mathcal {C}_3:=\left\{ \frac{1}{q-1}(\chi _a(x)\chi _b(y)\varphi (z))_{(x,y,z)\in D_3}:a,b\in \mathbb {F}_q, \varphi \in \widehat{\mathbb {F}_q^*}\right\} . \end{aligned}$$

We can derive the following Theorem.

Theorem 3.7

$\mathcal {C}_3$ is a codebook with $N_3=q^2(q-1)$, $K_3=(q-1)^2$ and $I_{max}(\mathcal {C}_3)=\frac{q}{(q-1)^2}$.

Proof

By the definition of $\mathcal {C}_3$, it contains $q^2(q-1)$ codewords of length $(q-1)^2$. Then it is easy to see $N_3=q^2(q-1)$ and $K_3=(q-1)^2$. Let $\mathbf {c}$ and $\mathbf {c}'$ be any different codewords in $\mathcal {C}_3$, $\mathbf {c}=\frac{1}{q-1}(\chi _{a_1}(x)\chi _{b_1}(y)\varphi _j(z))_{(x,y,z)\in D_3}$ and $\mathbf {c}'=\frac{1}{q-1}(\chi _{a_2}(x)\chi _{b_2}(y)\varphi _k(z))_{(x,y,z)\in D_3}$, where $a_1,a_2,b_1,b_2\in \mathbb {F}_q, \varphi _j,\varphi _k\in \widehat{\mathbb {F}_q^*}$. Then the correlation of $\mathbf {c}$ and $\mathbf {c}'$ is as follows.

$$\begin{aligned}&K_3\mathbf {c}\mathbf {c}'^H\\&\quad =\sum _{(x,y,z)\in D_3}\chi _{a_1}(x)\chi _{b_1}(y)\varphi _j(z)\overline{\chi _{a_2}(x)\chi _{b_2}(y)\varphi _k(z)}\\&\quad =\sum _{x,y\in \mathbb {F}_q^*}\chi ((a_1-a_2)x+(b_1-b_2)y)\varphi _j\overline{\varphi _k}(xy)\\&\quad =\sum _{x,y\in \mathbb {F}_q^*}\chi (ax+by)\varphi (xy) \ \ (\hbox {where } a=a_1-a_2, b=b_1-b_2, \hbox { and } \varphi =\varphi _j\overline{\varphi _k})\\&\quad = \sum _{x\in \mathbb {F}_q^*}\chi _a(x)\varphi (x)\sum _{y\in \mathbb {F}_q^*}\chi _b(y)\varphi (y)\\&\quad =G(\varphi ,\chi _a)G(\varphi ,\chi _b). \end{aligned}$$

When $\varphi$ is trivial, since $\mathbf {c}\ne \mathbf {c}'$, we get $a\ne 0$ or $b\ne 0$. By Lemma 2.1, we have

$$\begin{aligned} K_3\mathbf {c}\mathbf {c}'^H=G(\varphi ,\chi _a)G(\varphi ,\chi _b)=\left\{ \begin{array}{ll} -(q-1),&{} \hbox {if } a=0,b\ne 0\hbox { or } b=0,a\ne 0,\\ (-1)(-1),&{} \hbox {if } a,b\in \mathbb {F}_q^*, \end{array} \right. \end{aligned}$$

When $\varphi$ is nontrivial, by Lemmas 2.1 and 2.2, we have

$$\begin{aligned} K_3\mathbf {c}\mathbf {c}'^H=G(\varphi ,\chi _a)G(\varphi ,\chi _b)=\left\{ \begin{array}{ll} 0,&{} \hbox {if } a=0\hbox { or } b=0,\\ \overline{\varphi }(ab)g^2(\varphi ),&{} \hbox {if } a,b\in \mathbb {F}_q^*. \end{array} \right. \end{aligned}$$

Therefore, we have

$$\begin{aligned} I_{max}(\mathcal {C}_3)=max\{|\mathbf {c}\mathbf {c}'^H|:\mathbf {c},\mathbf {c}'\in \mathcal {C}, and\ \mathbf {c}\ne \mathbf {c}'\}=\frac{q}{K_3}=\frac{q}{(q-1)^2}. \end{aligned}$$

$\square$

Using Theorem 3.7, we can derive the ratio of $I_{max}(\mathcal {C}_3)$ of the proposed codebooks to that of the MWBE codebooks and show the asymptotic optimality of the proposed codebooks as in the following theorem.

Theorem 3.8

Let $I_{W3}$ be the Welch bound equality, for the given $N_3$, $K_3$ in the current section. We have

$$\begin{aligned} \lim _{{q}\rightarrow \infty }\frac{I_{max}(\mathcal {C}_3)}{I_{W3}}=1, \end{aligned}$$

then the codebooks we proposed are asymptotically optimal.

Proof

Note that $N_3=(q-1)q^2$ and $K_3=(q-1)^2$. Then the corresponding Welch bound is

$$\begin{aligned} I_{W3}=\sqrt{\frac{N_3-K_3}{(N_3-1)K_3}}=\sqrt{\frac{(q-1)q^2-(q-1)^2}{((q-1)q^2-1)(q-1)^2}}=\sqrt{\frac{q^2-q+1}{(q^3-q^2-1)(q-1)}}, \end{aligned}$$

we have

$$\begin{aligned} \frac{I_{max}(\mathcal {C}_3)}{I_{W3}}=\frac{q}{q-1}\sqrt{\frac{q^3-q^2-1}{(q^2-q+1)(q-1)}}. \end{aligned}$$

It is obvious that $\lim _{q\rightarrow +\infty }\frac{I_{max}(\mathcal {C}_3)}{I_{W3}}=1$. The codebook $\mathcal {C}_3$ asymptotically meets the Welch bound. This completes the proof. $\square$

In Table 3, we provide some explicit values of the parameters of the codebooks we proposed for some given q, and corresponding numerical data of the Welch bound for comparison. The numerical results show that the codebooks $\mathcal {C}_3$ asymptotically meet the Welch bound.

Table 3 Parameters of the $(N_3,K_3)$ codebook of Section III

The distribution of correlation magnitudes of $\mathcal {C}_3$ is given as follows.

Corollary 3.9

$$\begin{aligned} |\mathbf {c}\mathbf {c}'^H|=\left\{ \begin{array}{ll} 1,&{} (q-1)q^2\hbox { times},\\ \frac{q-1}{K_3},&{} 2(q-1)^2q^2\hbox { times},\\ \frac{1}{K_3},&{} q-1)^3q^2\hbox { times},\\ 0,&{} (q-2)(q-1)q^2(2q-1)\hbox { times},\\ \frac{q}{K_3},&{} (q-2)(q-1)^3q^2\hbox { times},\\ \end{array} \right. \end{aligned}$$

3.4 The fourth construction of codebooks

Let

$$\begin{aligned} D_4:=\{(x,y,z)\in \mathbb {F}_{q}^*\times \mathbb {F}_q^*\times \mathbb {F}_q^*: z=xy\}. \end{aligned}$$

Then $\#D_4=(q-1)^2$.

The codebook $\mathcal {C}_4$ is constructed as

$$\begin{aligned} \mathcal {C}_4:=\left\{ \frac{1}{q-1}(\varphi _i(x)\varphi _j(y)\chi _c(z))_{(x,y,z)\in D_4}:\varphi _i,\varphi _j\in \widehat{\mathbb {F}_q^*},\ c\in \mathbb {F}_q \right\} . \end{aligned}$$

We can derive the following Theorem.

Theorem 3.10

$\mathcal {C}_4$ is a codebook with $N_4=(q-1)^2q$, $K_4=(q-1)^2$ and $I_{max}(\mathcal {C}_4)=\frac{q}{(q-1)^2}$.

Proof

By the definition of $\mathcal {C}_4$, it contains $(q-1)^2q$ codewords of length $(q-1)^2$. Then it is easy to see $N_4=(q-1)^2q$ and $K_4=(q-1)^2$. Let $\mathbf {c}$ and $\mathbf {c}'$ be any different codewords in $\mathcal {C}_4$, $\mathbf {c}=\frac{1}{q-1}(\varphi _s(x)\varphi _t(y)\chi _{c_1}(z))_{(x,y,z)\in D_4}$ and $\mathbf {c}'=\frac{1}{q-1}(\varphi _s'(x)\varphi _t'(y)\chi _{c_2}(z))_{(x,y,z)\in D_4}$, where $\varphi _s,\varphi _t,\varphi _s',\varphi _t'\in \widehat{\mathbb {F}_q^*},c_1,c_2\in \mathbb {F}_q$. Then the correlation of $\mathbf {c}$ and $\mathbf {c}'$ is as follows.

$$\begin{aligned}&K_4\mathbf {c}\mathbf {c}'^H\\&\quad =\sum _{(x,y,z)\in D_4}\varphi _s(x)\varphi _t(y)\chi _{c_1}(z)\overline{\varphi _s'(x)\varphi _t'(y)\chi _{c_2}(z)}\\&\quad =\sum _{(x,y,z)\in D_4}\varphi _s\overline{\varphi _s'}(x)\varphi _t\overline{\varphi _t'}(y)\chi ((c_1-c_2)z)\\&\quad =\sum _{x,y\in \mathbb {F}_q^*}\varphi (x)\varphi '(y)\chi (cxy) \ \ (\hbox {where } \varphi =\varphi _s\overline{\varphi _s'}, \varphi '=\varphi _t\overline{\varphi _t'}, \hbox { and } c=c_1-c_2)\\&\quad = \sum _{x\in \mathbb {F}_q^*}\varphi (x)\chi _c(x)\sum _{y\in \mathbb {F}_q^*}\varphi '(y)\chi _c(y)\\&\quad =G(\varphi ,\chi _c)G(\varphi ',\chi _c). \end{aligned}$$

When $c=0$, since $\mathbf {c}\ne \mathbf {c}'$, at least one of $\varphi$ and $\varphi '$ is nontrivial. By Lemma 2.1, we have

$$\begin{aligned} K_4\mathbf {c}\mathbf {c}'^H=G(\varphi ,\chi _c)G(\varphi ',\chi _c)=0. \end{aligned}$$

When $c\ne 0$, by Lemmas 2.1 and 2.2, we have

$$\begin{aligned} K_4\mathbf {c}\mathbf {c}'^H=G(\varphi ,\chi _c)G(\varphi ',\chi _c)=\left\{ \begin{array}{ll} (-1)(-1), &{} \hbox {both } \varphi \hbox { and } \varphi ' \hbox { are trivial,} \\ (-1)\overline{\varphi '(c)}g(\varphi '),&{} \varphi \hbox { is trivial}, \hbox { and } \varphi ' \hbox { is nontrivial,}\\ (-1)\overline{\varphi (c)}g(\varphi ),&{} \varphi \hbox { is nontrivial}, \hbox { and } \varphi ' \hbox { is trivial,}\\ \overline{\varphi \varphi '}(c)g(\varphi )g(\varphi '),&{} \hbox {both } \varphi \hbox { and } \varphi ' \hbox { are nontrivial.} \end{array} \right. \end{aligned}$$

Therefore, we have

$$\begin{aligned} I_{max}(\mathcal {C}_4)=max\{|\mathbf {c}\mathbf {c}'^H|:\mathbf {c},\mathbf {c}'\in \mathcal {C}_4, and\ \mathbf {c}\ne \mathbf {c}'\}=\frac{q}{K_4}=\frac{q}{(q-1)^2}. \end{aligned}$$

$\square$

Using Theorem 3.10, we can derive the ratio of $I_{max}(\mathcal {C}_4)$ of the proposed codebooks to that of the MWBE codebooks and show the asymptotic optimality of the proposed codebooks as in the following theorem.

Theorem 3.11

Let $I_{W4}$ be the Welch bound equality, for the given $N_4$, $K_4$ in the current section. We have

$$\begin{aligned} \lim _{{q}\rightarrow \infty }\frac{I_{max}(\mathcal {C}_4)}{I_{W4}}=1, \end{aligned}$$

then the codebooks we proposed are asymptotically optimal.

Proof

Note that $N_4=(q-1)^2 q$ and $K_4=(q-1)^2$. Then the corresponding Welch bound is

$$\begin{aligned} I_{W4}=\sqrt{\frac{N_4-K_4}{(N_4-1)K_4}}=\sqrt{\frac{(q-1)^2 q-(q-1)^2}{((q-1)^2 q-1)(q-1)^2}}=\sqrt{\frac{q-1}{(q-1)^2q-1}}, \end{aligned}$$

we have

$$\begin{aligned} \frac{I_{max}(\mathcal {C}_4)}{I_{W4}}=\frac{q}{q-1}\sqrt{\frac{(q-1)^2q-1}{(q-1)^3}}. \end{aligned}$$

It is obvious that $\lim _{q\rightarrow +\infty }\frac{I_{max}(\mathcal {C}_4)}{I_{W4}}=1$. The codebook $\mathcal {C}_4$ asymptotically meets the Welch bound. This completes the proof. $\square$

In Table 4, we provide some explicit values of the parameters of the codebooks we proposed for some given q, and corresponding numerical data of the Welch bound for comparison. The numerical results show that the codebooks $\mathcal {C}_4$ asymptotically meet the Welch bound.

Table 4 Parameters of the $(N_4,K_4)$ codebook of Section III

The distribution of correlation magnitudes of $\mathcal {C}_4$ is given as follows.

Corollary 3.12

$$\begin{aligned} |\mathbf {c}\mathbf {c}'^H|=\left\{ \begin{array}{ll} 1,&{} (q-1)^2q\hbox { times},\\ 0,&{} (q-2)(q-1)^2q^2\hbox { times},\\ \frac{1}{K_4},&{} (q-1)^3q\hbox { times},\\ \frac{\sqrt{q}}{K_4} ,&{} 2(q-2)(q-1)^3q\hbox { times},\\ \frac{q}{K_4},&{} (q-2)^2(q-1)^3q\hbox { times}. \end{array} \right. \end{aligned}$$

3.5 The fifth construction of codebooks

Let

$$\begin{aligned} D_5:=\{(x,y,z)\in \mathbb {F}_{q}^*\times \mathbb {F}_q^*\times \mathbb {F}_q^*: z=x(1-y)\}. \end{aligned}$$

Then $\#D_5=(q-1)(q-2)$.

The codebook $\mathcal {C}_5$ is constructed as

$$\begin{aligned} \mathcal {C}_5:=\left\{ \frac{1}{\sqrt{(q-1)(q-2)}}(\chi _a(x)\varphi _i(y)\varphi _j(z))_{(x,y,z)\in D_5}:\varphi _i,\varphi _j\in \widehat{\mathbb {F}_q^*},\ a\in \mathbb {F}_q \right\} . \end{aligned}$$

We can derive the following Theorem.

Theorem 3.13

$\mathcal {C}_5$ is a codebook with $N_5=(q-1)^2q$, $K_5=(q-1)(q-2)$ and $I_{max}(\mathcal {C}_5)=\frac{q}{(q-1)(q-2)}$.

Proof

By the definition of $\mathcal {C}_5$, it contains $(q-1)^2q$ codewords of length $(q-1)(q-2)$. Then it is easy to see $N_5=(q-1)^2q$ and $K_5=(q-1)(q-2)$. Let $\mathbf {c}$ and $\mathbf {c}'$ be any different codewords in $\mathcal {C}_5$, $\mathbf {c}=\frac{1}{\sqrt{(q-1)(q-2)}}(\chi _{a_1}(x)\varphi _s(y)\varphi _t(z))_{(x,y,z)\in D_5}$ and $\mathbf {c}'=\frac{1}{\sqrt{(q-1)(q-2)}}(\chi _{a_2}(x)\varphi _s'(y)\varphi _t'(z))_{(x,y,z)\in D_5}$, where $\varphi _s,\varphi _t,\varphi _s',\varphi _t'\in \widehat{\mathbb {F}_q^*},a_1,a_2\in \mathbb {F}_q$. Then the correlation of $\mathbf {c}$ and $\mathbf {c}'$ is as follows.

$$\begin{aligned}&K_5\mathbf {c}\mathbf {c}'^H\\&\quad =\sum _{(x,y,z)\in D_5}\chi _{a_1}(x)\varphi _s(y)\varphi _t(z)\overline{\chi _{a_2}(x)\varphi _s'(y)\varphi _t'(z)}\\&\quad =\sum _{(x,y,z)\in D_5}\chi ((a_1-a_2)x)\varphi _s\overline{\varphi _s'}(y)\varphi _t\overline{\varphi _t'}(z)\\&\quad =\sum _{x,y\in \mathbb {F}_q^*,y\ne 1}\chi (ax)\varphi (y)\varphi '(x(1-y)) \ \ (\hbox {where } a=a_1-a_2, \varphi =\varphi _s\overline{\varphi _s'}, \hbox { and } \varphi '=\varphi _t\overline{\varphi _t'})\\&\quad = \sum _{x\in \mathbb {F}_q^*}\chi (ax)\varphi '(x)\sum _{y\in \mathbb {F}_q^*,y\ne 1}\varphi (y)\varphi '(1-y)\\&\quad =G(\varphi ',\chi _a)(J(\varphi ,\varphi ')-\varphi (0)\varphi '(1)-\varphi (1)\varphi '(0)). \end{aligned}$$

When $a=0$, since $\mathbf {c}\ne \mathbf {c}'$, at least one of $\varphi$ and $\varphi '$ is nontrivial, by Lemmas 2.1 and 2.3, we have

$$\begin{aligned} K_5\mathbf {c}\mathbf {c}'^H=\left\{ \begin{array}{ll} (-1)(q-1),&{} \varphi ' \hbox { is trivial, and } \varphi \hbox { is nontrivial,}\\ 0 ,&{} \varphi ' \hbox { are nontrivial.} \end{array} \right. \end{aligned}$$

When $a\ne 0$, by Lemmas 2.1 and 2.3, we have

$$\begin{aligned} K_5\mathbf {c}\mathbf {c}'^H=\left\{ \begin{array}{ll} (-1)(q-2), &{} \hbox {both } \varphi ' \hbox { and } \varphi \hbox { are trivial,} \\ (-1)(-1),&{} \varphi ' \hbox { is trivial, and } \varphi \hbox { is nontrivial,}\\ \overline{\varphi '}(a)g(\varphi ')(-1),&{} \varphi ' \hbox { is nontrivial, and } \varphi \hbox { is trivial,}\\ \overline{\varphi '}(a)g(\varphi ')J(\varphi ,\varphi '),&{} \hbox {both } \varphi ' \hbox { and } \varphi \hbox { are nontrivial.} \end{array} \right. \end{aligned}$$

Therefore, we have

$$\begin{aligned} I_{max}(\mathcal {C}_5)=max\{|\mathbf {c}\mathbf {c}'^H|:\mathbf {c},\mathbf {c}'\in \mathcal {C}, and\ \mathbf {c}\ne \mathbf {c}'\}=\frac{q}{K_5}=\frac{q}{(q-1)(q-2)}, \end{aligned}$$

the maximal value obtained when all of $\varphi , \varphi '$ and $\varphi \varphi '$ are nontrivial. $\square$

Using Theorem 3.13, we can derive the ratio of $I_{max}(\mathcal {C}_5)$ of the proposed codebooks to that of the MWBE codebooks and show the asymptotic optimality of the proposed codebooks as in the following theorem.

Theorem 3.14

Let $I_{W5}$ be the Welch bound equality, for the given $N_5$, $K_5$ in the current section. We have

$$\begin{aligned} \lim _{{q}\rightarrow \infty }\frac{I_{max}(\mathcal {C}_5)}{I_{W5}}=1, \end{aligned}$$

then the codebooks we proposed are asymptotically optimal.

Proof

Note that $N_5=(q-1)^2 q$ and $K_5=(q-1)(q-2)$. Then the corresponding Welch bound is

$$\begin{aligned} I_{W5}=\sqrt{\frac{N_5-K_5}{(N_5-1)K_5}}=\sqrt{\frac{(q-1)^2 q-(q-1)(q-2)}{((q-1)^2 q-1)(q-1)(q-2)}}=\sqrt{\frac{q^2-2q+2}{(q(q-1)^2-1)(q-2)}}, \end{aligned}$$

we have

$$\begin{aligned} \frac{I_{max}(\mathcal {C}_5)}{I_{W5}}=\frac{q}{q-1}\sqrt{\frac{q(q-1)^2-1}{(q^2-2q+2)(q-2)}}. \end{aligned}$$

It is obvious that $\lim _{q\rightarrow +\infty }\frac{I_{max}(\mathcal {C}_5)}{I_{W5}}=1$. The codebook $\mathcal {C}_5$ asymptotically meets the Welch bound. This completes the proof. $\square$

In Table 5, we provide some explicit values of the parameters of the codebooks we proposed for some given q, and corresponding numerical data of the Welch bound for comparison. The numerical results show that the codebooks $\mathcal {C}_5$ asymptotically meet the Welch bound.

Table 5 Parameters of the $(N_5,K_5)$ codebook of Section III

The distribution of correlation magnitudes of $\mathcal {C}_5$ is given as follows.

Corollary 3.15

$$\begin{aligned} |\mathbf {c}\mathbf {c}'^H|=\left\{ \begin{array}{ll} 1,&{} (q-1)^2q\hbox { times},\\ \frac{q-1}{K_5},&{} (q-2)(q-1)^2q\hbox { times},\\ 0,&{} (q-2)(q-1)^3q\hbox { times},\\ \frac{q-2}{K_5} ,&{} (q-1)^3q\hbox { times},\\ \frac{1}{K_5},&{} (q-2)(q-1)^3q\hbox { times},\\ \frac{\sqrt{q}}{K_5},&{} 2(q-2)(q-1)^3q\hbox { times},\\ \frac{q}{K_5},&{} (q-3)(q-2)(q-1)^3q\hbox { times}. \end{array} \right. \end{aligned}$$

3.6 The sixth construction of codebooks

Let

$$\begin{aligned} D_6:=\{(x,y,z)\in \mathbb {F}_{q}^*\times \mathbb {F}_q^*\times \mathbb {F}_q^*: z=(1-x)(1-y)\}. \end{aligned}$$

Then $\#D_6=(q-2)^2$.

The codebook $\mathcal {C}_6$ is constructed as

$$\begin{aligned} \mathcal {C}_6:=\left\{ \frac{1}{q-2}(\varphi _i(x)\varphi _j(y)\varphi _k(z))_{(x,y,z)\in D_6}:\varphi _i,\varphi _j, \varphi _k\in \widehat{\mathbb {F}_q^*} \right\} . \end{aligned}$$

We can derive the following Theorem.

Theorem 3.16

$\mathcal {C}_6$ is a codebook with $N_6=(q-1)^3$, $K_6=(q-2)^2$ and $I_{max}(\mathcal {C}_6)=\frac{q}{(q- 2)^2}$.

Proof

By the definition of $\mathcal {C}_6$, it contains $(q-1)^3$ codewords of length $(q-2)^2$. Then it is easy to see $N_6=(q-1)^3$ and $K_6=(q-2)^2$. Let $\mathbf {c}$ and $\mathbf {c}'$ be any different codewords in $\mathcal {C}_6$, $\mathbf {c}=\frac{1}{q-2}(\varphi _s(x)\varphi _u(y)\varphi _v(z))_{(x,y,z)\in D_6}$ and $\mathbf {c}'=\frac{1}{q-2}(\varphi _s'(x)\varphi _u'(y)\varphi _v'(z))_{(x,y,z)\in D_6}$, where $\varphi _s,\varphi _u,\varphi _v,\varphi _s',\varphi _u',\varphi _v'\in \widehat{\mathbb {F}_q^*}$. Then the correlation of $\mathbf {c}$ and $\mathbf {c}'$ is as follows.

$$\begin{aligned}&K\mathbf {c}\mathbf {c}'^H\\&\quad =\sum _{(x,y,z)\in D_6}\varphi _s(x)\varphi _u(y)\varphi _v(z)\overline{\varphi _s'(x)\varphi _u'(y)\varphi _v'(z)}\\&\quad =\sum _{(x,y,z)\in D_6}\varphi _s\overline{\varphi _s'}(x)\varphi _u\overline{\varphi _u'}(y)\varphi _v\overline{\varphi _v'}(z)\\&\quad =\sum _{x,y\in \mathbb {F}_q^*,x\ne 1,y\ne 1}\varphi _i(x)\varphi _j(y)\varphi _k((1-x)(1-y)) \\&\qquad \qquad (\hbox {where } \varphi _i=\varphi _s\overline{\varphi _s'}, \varphi _j=\varphi _u\overline{\varphi _u'}, \hbox { and } \varphi _k=\varphi _v\overline{\varphi _v'})\\&\quad = \sum _{x\in \mathbb {F}_q^*,x\ne 1}\varphi _i(x)\varphi _k(1-x)\sum _{y\in \mathbb {F}_q^*,y\ne 1}\varphi _j(y)\varphi _k(1-y)\\&\quad =(J(\varphi _i,\varphi _k)-\varphi _i(0)\varphi _k(1)-\varphi _i(1)\varphi _k(0))(J(\varphi _j,\varphi _k)-\varphi _j(0)\varphi _k(1) -\varphi _j(1)\varphi _k(0)). \end{aligned}$$

When $\varphi _k$ is trivial, since $\mathbf {c}\ne \mathbf {c}'$, at least one of $\varphi _i$ and $\varphi _j$ is nontrivial. By Lemma 2.3, we have

$$\begin{aligned} K_6\mathbf {c}\mathbf {c}'^H=\left\{ \begin{array}{ll} (-1)(q-2),&{} \varphi _i \hbox { is trivial, and } \varphi _j \hbox { is nontrivial,}\\ (-1)(q-2),&{} \varphi _i \hbox { is nontrivial, and } \varphi _j \hbox { is trivial,}\\ (-1)(-1) ,&{} \hbox {both } \varphi _i \hbox { and } \varphi _j \hbox { are nontrivial.} \end{array} \right. \end{aligned}$$

When $\varphi _k$ is nontrivial, by Lemma 2.3, we have

$$\begin{aligned} K_6\mathbf {c}\mathbf {c}'^H=\left\{ \begin{array}{ll} (-1)(-1), &{} \hbox {both } \varphi _i \hbox { and } \varphi _j \hbox { are trivial,} \\ (-1)J(\varphi _j,\varphi _k),&{} \varphi _i \hbox { is trivial, and } \varphi _j \hbox { is nontrivial,}\\ (-1)J(\varphi _i,\varphi _k),&{} \varphi _i \hbox { is nontrivial, and } \varphi _j \hbox { is trivial,}\\ J(\varphi _i,\varphi _k)J(\varphi _j,\varphi _k),&{} \hbox {both } \varphi _i \hbox { and } \varphi _j \hbox { are nontrivial.} \end{array} \right. \end{aligned}$$

Therefore, we have

$$\begin{aligned} I_{max}(\mathcal {C}_6)=max\{|\mathbf {c}\mathbf {c}'^H|:\mathbf {c},\mathbf {c}'\in \mathcal {C}, and\ \mathbf {c}\ne \mathbf {c}'\}=\frac{q}{K_6}=\frac{q}{(q-2)^2}, \end{aligned}$$

the maximal value obtained when all of $\varphi _i,\varphi _j,\varphi _k,\varphi _i\varphi _k$ and $\varphi _j\varphi _k$ are nontrivial. $\square$

Using Theorem 3.16, we can derive the ratio of $I_{max}(\mathcal {C}_6)$ of the proposed codebooks to that of the MWBE codebooks and show the asymptotic optimality of the proposed codebooks as in the following theorem.

Theorem 3.17

Let $I_{W6}$ be the Welch bound equality, for the given $N_6$, $K_6$ in the current section. We have

$$\begin{aligned} \lim _{{q}\rightarrow \infty }\frac{I_{max}(\mathcal {C}_6)}{I_{W6}}=1, \end{aligned}$$

then the codebooks we proposed are asymptotically optimal.

Proof

Note that $N_6=(q-1)^3$ and $K_6=(q-2)^2$. Then the corresponding Welch bound is

$$\begin{aligned} I_{W6}=\sqrt{\frac{N_6-K_6}{(N_6-1)K_6}}=\sqrt{\frac{(q-1)^3-(q-2)^2}{((q-1)^3-1)(q-2)^2}}=\frac{1}{q-2}\sqrt{\frac{q^3-4q^2+7q-5}{q^3-3q^2+3q-2}}, \end{aligned}$$

we have

$$\begin{aligned} \frac{I_{max}(\mathcal {C}_6)}{I_{W6}}=\frac{q}{q-2}\sqrt{\frac{q^3-3q^2+3q-2}{q^3-4q^2+7q-5}}. \end{aligned}$$

It is obvious that $\lim _{q\rightarrow +\infty }\frac{I_{max}(\mathcal {C}_6)}{I_{W6}}=1$. The codebook $\mathcal {C}$ asymptotically meets the Welch bound. This completes the proof. $\square$

In Table 6, we provide some explicit values of the parameters of the codebooks we proposed for some given q, and corresponding numerical data of the Welch bound for comparison. The numerical results show that the codebooks $\mathcal {C}_6$ asymptotically meet the Welch bound.

Table 6 Parameters of the $(N_6,K_6)$ codebook of Section III

The distribution of correlation magnitudes of $\mathcal {C}_6$ is given as follows.

Corollary 3.18

$$\begin{aligned} |\mathbf {c}\mathbf {c}'^H|=\left\{ \begin{array}{ll} 1,&{} (q-1)^3\hbox { times},\\ \frac{q-2}{K_6},&{} 2(q-2)(q-1)^3\hbox { times},\\ \frac{1}{K_6} ,&{} (q-2)(q-1)^3(q+2)\hbox { times},\\ \frac{\sqrt{q}}{K_6},&{} 4(q-3)(q-2)(q-1)^3\hbox { times},\\ \frac{q}{K_6},&{} (q-3)^2(q-2)(q-1)^3\hbox { times}. \end{array} \right. \end{aligned}$$

4 Another family of codebooks

Based on the six constructions of codebooks in Sect. 3, more new codebooks can be derived, which are also asymptotically optimal.

Let $\mathcal {E}_n$ denote the set formed by the standard basis of the n-dimensional Hilbert space:

$$\begin{aligned} \begin{array}{c} (1,0,0,...,0,0),\\ (0,1,0,...,0,0),\\ ......\\ (0,0,0,...,0,1). \end{array} \end{aligned}$$

Combining with the preceding six constructions, we get the following result.

Theorem 4.1

Let $\mathcal {C}'_i=\mathcal {C}_i\cup \mathcal {E}_{K_i}$. Then the codebooks $\mathcal {C}'_i$ are all asymptotically optimal, $i=1,2,3,4,5,6$, and the parameters of the new codebooks are as follows:

$N_i'=N_i+K_i$, $K_i'=K_i$ and $I_{max}(\mathcal {C}_i')=I_{max}(\mathcal {C}_i)$. Specifically,

• (1) $N_1'=N_1+K_1=q^3+q^2$, $K_1'=K_1=q^2$ and $I_{max}(\mathcal {C}_1')=I_{max}(\mathcal {C}_1)=\frac{1}{q}$;

• (2) $N_2'=N_2+K_2=(q^2-1)$, $K_2'=K_2=(q-1)q$ and $I_{max}(\mathcal {C}_2')=I_{max}(\mathcal {C}_2)=\frac{1}{q-1}$;

• (3) $N_3'=N_3+K_3=q^3-2q+1$, $K_3'=K_3=(q-1)^2$ and $I_{max}(\mathcal {C}_3')=I_{max}(\mathcal {C}_3)=\frac{q}{(q-1)^2}$;

• (4) $N_4'=N_4+K_4=q^3-q^2-q+1$, $K_4'=K_4=(q-1)^2$ and $I_{max}(\mathcal {C}_4')=I_{max}(\mathcal {C}_4)=\frac{q}{(q-1)^2}$;

• (5) $N_5'=N_5+K_5=q^3-q^2-2q+2$, $K_5'=K_5=(q-1)(q-2)$ and $I_{max}(\mathcal {C}_5')=I_{max}(\mathcal {C}_5)=\frac{q}{(q-1)(q-2)}$;

• (6) $N_5'=N_6+K_6=q^3-2q^2-q+3$, $K_6'=K_6=(q-2)^2$ and $I_{max}(\mathcal {C}_5')=I_{max}(\mathcal {C}_5)=\frac{q}{(q-2)^2}$.

Proof

We only prove the case when $i=2$, the other cases can be proved as a similar way. It is easy to see $N_2'=N_2+K_2=(q^2-1)q$ and $K_2'=K_2=(q-1)q$. Let $\mathbf {c}$ and $\mathbf {c}'$ be any different codewords in $\mathcal {C}_2'$, the correlation of $\mathbf {c}$ and $\mathbf {c}'$ can be discussed in the following cases.

Case 1: If $\mathbf {c}$, $\mathbf {c}'\in \ \mathcal {C}_2$, by Theorem 3.4, we have

$$\begin{aligned} max\{|\mathbf {c}\mathbf {c}'^H|:\mathbf {c},\mathbf {c}'\in \mathcal {C}_2, and\ \mathbf {c}\ne \mathbf {c}'\}=\frac{1}{q-1}. \end{aligned}$$

Case 2: If $\mathbf {c}\in \mathcal {C}_2$ and $\mathbf {c}'\in \mathcal {E}_{(q-1)q}$ (or $\mathbf {c}'\in \mathcal {C}_2$ and $\mathbf {c}\in \mathcal {E}_{(q-1)q}$), it is easy to see

$$\begin{aligned} |\mathbf {c}\mathbf {c}'^H|=\frac{1}{\sqrt{(q-1)q}}. \end{aligned}$$

Case 3: If $\mathbf {c}$ and $\mathbf {c}'\in \mathcal {E}_{(q-1)q}$, it is obvious that $\mathbf {c}\mathbf {c}'^H=0.$

From the above three cases, we have

$$\begin{aligned} I_{max}(\mathcal {C}_2')=\frac{1}{q-1}. \end{aligned}$$

Note that $N_2'=(q^2-1)q$ and $K_2'=(q-1)q$, then the corresponding Welch bound is

$$\begin{aligned} I_W=\sqrt{\frac{N_2'-K_2'}{(N_2'-1)K_2'}}=\sqrt{\frac{q}{q^3-q-1}}, \end{aligned}$$

we have

$$\begin{aligned} \frac{I_{max}(\mathcal {C}_2')}{I_W}=\sqrt{\frac{q^3-q-1}{(q-1)^2q}}. \end{aligned}$$

It is obvious that $\lim _{q\rightarrow +\infty }\frac{I_{max}(\mathcal {C}_2')}{I_W}=1$. The codebook $\mathcal {C}_2'$ asymptotically meets the Welch bound. This completes the proof. $\square$

5 Concluding remarks

In this paper, we presented six constructions of codebooks asymptotically achieve the Welch bounds with additive characters, multiplicative characters and character sums of finite fields. Actually, the first construction in our paper is equivalent to the measurement matrix in [21]. The advantage of our construction is that it can be generalized naturally to the other five constructions of codebooks which are also asymptotically optimal.