Codebooks from generalized bent $\mathbb{Z}_4$-valued quadratic forms

TL;DR

This paper introduces a new class of generalized bent $\

Contribution

It constructs optimal codebooks achieving the Levenshtein bound using generalized bent $\

Findings

Constructed codebooks with parameters $(2^{2m}+2^m,2^m)$ and alphabet size 6.

Achieved codebooks meet the Levenshtein bound.

Extended previous work on bent functions over $\

Abstract

Codebooks with small inner-product correlation have application in unitary space-time modulations, multiple description coding over erasure channels, direct spread code division multiple access communications, compressed sensing, and coding theory. It is interesting to construct codebooks (asymptotically) achieving the Welch bound or the Levenshtein bound. This paper presented a class of generalized bent $\mathbb{Z}_4$-valued quadratic forms, which contain functions of Heng and Yue (Optimal codebooks achieving the Levenshtein bound from generalized bent functions over $\mathbb{Z}_4$. Cryptogr. Commun. 9(1), 41-53, 2017). By using these generalized bent $\mathbb{Z}_4$-valued quadratic forms, we constructs optimal codebooks achieving the Levenshtein bound. These codebooks have parameters $(2^{2m}+2^m,2^m)$ and alphabet size $6$.

Tables

Table 1. Table 1 : The distribution of values in the multiset { χ Q ​ ( λ ) : λ ∈ F } conditional-set subscript 𝜒 𝑄 𝜆 𝜆 𝐹 \{\chi_{Q}(\lambda):\lambda\in F\}

value	frequency
$0$	$2^m-2^r$
$\pm 2^{m-r/2}$	$2^{r-1}\pm 2^{r/2-1}$

Equations

$$I_{max}(\mathcal{C})=ma{x}_{0\le i<j\le N-1}({\mathbf{c}}_i{{\mathbf{c}}_{\mathbf{j}}}^H),$$

$$f_{\gamma }\left(x_1,x_2\right)=\sum _{i=1}^{\frac{m-2}{2}}{\mathrm{tr}}_1^{m-1}\left(x_1^{2^i+1}\right)+\sum _{i=1}^{\frac{\ell -1}{2}}{\mathrm{tr}}_1^{m-1}\left({\left(\gamma x_1\right)}^{2^{ei}+1}\right)+x_2{\mathrm{tr}}_1^{m-1}\left(x_1\right)$$

$$\mu :R$$

$$z$$

$${\mathrm{Tr}}_1^m:R$$

$$z=a+2b$$

$$rad(B)=\{x\in F:B(x,y)=0\text{for all}y\in F\}.$$

$$rank(B)=m-di{m}_K(rad(B)).$$

$${\chi }_Q(\lambda )=\sum _{x\in F}(\sqrt{-1}{)}^{Q(x)+2{\mathrm{Tr}}_1^m(\lambda x)}$$

$${\mathcal{C}}_{QF}=\bigcup _{Q\in QF}{S}_Q\bigcup S_0\bigcup E_{2^m},$$

$$S_Q=\left\{\frac{1}{\sqrt{2^m}}{\left((\sqrt{-1}{)}^{Q(\xi )+2{\mathrm{Tr}}_1^m(\lambda \xi )}\right)}_{\xi \in F}:\lambda \in F\right\}$$

$$S_0=\left\{\frac{1}{\sqrt{2^m}}{\left((\sqrt{-1}{)}^{2{\mathrm{Tr}}_1^m(\lambda \xi )}\right)}_{\xi \in F}:\lambda \in F\right\}.$$

$$Q_j(x)={\mathrm{Tr}}_1^m(\sum _{i=1}^{\frac{f_j-1}{2}}{x}^{2^{i{e}_j}+1}).$$

$$f_a(x)={\mathrm{Tr}}_1^m(ax)+2\sum _{j=1}^{l-1}{Q}_j({\gamma }_{j}ax).$$

$$q_j(x)={\mathrm{tr}}_1^m(\sum _{i=1}^{\frac{f_j-1}{2}}{x}^{2^{i{e}_j}+1}).$$

$$B_{q_j}(x,y)=q_j(x)+q_j(y)+q_j(x+y),$$

$$2B_{f_a}(x,y)=$$

$$=$$

$$2B_{f_a-f_b}(x,y)=2{\mathrm{tr}}_1^m\left(\overline{y}\left((\overline{a^2}+\overline{b^2})\overline{x}+\sum _{j=1}^{l-1}\left(\overline{{\gamma }_{j}a}({\mathrm{tr}}_{e_j}^m(\overline{{\gamma }_{j}ax})+\overline{{\gamma }_{j}ax})+\overline{{\gamma }_{j}b}({\mathrm{tr}}_{e_j}^m(\overline{{\gamma }_{j}bx})+\overline{{\gamma }_{j}bx})\right)\right)\right).$$

$$(\overline{a^2}+\overline{b^2})\overline{x}+\sum _{j=1}^{l-1}\left(\overline{{\gamma }_{j}a}({\mathrm{tr}}_{e_j}^m(\overline{{\gamma }_{j}ax})+\overline{{\gamma }_{j}ax})+\overline{{\gamma }_{j}b}({\mathrm{tr}}_{e_j}^m(\overline{{\gamma }_{j}bx})+\overline{{\gamma }_{j}bx})\right)=0.$$

$$(1+\sum _{j=1}^{l-1}{\overline{{\gamma }_j}}^2)(\overline{a^2}+\overline{b^2})\overline{x}+\sum _{j=1}^{l-1}\overline{{\gamma }_j^2}\left(\overline{a}{\mathrm{tr}}_{e_j}^m(\overline{ax})+\overline{b}{\mathrm{tr}}_{e_j}^m(\overline{bx})\right)=0$$

$$(1+\sum _{j=1}^{l-1}{\overline{{\gamma }_j}}^2)(\overline{a^2}+\overline{b^2}){\overline{x}}^2+\sum _{j=1}^{l-1}\overline{{\gamma }_j^2}\overline{x}\left(\overline{a}{\mathrm{tr}}_{e_j}^m(\overline{ax})+\overline{b}{\mathrm{tr}}_{e_j}^m(\overline{bx})\right)=0.$$

$$A=(1+\sum _{j=1}^{l-1}{\overline{{\gamma }_j}}^2)(\overline{a^2}+\overline{b^2}){\overline{x}}^2+\sum _{j=1}^{l-1}\overline{{\gamma }_j^2}\overline{x}\left(\overline{a}{\mathrm{tr}}_{e_j}^m(\overline{ax})+\overline{b}{\mathrm{tr}}_{e_j}^m(\overline{bx})\right).$$

$${\mathrm{tr}}_{e_{l-1}}^m(A)=$$

$$+\overline{{\gamma }_{l-1}^2}{\mathrm{tr}}_{e_{l-1}}^m\left((\overline{a^2}+\overline{b^2}){\overline{x}}^2\right)$$

$$=$$

$${\mathrm{tr}}_{e_1}^m\left({\mathrm{tr}}_{e_2}^m\left(\cdots {\mathrm{tr}}_{e_{l-1}}^m\left(A\right)\cdots \right)\right)={\mathrm{tr}}_{e_1}^m(({\overline{a}}^2+{\overline{b}}^2){\overline{x}}^2)={\left({\mathrm{tr}}_{e_1}^m((\overline{a}+\overline{b})\overline{x})\right)}^2=0.$$

$${\mathrm{tr}}_{e_{t+1}}^m\left(\overline{{\gamma }_{j}ax}({\mathrm{tr}}_{e_j}^m(\overline{{\gamma }_{j}ax})+\overline{{\gamma }_{j}ax})\right)=$$

$$=$$

$${\mathrm{tr}}_{e_{t+1}}^m(A)=$$

Full text

1 Introduction

Applied in unitary space-time modulations, multiple description coding over erasure channels, direct spread code division multiple access communications, compressed sensing, and coding theory [2, 24], an $(N,K)$ codebook is a signal set $(\mathbf{c}_{0},\ldots,\mathbf{c}_{N-1})$, where $\mathbf{c}_{0}\ldots,\mathbf{c}_{N-1}$ are unit norm $1\times K$ complex vectors over an alphabet. As a performance measure of a codebook in practical applications, the maximum crosscorrelation amplitude of an $(N,K)$ codebook $\mathcal{C}$ is defined by

[TABLE]

where $\mathbf{c}_{j}^{H}$ is the conjugate transpose of $\mathbf{c}_{j}$. There are two well known bounds of codebooks:

the Welch bound [30]: $I_{max}(\mathcal{C})\geq\sqrt{\frac{N-K}{(N-1)K}}$, where $N\geq K$;

• 2.

the Levenstein bounds [18, 20]:

• (a)

$I_{\max}(\mathcal{C})\geq\sqrt{\frac{3N-K^{2}-2K}{(N-K)(K+2)}}$ for any real-valued codebook, where $N>K(K+1)/2$;

• (b)

$I_{\max}(\mathcal{C})\geq\sqrt{\frac{2N-K^{2}-K}{(N-K)(K+1)}}$ for any complex-valued codebook, where $N>K^{2}$.

A codebook achieving the Welch bound is also called a maximum-Welch-bound-equality (MWBE) codebook, which is known as an equiangular tight frame [4]. The construction of MWBE codebooks is equivalent to line packing in Grassmannian spaces [29]. It is difficult to construct MWBE codebooks [26]. Known $(N,K)$ MWBE codebooks are listed:

$(N,N)$ orthogonal MWBE codebooks [26, 32], where $N>1$;

• 2.

$(N,N-1)$ MWBE codebooks from discrete Fourier transformation matrices [26, 32] or m-sequences [26], where $N>1$;

• 3.

$(N,K)$ MWBE codebooks from conference matrices [5, 29], where $N=2K$, $N=2^{d+1}$ or $N=p^{d}+1$ for a prime $p$;

• 4.

$(N,K)$ MWBE codebooks from difference sets in cyclic groups [32] and abelian group [6, 7], or Steiner systems [11].

Almost optimal codebooks asymptotically achieving the Welch bound are constructed in [8, 16, 22, 34, 35]. When $N>K(K+1)/2$ (resp. $N>K^{2}$), there are no real-valued (complex-valued) MWBE codebooks. The Levenshtein bounds hold for $N>K(K+1)/2$ or $N>K^{2}$. Some known $(N,K)$ codebooks achieving the Levenstein Bound are listed:

$(2^{2m-1}+2^{m},2^{m})$ codebooks with alphabet $4$ [2, 31, 33, 36], where $m$ is even;

• 2.

$(p^{2m}+p^{m},p^{m})$ codebooks with alphabet size $p+2$ [10, 31, 33], where $p$ is odd;

• 3.

$(2^{2m}+2^{m},2^{m})$ codebooks with alphabet $6$ [2, 15].

These codebooks can constructed from binary Kerdock codes [2, 31, 33], perfect nonlinear functions [10, 31], bent functions [36], $\mathbb{Z}_{4}$-Kerdock codes [2], and $\mathbb{Z}_{4}$-valued quadratic forms [15]. It is interesting to construct optimal codebooks achieving the Levenstein Bound with different parameters from different methods. Zhou et al. [36] presented codebooks achieving the Levenstein bound from bent functions of the form:

[TABLE]

where $x_{1}\in\mathbb{F}_{2^{m-1}}$, $x_{2}\in\mathbb{F}_{2}$, and $\mathrm{tr}_{1}^{m-1}$ is the trace function from $\mathbb{F}_{2^{m-1}}$ to $\mathbb{F}_{2}$. Heng and Yue [15] generalized their results to codebooks from generalized bent $\mathbb{Z}_{4}$-valued quadratic forms. Ding et al. [9] presented the notation of cyclic bent functions, gave a class of cyclic bent functions containing bent functions in [36], and used cyclic bent functions in the construction of good mutually unbiased bases (MUBs), codebooks and sequence families.

Motivated by their methods, this paper generalizes the construction of Heng and Yue [15] and constructs optimal codebooks from a class of generalized bent $\mathbb{Z}_{4}$-valued quadratic forms.

The rest of the paper is organized as follows. Section 2 introduces some basic results on Galois rings, $\mathbb{Z}_{4}$-valued quadratic forms and codebooks from generalized bent functions. Section 3 presents the construction of optimal codebooks from a class of generalized bent $\mathbb{Z}_{4}$-valued quadratic forms. Section 4 makes a conclusion.

2 Preliminaries

In this section, we introduce some results on Galois rings, $\mathbb{Z}_{4}$-valued quadratic forms, and codebooks from generalized bent $\mathbb{Z}_{4}$-quadratic forms.

2.1 Galois rings

Let $m$ be a positive integer. Let $R=\mathbb{GR}(4,m)$ be the Galois ring $\mathbb{Z}_{4}[x]/(f)$, where $\mathbb{Z}_{4}$ is the ring of integers modulo $4$ and $f$ is a monic basic irreducible polynomial of degree $m$ in $\mathbb{Z}_{4}[x]$. More results on Galois rings can be found in [13, 14, 23, 27].

Define $F=\{z\in R:z^{2^{m}}=z\}$, which is called the set of Teichmuller representatives in $R$. Then any $z\in R$ can be uniquely represented by $z=a+2b$, where $a,b\in F$. For two elements $a,b\in F$, Define $a\oplus b=(a+b)^{2^{m}}=a+b+2\sqrt{ab}$. Then $(F,\oplus,\cdot)$ is a Galois field of size $2^{m}$.

Let $\mu:\mathbb{Z}_{4}\longrightarrow\mathbb{Z}_{2}$ denote the modulo-$2$ reduction. This mapping induces the following mapping from $R$ to $\mathbb{F}_{2^{m}}$

[TABLE]

Further, $\mu:F\longrightarrow\mathbb{F}_{2^{m}}$ is an isomorphism.

The Frobenius automorphism $\sigma$ on $F$ is given by $\sigma(x)=x^{2}$. The trace function from $R$ to $\mathbb{Z}_{4}$ is defined by

[TABLE]

Then $\mathrm{Tr}_{1}^{m}(\alpha z_{1}+\beta z_{2})=\alpha\mathrm{Tr}_{1}^{m}(z_{1})+\beta\mathrm{Tr}_{1}^{m}(z_{2})$, where $z_{1},z_{2}\in R$ and $\alpha,\beta\in\mathbb{Z}_{4}$. Let $\mathrm{tr}_{1}^{m}$ be the trace function from $\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2}$. For any $x\in F$, $2\mathrm{Tr}_{1}^{m}(x)=2\mathrm{tr}_{1}^{m}(x)$. The follow lemma [12] shows that $\overline{x_{1}}\neq\overline{x_{2}}$ for two different elements $x_{1},x_{2}\in F$.

Lemma 1**.**

Let $F^{*}$ be the set of nonzero elements of $F$, which is a multiplicative group. Let $F^{*}=\langle\xi\rangle$. For $0\leq i<j<2^{m}-1$, $\xi^{i}\pm\xi^{j}$ is invertible.

2.2 $\mathbb{Z}_{4}$-valued

quadratic forms

Let $K=\{z\in\mathbb{Z}_{4}:z=z^{2}\}$. Then any $z\in\mathbb{Z}_{4}$ can be uniquely represented by $z=x+2y$, where $x,y\in K$. An operation $\oplus$ on $K$ can be denoted by $x\oplus y=(x+y)^{2}$. Then, $(K,\oplus,+)$ is the finite field of size $2$.

A symmetric bilinear form on $F$ is a mapping $B:F\times F\longrightarrow K$ satisfying

the symmetry condition: $B(x,y)=B(y,x)$;

• 2.

the bilinearity condition: $B(\alpha x_{1}\oplus\beta x_{2},y)=\alpha B(x_{1},y)+\beta B(x_{2},y)$, where $\alpha,\beta\in K$.

A symmetric bilinear for $B$ is alternating if $B(x,x)=0$ for all $x\in F$. Otherwise, it is nonalternating. The radical $rad(B)$ is the set

[TABLE]

The radical is a vector space over $K$. The rank of $B$ is defined by

[TABLE]

A $\mathbb{Z}_{4}$-valued quadratic form on $F$ [1] is a mapping $Q:F\longrightarrow\mathbb{Z}_{4}$ satisfying

$Q(\alpha x)=\alpha^{2}Q(x)$ for $\alpha\in K$;

• 2.

$Q(x\oplus y)=Q(x)+Q(y)+2B(x,y)$, where $B$ is a symmetric bilinear form.

A $\mathbb{Z}_{4}$-quadratic form $Q$ is alternating if its associated bilinear form $B$ is alternating. Otherwise, $Q$ is nonalternating. The rank of $Q$ is the rank of its associated bilinear form. Note that $Q(0)=0$ and $2Q(x)=2B(x,x)$. The Walsh transform of $Q$ is defined by

[TABLE]

where $\lambda\in F$. The multiset $\{\chi_{Q}(\lambda):\lambda\in F\}$ depends only on the rank of $Q$. For an alternating $Q$, the following theorem [13] gives the distribution of values in the multiset $\{\chi_{Q}(\lambda):\lambda\in F\}$.

Theorem 2**.**

Let $Q:F\longrightarrow\mathbb{Z}_{4}$ be an alternating $\mathbb{Z}_{4}$-valued quadratic form of rank $r$. The distribution of values in the multiset $\{\chi_{Q}(\lambda):\lambda\in F\}$ is given in Tabel 1.

For a nonalternating $Q$, the distribution of values in the multiset $\{\chi_{Q}(\lambda):\lambda\in F\}$ is given in Theorem 5 [27].

A $\mathbb{Z}_{4}$-quadratic form $Q$ is generalized bent if $\left|\chi_{Q}(\lambda)\right|=2^{m/2}$ for all $\lambda\in F$ [3, 25]. Let $\mathcal{GF}_{m}$ be the set of all generalized bent $\mathbb{Z}_{4}$-quadratic forms over $F$. The following lemma [21] gives a characterization of generalized bent $\mathbb{Z}_{4}$-quadratic forms.

Lemma 3**.**

A $\mathbb{Z}_{4}$-quadratic form $Q(x)$ has full rank $m$ if and only if it is generalized bent.

2.3 Codebooks from generalized bent

$\mathbb{Z}_{4}$-quadratic forms

Let $QF$ be an $n$-set of generalize bent $\mathbb{Z}_{4}$-quadratic forms from $F$ to $\mathbb{Z}_{4}$ such that the difference of arbitrary two distinct quadratic forms in $QF$ is generalized bent. Let $E_{n}=\{\mathbf{e}_{i}:1\leq n\}$ be the standard basis of the $2^{m}$-dimensional Hilbert space, where $\mathbf{e}_{i}=(0,\ldots,0,1,0,\ldots,0)$ is a vector with only the $i$-th entry being nonzero. Construct the following codebook from $QF$:

[TABLE]

where

[TABLE]

and

[TABLE]

The following theorem [15] gives parameters of the codebook constructed from $QF$.

Theorem 4**.**

Let $QF$ be an $n$-set of generalize bent $\mathbb{Z}_{4}$-quadratic forms from $F$ to $\mathbb{Z}_{4}$ such that the difference of arbitrary two distinct quadratic forms in $QF$ is generalized bent. Let $\mathcal{C}_{QF}$ be the codebook constructed in (1). Then $\mathcal{C}_{QF}$ is a $((n+1)2^{m}+2^{m},2^{m})$ codebook with $I_{max}(\mathcal{C}_{QF})=\frac{1}{\sqrt{2^{m}}}$ and alphabet size $6$. Further, the codebook $\mathcal{C}_{QF}$ is an optimal codebook achieving the Levenshtein bound if and only if $n=2^{m}-1$.

From Theorem 4, in order to construct optimal codebooks, we just need to give the set $QF$ of size $2^{m}-1$. Some known results on the set $QF=\{Q_{a}:a\in F^{*}\}$ [15] have been given below:

$Q_{a}(x)=\operatorname{Tr}_{1}^{m}(a(1+2\eta)x)$;

• 2.

$Q_{a}(x)=\operatorname{Tr}_{1}^{m}(ax)+2P(\gamma ax)$, where $P(x)={\sum_{i=1}^{\frac{s-1}{2}}\operatorname{Tr}_{1}^{m}\left(x^{2^{eki}+1}\right)}$ and $s$ is odd;

• 3.

$Q_{a}(x)=\operatorname{Tr}_{1}^{m}(a(1+2\eta)x)+2P(\gamma ax)$, where $P(x)={\sum_{i=1}^{\frac{s-1}{2}}\operatorname{Tr}_{1}^{m}\left(x^{2^{eki}+1}\right)}$ and $s$ is odd.

3 Codebooks from a class of generalize bent

$\mathbb{Z}_{4}$-quadratic forms

In this section, we present a class of generalize bent $\mathbb{Z}_{4}$-quadratic forms and construct codebooks from these quadratic forms.

Let $l$ be an positive integer. Let $e_{0}\leq e_{1}<\cdots<e_{l}$, where $e_{0}=1$, $e_{i}\mid e_{i+1}$ for $i\in\{0,\ldots,l-1\}$, and $e_{l}=m$. Let $f_{i}=\frac{m}{e_{i}}$ for $i\in\{1,\ldots,l-1\}$, where $f_{i}$ is odd. For $j\in\{1,\ldots,l-1\}$, define

[TABLE]

Take $\gamma_{0},\gamma_{1},\ldots,\gamma_{l-1}\in F$, where $\overline{\gamma_{0}}=1$, $\overline{\gamma_{j}}\in\mathbb{F}_{2^{e_{j}}}$, and $1+\sum_{j=1}^{t}\overline{\gamma_{j}}^{2}\neq 0$ for $t\leq l-1$. For any $a\in F^{*}$, define

[TABLE]

For $j\in\{1,\ldots,l-1\}$, define the Boolean function over $\mathbb{F}_{2^{m}}$

[TABLE]

It is a quadratic form and its symplectic form is

[TABLE]

where $x,y\in\mathbb{F}_{2^{m}}$. Further, $B_{q_{j}}(x,y)=\mathrm{tr}_{1}^{m}\left(y\left(\mathrm{tr}_{e_{j}}^{m}(x)+x\right)\right)$ [19]. We will use these functions defined in (2) to construct codebooks. Some lemmas are given first.

Lemma 5**.**

Let $f_{a}$ be defined in (2). Then $2B_{f_{a}}(x,y)=2\mathrm{tr}_{1}^{m}\left(\overline{y}\left(\overline{a}^{2}\overline{x}+\sum_{j=1}^{l-1}\overline{\gamma_{j}a}(\mathrm{tr}_{e_{j}}^{m}(\overline{\gamma_{j}ax})+\overline{\gamma_{j}ax})\right)\right)$.

Proof.

We have

[TABLE]

Hence, this lemma follows. ∎

Lemma 6**.**

Let $a,b\in F$ and $f_{a},f_{b}$ be defined in (2), where $a\neq b$. Then

[TABLE]

Proof.

Note that $2B_{f_{a}-f_{b}}(x,y)=2B_{f_{a}}(x,y)-2B_{f_{b}}(x,y)$. From Lemma 5, this lemma follows. ∎

Lemma 7**.**

Let $a,b\in F$ and $f_{a},f_{b}$ be defined in (2), where $a\neq b$. Then $rad(B_{f_{a}-f_{b}})=\{0\}$.

Proof.

Let $x\in rad(B_{f_{a}-f_{b}})$, we just need to prove that $x=0$.

Suppose that $x\in rad(B_{f_{a}-f_{b}})$ and $x\neq 0$. By Lemma 6, $\overline{x}$ is a nonzero solution of

[TABLE]

We have

[TABLE]

and

[TABLE]

Note that $\mathrm{tr}_{e_{j}}^{m}(y^{2})=(\mathrm{tr}_{e_{j}}^{m}(y))^{2}$ for any $y\in\mathbb{F}_{2^{m}}$. Let

[TABLE]

Then we have

[TABLE]

Since $f_{j+1}$ is odd, then $\mathrm{tr}_{e_{j}}^{m}(\mathrm{tr}_{e_{j+1}}^{m}(y))=\mathrm{tr}_{e_{j}}^{m}(y)$ for any $y\in\mathbb{F}_{2^{m}}$. We have

[TABLE]

Hence, $\mathrm{tr}_{e_{1}}^{m}((\overline{a}+\overline{b})\overline{x})=0$. There exists a $t$ such that \left\{\begin{array}[]{l}\mathrm{tr}_{e_{t+1}}^{m}((\overline{a}+\overline{b})\overline{x})\neq 0,\\ \mathrm{tr}_{e_{j}}^{m}((\overline{a}+\overline{b})\overline{x})=0,~{}\text{for any}~{}1\leq j\leq t\,.\end{array}\right. For any $j\geq t+1$, we have

[TABLE]

Since $\mathrm{tr}_{e_{j}}^{m}((\overline{a}+\overline{b})\overline{x})=0$ for any $1\leq j\leq t$, let $u_{j}=\mathrm{tr}_{e_{j}}^{m}(\overline{a}\overline{x})=\mathrm{tr}_{e_{j}}^{m}(\overline{b}\overline{x})$, where $1\leq j\leq t$. Then

[TABLE]

Note that $\mathrm{tr}_{e_{t+1}}^{m}((\overline{a}+\overline{b})\overline{x})\neq 0$ and $1+\sum_{j=1}^{t}\overline{\gamma_{j}}^{2}\neq 0$. We have

[TABLE]

and

[TABLE]

Note that $f_{j}$ is odd. We have

[TABLE]

which makes a contradiction with the definition of $t$. Hence, $x=0$ if $x\in rad(B_{f_{a}-f_{b}})$. This lemma follows. ∎

Theorem 8**.**

The set $S=\{f_{a}:a\in F^{*}\}$ is an $n$-set of generalize bent $\mathbb{Z}_{4}$-quadratic forms from $F$ to $\mathbb{Z}_{4}$ such that the difference of arbitrary two distinct quadratic forms in $S$ is generalized bent, where $f_{a}$ is defined in (2) and $n=2^{m}-1$.

Proof.

We just need to prove $f_{a}-f_{b}$ is generalized bent for two distinct $a$ and $b$ in $S$. From Lemma 3, $f_{a}-f_{b}$ is generalized bent if and only if $rank(f_{a}-f_{b})=m$. From Lemma 7, we have $rank(f_{a}-f_{b})=rank(B_{f_{a}-f_{b}})=m-dim(rad(B_{f_{a}-f_{b}}))=m$. Hence, this theorem follows. ∎

We construct optimal codebooks from the set $S=\{f_{a}:a\in F^{*}\}$ in the following theorem.

Theorem 9**.**

Let $S=\{f_{a}:a\in F^{*}\}$, where $f_{a}$ is defined in (2). Then the codebook $\mathcal{C}_{S}$ constructed in (1) is a $(2^{2m}+2^{m},2^{m})$ codebook with $I_{max}(\mathcal{C}_{S})=\frac{1}{\sqrt{2^{m}}}$ and alphabet size $6$, which is an optimal codebook achieving the Levenshtein bound.

Proof.

From Theorem 4 and Theorem 8, this theorem follows. ∎

Remark 1**.**

Let $\eta\in F$. Then for any $a\in F^{*}$, we have the following $\mathbb{Z}_{4}$-valued quadratic form:

[TABLE]

From similar proof, we have that the set $S^{\prime}=\{f^{\prime}_{a}(x):a\in F^{*}\}$ also satisfies properties in Theorem 8. Then $\mathcal{C}_{S^{\prime}}$ constructed in (1) is a $(2^{2m}+2^{m},2^{m})$ codebook with $I_{max}(\mathcal{C}_{S^{\prime}})=\frac{1}{\sqrt{2^{m}}}$ and alphabet size $6$, which is an optimal codebook achieving the Levenshtein bound. When $l=2$, $e_{1}=t$ and $f_{1}=s$, we have the functions $\mathrm{Tr}_{1}^{m}(a(1+2\eta)x)+2Q_{1}(\gamma_{1}ax)$, which are functions in [15]. These functions are used to construct optimal families of quadriphase sequences [17].

We will give a connection between generalized bent $f^{\prime}_{a}(x)$ and Boolean bent functions. A $\mathbb{Z}_{4}$-valued form $Q(x)$ has the representation

[TABLE]

where $q_{1}$ and $q_{2}$ are two Boolean functions. We have the following Gray map $\phi(Q)=q_{1}(u)v+q_{2}(u)$, which is a Boolean function defined over $\mathbb{F}_{2^{m}}\times\mathbb{F}_{2}$. Then we have the following proposition.

Proposition 10**.**

Let $m$ be odd and $f_{a}^{\prime}(x)$ be a $\mathbb{Z}_{4}$-valued quadratic form defined in (3). Then $\phi(f_{a}^{\prime})$ is a Boolean bent function over $\mathbb{F}_{2^{m}}\times\mathbb{F}_{2}$.

Proof.

Note that $\mathrm{Tr}_{1}^{m}(x)=\mathrm{tr}_{1}^{m}(\overline{x})+2q(\overline{x})$ [12], where $q(\overline{x})=\sum_{i=1}^{\frac{m-1}{2}}\mathrm{tr}_{1}^{m}(x^{2^{i}+1})$. Hence, $f_{a}^{\prime}(x)$ has the representation $f_{a}^{\prime}(x)=f_{1}(\overline{x})+2f_{2}(\overline{x})$, where $f_{1}$ and $f_{2}$ are two Boolean functions. In [28], $Q(x)=q_{1}(\overline{x})+2q_{2}(\overline{x})$ is generalized bent if and only if $\phi(Q)$ is bent. Hence, this proposition follows. ∎

4 Conclusion

We present an $n$-set of generalized bent $\mathbb{Z}_{4}$-valued quadratic forms such that any difference of two different elements in this set is also generalized bent, where $n=2^{m}-1$. Using this $n$-set, we construct optimal codebooks achieving the Levenshtein bound, which have parameters $(2^{2m}+2^{m},2^{m})$ and alphabet $6$. It is interesting to construct optimal or almost optimal codebooks with different parameters from other tools.

Acknowledgement. This work was supported by SECODE project and the National Natural Science Foundation of China (Grant No. 11871058, 11531002, 11701129). C. Tang also acknowledges support from 14E013, CXTD2014-4 and the Meritocracy Research Funds of China West Normal University. Y. Qi also acknowledges support from Zhejiang provincial Natural Science Foundation of China (LQ17A010008, LQ16A010005).