Non-norm criteria and optimal $2\times 2$ space-time block codes over rings of integers of imaginary quadratic fields

TL;DR

This paper investigates optimal $2\times 2$ space-time block codes over rings of integers of imaginary quadratic fields, proving the Eisenstein construction over $\mathbb{Z}[\zeta_3]$ is optimal among such codes.

Contribution

It establishes the optimality of the Eisenstein code within this family and rules out improvements over other quadratic fields using explicit arithmetic analysis.

Findings

Eisenstein construction attains the largest normalized density among considered codes.

No other codes over rings of integers of $\mathbb{Q}(\sqrt{-d})$ with $d\in\{2,7,11\}$ outperform the Eisenstein code.

Derived effective non-norm criteria using local methods for quadratic extensions of imaginary quadratic fields.

Abstract

Codes arising from algebraic structures over number fields lead naturally to determinant optimization problems governed by arithmetic invariants. In this paper, we investigate $2\times 2$ space-time block codes defined over rings of integers of imaginary quadratic fields, combining tools from algebraic number theory, cyclic algebras, and lattice theory. We prove that the Eisenstein construction over $\mathbb{Z}[\zeta_3]$ is optimal within the family considered here: it attains the largest normalized density among the $2\times 2$ space-time block codes arising from rings of integers of imaginary quadratic fields. As a first step, we show that any code that could improve upon the Eisenstein construction must be defined over the ring of integers of $\mathbb{Q}(\sqrt{-d})$ with $d\in\{2,7,11\}$, apart from the classical Gaussian and Eisenstein cases. We then analyze these remaining fields by explicit arithmetic arguments, determine the optimal constructions over them, and show that none of them improves upon the Eisenstein code. A key ingredient in our approach is the derivation of effective non-norm criteria for quadratic extensions of imaginary quadratic fields. These criteria are obtained by local methods involving $2$-adic and $3$-adic valuations together with Hensel's lemma, and they ensure the division algebra property required for full diversity. They may also be of independent interest in the study of division algebras and their applications to coding theory and lattice-based communication.

Tables

Table 1. TABLE I: Summary of algebraic extensions, algebras and normalized density of all optimal 2 × 2 2 2 2\times 2 space-time block codes over 𝒪 ℚ ​ ( d ) subscript 𝒪 ℚ 𝑑 \mathcal{O}_{\mathbb{Q}(\sqrt{d})} , d = − 1 , − 2 , − 3 , − 7 , − 11 𝑑 1 2 3 7 11 d=-1,-2,-3,-7,-11

Equations

$$\delta (\Lambda )=\frac{de{t}_{min}(\Lambda )}{|\det (\mathcal{G}){|}^{\frac{1}{2n}}}$$

$$\rho (\Lambda )=\frac{de{t}_{min}(\Lambda {)}^{2n}}{|\det (\mathcal{G})|}.$$

$${\Lambda }_n(M)$$

$${\Gamma }_n(G)$$

$$G=\left(\begin{array}[]{cccc}g_{11}&g_{12}&\ldots&g_{1n}\\ g_{21}&g_{22}&\ldots&g_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ g_{n1}&g_{n2}&\ldots&g_{nn}\end{array}\right),$$

$$\mathcal{G}=\left(\begin{array}[]{ccccc}Re(g_{11})&-Im(g_{11})&\ldots&Re(g_{1n})&-Im(g_{1n})\\ Im(g_{11})&Re(g_{11})&\ldots&Im(g_{1n})&Re(g_{1n})\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ Re(g_{n1})&-Im(g_{n1})&\ldots&Re(g_{nn})&-Im(g_{nn})\\ Im(g_{n1})&Re(g_{n1})&\ldots&Im(g_{nn})&Re(g_{nn})\end{array}\right)$$

$$(y_1,\cdots ,y_n{)}^t=G(x_1,\cdots ,x_n{)}^t.$$

$$\left(\begin{array}{c}Re(y_1)\\ Im(y_1)\\ ⋮\\ Re(y_n)\\ Im(y_n)\end{array}\right)$$

$$\left(\begin{array}{c}{x}_{k1}\\ x_{k2}\end{array}\right)=M\left(\begin{array}{c}{z}_{k1}\\ z_{k2}\end{array}\right).$$

$$|det({\mathcal{G}}_M)|$$

$$\left(\begin{array}{c}{y}_{(l-1)n+1}\\ \cdots \\ y_{ln}\end{array}\right)=G_l\left(\begin{array}{c}{x}_{(l-1)n+1}\\ \cdots \\ x_{ln}\end{array}\right),$$

$$|det({\mathcal{G}}_{M_1,\cdots ,M_n})|=\prod _{l=1}^L|det(G_l){|}^2\cdot |det(M_l){|}^n.$$

$$\rho (\Lambda )=\frac{de{t}_{min}{\left({\mathcal{G}}_{M_1,\cdots ,M_n}\right)}^{2n}}{\prod _{l=1}^L|det(G_l){|}^2\cdot |det(M_l){|}^n}.$$

$${\alpha }_1=\frac{-p+\sqrt{p^2-4q}}{2}\notin \mathbb{F},{\alpha }_2=\frac{-p-\sqrt{p^2-4q}}{2}\notin \mathbb{F}.$$

$$\mathcal{C}$$

$$de{t}_{min}(\mathcal{C}(\mathbb{F},{\alpha }_1,{\alpha }_2,\gamma ))=\underset{X\in \mathcal{C},X\ne 0}{\min }|det(X)|.$$

$$det(X)=(a+{\alpha }_{1}b)(a+{\alpha }_{2}b)-\gamma (c+{\alpha }_{1}d)(c+{\alpha }_{2}d),$$

$$M=\left(\begin{array}{cc}1& 0\\ 0& \sqrt{d}\end{array}\right)\text{mbox}orM=\left(\begin{array}{cc}1& \frac{1}{2}\\ 0& \frac{\sqrt{d}}{2}\end{array}\right),$$

$$\left(\begin{array}{c}a+b{\alpha }_1\\ a+b{\alpha }_2\end{array}\right)=\left(\begin{array}{cc}1& {\alpha }_1\\ 1& {\alpha }_2\end{array}\right)\left(\begin{array}{c}a\\ b\end{array}\right)\text{mbox}and$$

$$\left(\begin{array}{c}c+d{\alpha }_1\\ \gamma \left(c+d{\alpha }_2\right)\end{array}\right)=\left(\begin{array}{cc}1& {\alpha }_1\\ \gamma & \gamma {\alpha }_2\end{array}\right)\left(\begin{array}{c}c\\ d\end{array}\right),$$

$$G_1$$

$$G_2$$

$$|det({\mathcal{G}}_{M,M})|$$

$$\rho ({\Lambda }_{-d})=\frac{1}{|\gamma {|}^2|{\alpha }_1-{\alpha }_2{|}^4|det(M){|}^4}.$$

$$de{t}_{min}(\mathcal{C}({\mathbb{F}}_1,{\alpha }_1,{\alpha }_2,{\gamma }_1)$$

$$c_{\det }(\mathcal{C}({\mathbb{F}}_1,{\alpha }_1,{\alpha }_2,{\gamma }_1))<c_{\det }(\mathcal{C}({\mathbb{F}}_2,{\beta }_1,{\beta }_2,{\gamma }_2),i.e.,$$

$$|{\gamma }_1||{\alpha }_1-{\alpha }_2{|}^2|det(M_1){|}^2<|{\gamma }_2||{\beta }_1-{\beta }_2{|}^2|det(M_2){|}^2,$$

$$c_{\det }({\mathcal{C}}_1)$$

$$c_{det}({\mathcal{C}}_2)$$

$$c_{\det }(\mathcal{C}(\mathbb{F},{\alpha }_1+p_0,{\alpha }_2+p_0,\gamma ))=c_{\det }(\mathcal{C}(\mathbb{F},{\alpha }_1,{\alpha }_2,\gamma )),$$

$$|det(M){|}^2|{\alpha }_1-{\alpha }_2{|}^2|\gamma |$$

Full text

i Introduction

The use of several antennas at both the transmitter and receiver ends of a wireless channel increases the data rate. Good transmission over such channels can only be achieved through Space-Time coding. We consider the coherent case where the receiver has perfect knowledge of all the channel coefficients.

There have been several kinds of space-time block code designs, for example, orthogonal space-time code designs [1, 2, 3, 4], unitary space-time designs [5, 6, 7], lattice based diagonal space-time code designs using algebraic number theory [8, 9, 10] and lattice based space-time block code designs from algebraic number fields, that have attracted much attention [11, 12, 13, 14, 15, 16].

A * lattice* $\Lambda$ is a discrete finitely generated free abelian subgroup $\Lambda$ of a real or complex finite dimensional vector space, called the ambient space. In the space-time setting, a natural ambient space is the space $M_{n}(\mathbb{C})$ of complex $n\times n$ matrices. Due to the symmetric situation, we only consider full-rank lattices that have a basis $x_{1},x_{2},\cdots,x_{2n^{2}}$ consisting of matrices that are linearly independent over $\mathbb{R}.$ We can form a $2n^{2}\times 2n^{2}$ matrix $\mathcal{G}$ having rows consisting of the real and imaginary parts of all the basis elements.

From the pairwise error probability (PEP) point of view [17] the performance of a space-time code is dependent on two parameters: full diversity * and minimum determinant (or coding gain or * diversity product or determinant distance, or minimum product distance). Since we are considering lattices with full-diversity, the challenge is to maximize the second parameter.

By [18] the * normalized minimum determinant* of $\Lambda$ is defined as

[TABLE]

where $\mathcal{G}$ is a generator matrix of $\Lambda$ and $det_{min}(\Lambda)=min_{0\neq X\in\Lambda}|det(X)|$. We can as well use the normalized density of the lattice

[TABLE]

Since $\delta(\Lambda)=(\rho(\Lambda))^{\frac{1}{2n}},$ we can conclude that maximizing the normalized minimum determinant is equivalent to maximizing the normalized density of the corresponding lattice, i.e., minimizing the absolute value of the generator matrix $\mathcal{G}.$

In [19] and [20], a space-time code design of full diversity is proposed by using cyclotomic extensions and in [21] is proposed space-time block codes designs based on quadratic extension. In this paper we present a structure of $2\times 2$* space-time block code* with full diversity similar to the one defined in [21], over the ring of integers of a imaginary quadratic field $\mathbb{F}$ for two-transmitter antennas. Here, contrary to what is presented in [21] the informations symbols are not only in $\mathbb{Z}[i]$ or $\mathbb{Z}[\zeta_{3}]$.

Our main contribution of this paper is to prove that the $2\times 2$ space-time block code design over $\mathbb{Z}[\zeta_{3}]$ proposed in [21] is an optimal $2\times 2$ space-time block code compared to any $2\times 2$ space-time block code over $\mathcal{O}_{\mathbb{F}},$ where $\mathbb{F}$ is any imaginary quadratic field. We also find optimal $2\times 2$ space-time block codes designs over the ring of integers of $\mathbb{Q}(\sqrt{-d}),$ with $d=2,7$ and $11.$ The optimality is the sense that the normalized density (or normalized minimum determinant) is maximal when the mean transmission signal power is fixed. When $d=7$ we will see that the optimal code obtained has the same normalized minimum determinant of the Silver code [22].

The motivation to consider information symbols in another quadratic imaginary field instead $\mathbb{Z}[i]$ and $\mathbb{Z}[\zeta_{3}]$, which are considered in most communication problems, is that there are some recently researches that consider as base field a general ring of algebraic integer, for example [23] and [24]. Moreover, the lattice reduction has also been generalized to a ring of imaginary quadratic integers [25] and only the rings from $\mathbb{Q}(\sqrt{-d}),$ where $d$ takes the values $1,2,3,7$ and $11$ can be used to define Lovász condition [26].

This paper is organized as follows. In Section II, we define real and complex lattices and briefly introduce the normalized density to composed lattices. In Section III, we propose the $2\times 2$ space-time block code designs scheme and some properties. In Section IV, we give a criterion to compare two $2\times 2$ space-time block codes and some results that will be useful in the proofs of the next section. In Section V, we dedicate to optimal $2\times 2$ space-time block codes designs over the ring of integers of imaginary quadratic fields. We present optimal codes over the ring of integers of $\mathbb{Q}(\sqrt{-d})$ with $d$ a positive squarefree integer. We also prove that the $2\times 2$ space-time block code over $\mathbb{Z}[\zeta_{3}]$ is an optimal $2\times 2$ space-time block code compared to any $2\times 2$ space-time block code over the ring of integers of any imaginary quadratic field.

ii Real and Complex Lattices

In this section, we first define real and complex lattices, and next we describe how a complex lattice can be represented by a real lattice.

Definition** ii.1**

An $n$-dimensional real lattice $\Lambda=\Lambda_{n}(M)$ is a subset in $\mathbb{R}^{n}:$

[TABLE]

where t stands for the transpose and $M$ is an $n\times n$ real matrix of full rank and called the generator matrix of the real lattice $\Lambda_{n}(M)$ and $det(\Lambda)=det(M)^{2}$ .

Definition** ii.2**

An $n$-dimensional complex lattice $\Gamma=\Lambda_{n}(L)$ over a 2-dimensional real lattice $\Lambda_{2}(M)$ is a subset of $\mathbb{C}^{n}$:

[TABLE]

where t stands for the transpose and $G$ is an $n\times n$ complex matrix of full rank and called the generator matrix of the complex lattice $\Gamma_{n}(G)$ and $M$ is a generator matrix of the lattice $\Lambda_{2}(M).$

Let $G$ be an $n\times n$ complex matrix

[TABLE]

with $|det(G)|>0$ and $\mathcal{G}$ be a $2n\times 2n$ real matrix, which is from the real and imaginary parts of $G$ as follows:

[TABLE]

where $Re(z)$ and $Im(z)$ means the real and imaginary parts of $z$, respectively.

If $(y_{1},\ldots,y_{n})^{t}\in\Gamma_{n}(G)$ over $\Lambda_{2}(M)$ then

[TABLE]

Rewritten $y_{k}$ with its real part and imaginary part it follows that $y_{k}=Re(y_{k})+iIm(y_{k}),$ for $1\leq k\leq n.$ Then $(y_{1},\cdots,y_{n})^{t}$ can be rewritten as

[TABLE]

where $z_{k1},z_{k2}\in\mathbb{Z}$ with

[TABLE]

Set $\mathcal{G}_{M}=\mathcal{G}\,\mbox{diag}(M,\ldots,M)$. We need to show that $|det(\mathcal{G}_{M})|\neq 0$, i.e, $\mathcal{G}_{M}$ has full rank and therefore $\mathcal{G}_{M}$ is a real generator matrix of a $2n$-dimensional real lattice. Since $M$ is the real generator matrix of a two-dimensional real lattice $\Lambda_{2}(M),\,|det(M)|\neq 0.$ We can conclude that $|det(\mathcal{G}_{M})|\neq 0$ by the following proposition.

Proposition 1

[19]** Let $G$ be an $n\times n$ complex matrix defined in (1) and $\mathcal{G}$ be the $2n\times 2n$ real matrix defined in (2). Then $|det(G)|^{2}=|det(\mathcal{G})|$.

Proposition 1 tell us that an $n$-dimensional complex lattice $\Gamma_{n}(G)$ over $\Lambda_{2}(M)$ can be equivalently represented as a $2n$-dimensional real lattice $\Lambda_{2n}(\mathcal{G}_{M}).$ Furthermore, the determinant of their generator matrices have the following relationship:

[TABLE]

Now we present the definition of composed complex lattice given in [20]. It is useful for $L$-layer space-time codes.

Definition** ii.3**

*An $nL$-dimensional composed complex lattice $\Gamma_{nL}(G_{1},\cdots,G_{L})$ over $\Lambda_{2}(M_{1})\times\cdots\times\Lambda_{2}(M_{L})$ consists of all points $[y_{1},\cdots,y_{n},y_{n+1},\cdots,y_{nL}]^{t},$ where each segment $[y_{(l-n)n+1},\cdots,y_{ln}]^{t}$ of length $n$ belongs to complex lattice $\Gamma_{n}(G_{l})$ over $\Lambda_{2}(M_{l}),$ i.e.

[TABLE]

$\mbox{for}\,\,x_{(l-1)n+j}\in\Lambda_{2}(M_{l}),\,\,1\leq j\leq n,\,1\leq l\leq L.$**

Similarly to a complex lattice, an $nL$-dimensional composed complex lattice can also be represented by a $2nL$-dimensional real lattice $\Lambda$ of generator matrix $\mathcal{G}_{M_{1},\cdots,M_{n}}$ and the following determinant relationship holds:

[TABLE]

According to the theory presented here, the normalized density can be written as follows:

[TABLE]

In this paper we focus when $L=2$ and $n=2.$

iii $2\times 2$ Space-Time Block Codes

Let $\mathbb{F}$ be a field, $x^{2}+px+q$ is an irreducible polynomial over $\mathbb{F}$, with $p,q\in\mathcal{O}_{\mathbb{F}}$, the ring of algebraic integers of $\mathbb{F}$. The polynomial $x^{2}+px+q$ has two roots:

[TABLE]

Let $\mathbb{K}=\mathbb{F}(\alpha_{1})$, so $[\mathbb{K}:\mathbb{F}]=2$ and $\{1,\alpha_{1}\}$ is a basis of $\mathbb{K}$ over $\mathbb{F}$. Let $\sigma_{1}$ and $\sigma_{2}$, be the two embeddings of $\mathbb{K}$ to $\mathbb{C}$ such that $\sigma_{i}(y)=y,$ for any $y\in\mathbb{F},$ $i=1,\,2$ and $\sigma_{1}(\alpha_{1})=\alpha_{1},\ \sigma_{2}(\alpha_{1})=\alpha_{2}$.

Definition** iii.1**

A $2\times 2$ space-time block code $\mathcal{C}:=\mathcal{C}(\mathbb{F},\alpha_{1},\alpha_{2},\gamma)$ based on an irreducible quadratic polynomial $x^{2}+px+q\in\mathcal{O}_{\mathbb{F}}[x]$ over $\mathbb{F}$, with roots $\alpha_{1},\,\alpha_{2}$ defined by

[TABLE]

where $\gamma\in\mathcal{O}_{\mathbb{F}},$ the ring of integers of $\mathbb{F},$ is a complex number chosen so that $\gamma\neq N_{\mathbb{F}(\alpha_{1})/\mathbb{F}}(x)$, $\forall x\in\mathbb{F}(\alpha_{1})$.

This code can be defined in terms of the quaternion division algebra $\mathcal{A}=(p^{2}-4q,\gamma)_{\mathbb{F}}$, $i^{2}=p^{2}-4q,$ $j^{2}=\gamma$ with $\gamma\neq N_{\mathbb{F}(\alpha_{1})/\mathbb{F}}(x)$, $\forall x\in\mathbb{F}(\alpha_{1})$. We refer readers to [27], [28] for a detailed exposition of the theory of simple algebras, cyclic algebras, their matrix representations and their use in space-time coding.

Definition** iii.2**

Let $\mathcal{C}(\mathbb{F},\alpha_{1},\alpha_{2},\gamma)$ be an $2\times 2$ space-time block code, we define the minimum determinant of the code by

[TABLE]

The next lemma tells us that when we consider quadratic imaginary fields, the minimum determinant of a $2\times 2$ space-time block codes $1$.

Lemma 1

If $\mathbb{F}=\mathbb{Q}(\sqrt{-d})$, with $d$ a positive squarefree integer then any $\mathcal{C}(\mathbb{F},\alpha_{1},\alpha_{2},\gamma)$ has $det_{min}(\mathcal{C}(\mathbb{F},\alpha_{1},\alpha_{2},\gamma))=1.$

Proof:

If $X\in\mathcal{C}(\mathbb{F},\alpha_{1},\alpha_{2},\gamma),$ then

[TABLE]

and thus, taking $x_{1}=a+\alpha_{1}b$ and $x_{2}=c+\alpha_{1}d$, we have

$det(X)=N_{\mathbb{F}(\alpha_{1})/\mathbb{F}}(x_{1})-\gamma N_{\mathbb{F}(\alpha_{1})/\mathbb{F}}(x_{2})$.

Since $\gamma$ is not the algebraic norm of $\mathbb{F}(\alpha_{1})$ on $\mathbb{F}$ it follows that if either $x_{1}\neq 0$ or $x_{2}\neq 0$ then $N(x_{1})\neq\gamma N(x_{2}),$ i.e., $det(X)\neq 0$, if $X\neq 0.$ Also, as $\gamma\in\mathcal{O}_{\mathbb{F}}$, it follows that $det(X)\in\mathcal{O}_{\mathbb{F}}$. Thus we conclude that $|det(X)|\geq 1.$

Now, taking $X$ such that $a=1,\,b=c=d=0$, it follows that $det(X)=1$. Therefore, $det_{min}(\mathcal{C}(\mathbb{F},\alpha_{1},\alpha_{2},\gamma))=1$. ∎

In order to design a $2\times 2$ space-time block code with a large normalized density and nonvanishing determinant, we consider $\mathbb{F}=\mathbb{Q}(\sqrt{-d}),$ with $d$ a positive squarefree integer.

Lattice theory has a close relation to number theory. We refer readers to [29, 30] for some concepts of lattices and rings that will be used throughout this paper.

If $-d\equiv 1(mod\,4)\,$ then $\mathcal{O}_{\mathbb{F}}=\{1,\frac{1+\sqrt{-d}}{2}\}$ with integral basis $B_{\mathbb{F}}=\{1,\frac{1+\sqrt{-d}}{2}\}.$ Otherwise, if $-d\equiv 2,3(mod\,4)\,$ then $\mathcal{O}_{\mathbb{F}}=\{1,\sqrt{-d}\}$ with integral basis $B_{\mathbb{F}}=\{1,\sqrt{-d}\}$. Then the corresponding generator matrix of 2-dimensional lattice $\Lambda_{2}(M)$ is

[TABLE]

respectively.

A two-layer $2\times 2$ space-time block code is a lattice code over $\Lambda_{2}(M)\times\Lambda_{2}(M)$ and by Definition ii.3:

[TABLE]

[TABLE]

with

[TABLE]

By (3), if $\mathcal{G}_{M,M}$ is a generator matrix of the 8-dimensional real lattice that we denote by $\Lambda_{-d},$ then

[TABLE]

and by (4)

[TABLE]

Remark 1

We denote $|\gamma||\alpha_{1}-\alpha_{2}|^{2}|det(M)|^{2}$ by ${c}_{\det}(\mathcal{C}(\mathbb{F},\alpha_{1},\alpha_{2},\gamma)).$ Note that the smaller ${c}_{\det}(\mathcal{C}(\mathbb{F},\alpha_{1},\alpha_{2},\gamma))$ is, the better normalized density of $\Lambda_{-d}$ is.

iv Lattice Design Criterion

With the argument of Section III we present a criterion [20] to compare two $2\times 2$ space-time block codes.

Definition** iv.1**

(Determinant Criteria) Let $\mathcal{C}(\mathbb{F}_{1},\alpha_{1},\alpha_{2},\gamma_{1})$ and $\mathcal{C}(\mathbb{F}_{2},\beta_{1},\beta_{2},\gamma_{2})$ be two $2\times 2$ space-time block codes with

[TABLE]

We say that $\mathcal{C}(\mathbb{F}_{1},\alpha_{1},\alpha_{2},\gamma_{1})$ is better than $\mathcal{C}(\mathbb{F}_{2},\beta_{1},\beta_{2},\gamma_{2})$ if

[TABLE]

[TABLE]

where $M_{1}$ a generator matrix of $\Lambda_{2}(M_{1})$ and $M_{2}$ a generator matrix of $\Lambda_{2}(M_{2})$.

Note that codes from same algebras may have different $c_{\det}(\cdot)$ as we can see in the Example 1.

Example 1

Let $\mathcal{C}_{1}=\mathcal{C}\left(\mathbb{Q}(\sqrt{-2}),\sqrt{-3},-\sqrt{-3},\right.$ $\left.\gamma\right)$ and $\mathcal{C}_{2}=\mathcal{C}\left(\mathbb{Q}(\sqrt{-2}),\frac{1+\sqrt{-3}}{2},\frac{1-\sqrt{-3}}{2},\gamma\right)$ be two $2\times 2$ space-time block codes and $M$ a generator matrix of $\Lambda_{2}(M)$. Note that $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ come from the algebra $\mathcal{A}=(-3,\gamma)_{\mathbb{Q}(\sqrt{-2})}$ and

[TABLE]

while

[TABLE]

Soon, $c_{det}(\mathcal{C}_{1})>c_{det}(\mathcal{C}_{2})$.

Definition** iv.2**

Let $S$ be a set of $2\times 2$ space-time block codes, where $det_{min}\mathcal{C}=det_{min}\overline{\mathcal{C}}$ for all $\mathcal{C},\overline{\mathcal{C}}\in S$. We say that $\mathcal{C}$ is an optimal $2\times 2$ space-time block codes in $S$, if

$c_{\det}(\mathcal{C})\leq c_{\det}(\overline{\mathcal{C}})$,

for all $\overline{\mathcal{C}}\in S$.

Now we stablish a strategy to minimize $c_{det}(\cdot).$

In [21] it was shown that if $p_{0}\in\mathcal{O}_{\mathbb{F}}$ then

[TABLE]

i.e.,

[TABLE]

Note that the irreducible polynomial of $\alpha_{1}$ over $\mathbb{F}$ is $(x-\alpha_{1})(x-\alpha_{2})=x^{2}+px+q$, where $p=-(\alpha_{1}+\alpha_{2}),$ $q=\alpha_{1}\alpha_{2}$ and the irreducible polynomial of $\alpha_{1}+p_{0}$ over $\mathbb{F}$ is

[TABLE]

where $p^{\prime}=p-2p_{0}$ and $q^{\prime}=q-p_{0}p+p_{0}^{2}.$

We can see by (6) that minimize $|\alpha_{1}-\alpha_{2}|^{2}=|p^{2}-4q|$ is the same that minimize $|(\alpha_{1}+p_{0})-(\alpha_{2}+p_{0})|^{2}=|(p-2p_{0})^{2}-4(q-pp_{0}+p^{2})|.$

One strategy to minimize $|(p-2p_{0})^{2}-4(q-pp_{0}+p^{2})|\leq|p-2p_{0}|^{2}+|-4(q-pp_{0}+p^{2})|$ (triangular inequality) is first minimize $|p-2p_{0}|.$

Proposition 2

Let $\mathbb{F}=\mathbb{Q}(\sqrt{-d})$, with $d$ a positive squarefree integer and $\mathcal{O}_{\mathbb{F}}$ its ring of integer. If $p\in\mathcal{O}_{\mathbb{F}}$ then there exists $p_{0}\in\mathbb{F}$ such that

[TABLE]

Proof:

If $d\equiv 1(mod\,4)$, then $p=a+b\left(\frac{1+\sqrt{-d}}{2}\right)$ with $a,b\in\mathbb{Z}$. Taking $p_{0}=r+s\left(\frac{1+\sqrt{-d}}{2}\right)\in\mathbb{F}$ we need to check the following cases:

• (i)

if $a,b$ are even, then $min\{|p-2p_{0}|\}=0$ (take $r=\frac{a}{2}$ and $s=\frac{b}{2}$).

• (ii)

if $a$ is even and $b$ is odd, then $min\{|p-2p_{0}|\}=\left|\frac{1+\sqrt{-d}}{2}\right|$ (take $r=\frac{a}{2}$ and $s=\frac{b-1}{2}$).

• (iii)

if $a$ is odd and $b$ is even, then $min\{|p-2p_{0}|\}=1$ (take $r=\frac{a+1}{2}$ and $s=\frac{b}{2}$).

• (iv)

if $a,b$ are odd, then $min\{|p-2p_{0}|\}=\left|\frac{-1+\sqrt{-d}}{2}\right|$ (take $r=\frac{a+1}{2}$ and $s=\frac{b-1}{2}$).

So, given $p\in\mathcal{O}_{\mathbb{F}}$, there exists $p_{0}\in\mathcal{O}_{\mathbb{F}}$ such that $|p-2p_{0}|\leq\left|\frac{1+\sqrt{-d}}{2}\right|$.

Otherwise, if $d\equiv 2,3(mod\,4)$, then $p=a+b\sqrt{-d}$, with $a,b\in\mathbb{Z}$. Analogous, taking $p_{0}=r+s\sqrt{-d}\in\mathbb{F}$ and checking the same cases as above we conclude that there exists $p_{0}\in\mathcal{O}_{\mathbb{F}}$ such that $|p-2p_{0}|\leq\left|1+\sqrt{-d}\right|$. ∎

v Optimal $2\times 2$ Space-Time Block Codes designs

The Golden Code $\mathcal{C}_{GC}=\mathcal{C}(\mathbb{Q}(\sqrt{-1}),\frac{1+\sqrt{5}}{2},\frac{1-\sqrt{5}}{2},\sqrt{-1})$ is a perfect space-time block code for two-transmitter antennas with minimum determinant 1 [31]. We can to verify by (5) that $\rho(\Lambda_{GC})=0.04.$

In [21], the authors proved that the $2\times 2$ space-time block code

In addition, they also have proved that $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-1})}}=\mathcal{C}(\mathbb{Q}(\sqrt{-1}),\frac{-i-\sqrt{3}}{2},\frac{-i+\sqrt{3}}{2},\sqrt{1+i})$ is an optimal $2\times 2$ space-time block code over $\mathbb{Z}[i]$ with $\rho(\Lambda_{-1})\approx 0.056.$

As we can see, in terms of normalized density analysis the $2\times 2$ space-time block code $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-3})}}$ is better than $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-1})}}.$

According to these considerations, there is a question unanswered. Is $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-3})}}$ an optimal $2\times 2$ space-time block code when compared with any $2\times 2$ space-time block code $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-d})}}$, with $d$ a positive squarefree integer? In order to give an answer to this question, we will see that is enough to find optimal $2\times 2$ space-time block codes over $\mathcal{O}_{\mathbb{Q}(\sqrt{-d})},$ with $d=2,7,11.$

Suppose that there exists a $2\times 2$ space-time block code better than $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-3})}}$, i.e., there exists a $2\times 2$ space-time block code $\mathcal{C}(\mathbb{Q}(\sqrt{-d}),\beta_{1},\beta_{2},\gamma)$, with $d$ a positive squarefree integer, such that $\rho(\Lambda_{-3})<\rho(\Lambda_{-d}),$ in other words, by Definition (iv.1)

$|det(M)|^{2}|\beta_{1}-\beta_{2}|^{2}|\gamma|<c_{det}(\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-3})}})\approx 3.44$,

where $M$ is a generator matrix of the lattice obtained via $\mathcal{O}_{\mathbb{Q}(\sqrt{-d})}.$

Let’s to analyse when $|det(M)|^{2}<3.44$.

In the Section III we have seen that,

• i)

if $d<0,\,d\equiv 1(mod\,4)$, then $|det(M)|^{2}=\frac{|d|}{4}$. Soon $d\in\{-3,-7,-11\}$ are the only $d$ satisfying $|det(M)|^{2}<3.44$.

• ii)

if $d<0,\,d\equiv 2,\,3(mod\,4)$ then $|det(M)|^{2}=|d|$. Soon, $d\in\{-1,-2\}$ are the only $d$ satisfying $|det(M)|^{2}<3.44$.

So, we only need to consider $d\in\{-2,-7,-11\}$.

Therefore, from now on we will find the optimal $2\times 2$ space-time block codes over $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-d})}},$ with $d=2,7,11$ in order to compare with $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-3})}}$. To do it we need to find convenient $\alpha_{1},\alpha_{2}$ and $\gamma$, where $\gamma$ is not algebraic norm of $\mathbb{Q}(\sqrt{d},\alpha_{1})$ over $\mathbb{Q}(\sqrt{d})$. The corollaries bellow will be useful in this search.

Corollary 1

Let $d\in\mathbb{Z}$. If $d\equiv 1(mod\,3)$, then $-1$ is not algebraic norm of $\mathbb{K}=\mathbb{Q}(\sqrt{d},\sqrt{-3})$ over $\mathbb{F}=\mathbb{Q}(\sqrt{d})$.

Proof:

Our goal is to show that the equation $\mathcal{N}_{\mathbb{K}/\mathbb{F}}(x)=-1$ has no solution in the field of 3-adic numbers $Q_{3}$, and thus, no solution for all $x\in\mathbb{K}.$

Let $Z_{3}=\{x\in Q_{3}\mid v_{3}(x)\geq 0\}$ be the valuation ring of $Q_{3}$, where $v_{3}(x)$ denotes the 3-adic valuation of $x$. Since $d\in\mathbb{Z}$, it follows that $v_{3}(d)\geq 0$, and then $d\in Z_{3}$. Taking $f(x)=x^{2}-d$ obviously $f(x)\in Z_{3}[x]$. Note that $f(1)=1-d\equiv 0(mod\,3)$ and $f^{\prime}(1)=2\not\equiv 0(mod\,3).$ By Hensel’s Lemma [32], it follows that $\exists\gamma\in Z_{3}$ such that $f(\gamma)=0$, i.e., $\gamma^{2}-d=0$. It means that $\sqrt{d}\in Z_{3}\subset Q_{3}$. Thus we can view the field $\mathbb{F}$ as a subfield of $Q_{3}$. Analogously, the field $\mathbb{K}$ can be viewed as a subfield of $Q_{3}(\sqrt{-3})$.

Furthermore, the norm map $\mathcal{N}_{\mathbb{K}/\mathbb{F}}:\mathbb{K}\rightarrow\mathbb{F}$ is then a restriction of the norm maps $\mathcal{N}:Q_{3}(\sqrt{-3})\rightarrow Q_{3}$, where $\mathcal{N}(r+s(\frac{1+\sqrt{-3}}{2}))=(r+\frac{s}{2})^{2}+3(\frac{s}{2})^{2},\forall r,s\in Q_{3}$.

Thus, in order to prove our claim, it is enough to show that $-1$ is not in the image of the map $\mathcal{N}.$

Set $a=r+\frac{s}{2}$ and $b=\frac{s}{2}$. Suppose on the contrary, that there are $3$-adic numbers $a$ and $b$ such that $a^{2}+3b^{2}=-1$. We first show that $a$ and $b$ are in $Z_{3},$ i.e., $v_{3}(a)>0$ and $v_{3}(b)>0$. Assume that at least one of them has a negative exponent 3-adic valuation, i.e.,

• i)

if $v_{3}(a)<0$, then $v_{3}(a^{2})=2v_{3}(a)<0$ and $|a^{2}|_{3}>1$. Since $|a^{2}+3b^{2}|_{3}=|-1|_{3}=1$ the non-archimedean property (see [32]) implies that $v_{3}(a^{2})=v_{3}(3b^{2})$.

• ii)

if $v_{3}(b)<0$, then $2v_{3}(b)<0$. Since $v_{3}(3b^{2})=v_{3}(3)+v_{3}(b^{2})=1+2v_{3}(b)$ and $v_{3}(n)\in\mathbb{Z},\,\forall n\in\mathbb{Q}$, it follows that

$v_{3}(b)<0\Rightarrow v_{3}(b)\leq-1\Rightarrow 2v_{3}(b)\leq-2$

$\Rightarrow 2v_{3}(b)+1\leq-1.$

Thus $|3b^{2}|=3^{-(2v_{3}(b)+1)}>3^{-(-1)}=3>1=|-1|=|a^{2}+3b^{2}|$. Again, the non-archimedean property implies that $v_{3}(a^{2})=v_{3}(3b^{2})$.

In both cases, $2v_{3}(a)=v_{3}(a^{2})=v_{3}(3b^{2})=1+2v(b)$. It means that, there exists $t\in\mathbb{Z}$, with $t<0$, such that $a=3^{t}a^{\prime}$ and $b=3^{t-1/2}b^{\prime}$, with $a^{\prime},$ $b^{\prime}$ 3-adic units. This implies that $a^{2}+3b^{2}=3^{2t}a^{\prime 2}+3^{2t}b^{\prime}=3^{2t}(a^{\prime 2}+b^{\prime 2})$.

Since $a^{\prime}$ and $b^{\prime}$ are units, it follows that $v_{3}(a^{\prime})=v_{3}(b^{\prime})=0$, and then

$a^{\prime},b^{\prime}\equiv 1,2(mod\,3)\Rightarrow a^{\prime 2},b^{\prime 2}\equiv 1(mod\,3)\Rightarrow a^{\prime 2}+b^{\prime 2}\equiv 2(mod\,3)$.

In terms of valuation, we have $v_{3}(a^{2}+b^{2})=2t<0$ since $t<0$. Thus, $0>v_{3}(a^{2}+3b^{2})=v_{3}(-1)=0$, which is a contradiction. Therefore $v_{3}(a)\geq 0$ and $v_{3}(b)\geq 0$, and hence $a$ and $b$ are in $Z_{3}$. In this case the result follows from the fact that the square of an integer is always congruent to either [math] or $1$ module $3$.

Using the fact that $3b^{2}\equiv 0(mod\,3)$ and $a^{2}\not\equiv 2(mod\,3)$ we have that $a^{2}+3b^{2}$ can be not equal to $-1$, since $-1\equiv 2(mod\,3).$ Therefore, $-1$ is not in the image of the map $\mathcal{N}$ and consequently $\mathcal{N}_{\mathbb{K}/\mathbb{F}}(x)\neq-1,$ for all $x\in\mathbb{K}.$ ∎

Corollary 2

If $d\in\mathbb{Z}$ with $d\equiv 1(mod\,8)$, then the integer $-1$ is not an algebraic norm of the extension $\mathbb{Q}(\sqrt{d},i)$ over $\mathbb{Q}(\sqrt{d})$.

Proof:

Analogous to Corollary 1, here consider the 2-adic field $Q_{2}$. ∎

In what follows we present the optimum $2\times 2$ space-time block code over $\mathcal{O}_{\mathbb{F}}$ when $\mathbb{F}=\mathbb{Q}(\sqrt{-d})$, with $d=2,7,11$.

v-A Optimal $2\times 2$ Space-Time Block Codes over $\mathbb{Z}[\sqrt{-2}]$

We are interested in finding the optimal $2\times 2$ space-time block code $\mathcal{C}(\mathbb{Q}(\sqrt{-2}),\alpha_{1},\alpha_{2},\gamma)$ over $\mathbb{Z}[\sqrt{-2}]$. By Corollary 1, -1 is not an algebraic norm of $\mathbb{Q}(\sqrt{-2},\sqrt{-3})$ over $\mathbb{Q}(\sqrt{-2})$, then we can take $\gamma=-1.$ An irreducible polynomial of $\mathbb{Q}(\sqrt{-2},\sqrt{-3})$ over $\mathbb{Q}(\sqrt{-2})$ is $x^{2}-x+1,$ so $\alpha_{1}=\frac{1+\sqrt{-3}}{2}$ and $\alpha_{2}=\frac{1-\sqrt{-3}}{2}.$ From this we can establish the following theorem.

Theorem** v.1**

The code $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-2})}}=\mathcal{C}(\mathbb{Q}(\sqrt{-2}),\frac{1+\sqrt{-3}}{2},\frac{1-\sqrt{-3}}{2},-1)$ is an optimal $2\times 2$ space-time block code among all $2\times 2$ space-time block codes over $\mathbb{Z}[\sqrt{-2}]$ with minimum determinant $1$.

Proof:

By Lemma 1 we know that $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-2})}}$ is a $2\times 2$ space-time block code with minimum determinant 1. According to Section III, since $-2\equiv 2(mod\,4)$, it follows that $|det(M)|=\sqrt{2.}$ Thus, by (5),

[TABLE]

The value $|det(M)|^{2}$ is invariant because $\mathbb{F}=\mathbb{Q}(\sqrt{-2})$ is the base field, then we can remove it in our analysis.

In this case, by according Remark 1, $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-2})}}$ is optimal if $c_{\det}(\mathcal{C}(\mathbb{Q}(\sqrt{-2}),\beta_{1},\beta_{2},\gamma))\geq 3$, to any $2\times 2$ space-time block code $\mathcal{C}(\mathbb{Q}(\sqrt{-2}),\beta_{1},\beta_{2},\gamma).$

Suppose that there exists a $2\times 2$ space-time block code $\tilde{\mathcal{C}}=\mathcal{C}(\mathbb{Q}(\sqrt{-2}),\beta_{1},\beta_{2},\gamma)$ with minimum determinant 1 based on irreducible polynomial $x^{2}+px+q$, with $p,q\in\mathbb{Z}[\sqrt{-2}]$ whose roots are $\beta_{1}$ and $\beta_{2}$ such that

[TABLE]

By Proposition 2, without loss of generality, we can always assume that $|p|\leq|1+\sqrt{-2}|=\sqrt{3}.$ Since $p\in\mathbb{Z}[\sqrt{-2}]$ we have that $p=a+b\sqrt{-2},\,a,b\in\mathbb{Z}$ and then

[TABLE]

[TABLE]

Again, since $q\in\mathbb{Z}[\sqrt{-2}]$ we have that $q=c+d\sqrt{-2},\,c,d\in\mathbb{Z}$ and then to each pair $(a,b)$ of (7) we analyse when $|p^{2}-4q|=|(a+b\sqrt{-2})^{2}-4(c+d\sqrt{-2})|<3$. Thus, we consider the following cases.

• (i)

$p^{2}=0.$ In this case, $|p^{2}-4q|<3\Leftrightarrow c=d=0,$ i.e., $q=0.$ When $q=0$ we have $x^{2}+px+q=x^{2}$, which is reducible over $\mathbb{Q}(\sqrt{-2}).$

• (ii)

$p^{2}=-2$. In this case, $|p^{2}-4q|<3\Leftrightarrow c=d=0$ or $c=-1,d=0,$ i.e., $q=0$ or $q=-1.$ When $q=0$ we have $x^{2}+px+q=x^{2}\pm\sqrt{-2}x$, which are reducible over $\mathbb{Q}(\sqrt{-2}).$ When $q=-1$ we have $x^{2}+px+q=x^{2}\pm\sqrt{-2}x-1$, which are irreducible over $\mathbb{Q}(\sqrt{-2}).$

• (iii)

$p^{2}=1$. In this case, $|p^{2}-4q|<3\Leftrightarrow c=d=0,$ i.e., $q=0.$ When $q=0$ we have $x^{2}+px+q=x^{2}\pm x$, which are reducible over $\mathbb{Q}(\sqrt{-2}).$

Considering the irreducible polynomials obtained in (ii) we have that

[TABLE]

[TABLE]

However, if $\mathbb{K}=\mathbb{Q}(\sqrt{-2},\sqrt{2})$ and $\mathbb{F}=\mathbb{Q}(\sqrt{-2})$, then

$N_{\mathbb{K}/\mathbb{F}}(0)=0,$ $N_{\mathbb{K}/\mathbb{F}}(1)=1,$ $N_{\mathbb{K}/\mathbb{F}}\left(\frac{-\sqrt{-2}+\sqrt{2}}{2}\right)=-1,$

$N_{\mathbb{K}/\mathbb{F}}\left(-\left(\frac{-\sqrt{-2}+\sqrt{2}}{2}\right)-\sqrt{-2}-1\right)=\sqrt{-2},$ and $N_{\mathbb{K}/\mathbb{F}}\left(\left(\frac{-\sqrt{-2}+\sqrt{2}}{2}\right)\left(\sqrt{-2}+1\right)-1\right)=-\sqrt{-2}.$

It means that $\gamma$ does not satisfy the conditions of Definition iii.1.

In conclusion, there is no $\tilde{\mathcal{C}}$ in the conditions above such that $c_{det}(\tilde{\mathcal{C}})<3$. Therefore $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-2})}}$ is an optimal $2\times 2$ space-time block code over $\mathbb{Z}[{\sqrt{-2}}].$ ∎

v-B Optimal $2\times 2$ Space-Time Block Codes over $\mathbb{Z}[\frac{1+\sqrt{-7}}{2}]$

We are interested in finding the optimal $2\times 2$ space-time block code $\mathcal{C}(\mathbb{Q}(\sqrt{-7}),\alpha_{1},\alpha_{2},\gamma)$ over $\mathbb{Z}[\frac{1+\sqrt{-7}}{2}]$. By Corollary 2, -1 is not an algebraic norm of $\mathbb{Q}(\sqrt{-7},\sqrt{-1})$ over $\mathbb{Q}(\sqrt{-7})$, so we can take $\gamma=-1.$ An irreducible polynomial of $\mathbb{Q}(\sqrt{-7},\sqrt{-1})$ over $\mathbb{Q}(\sqrt{-7})$ is $x^{2}+1,$ so $\alpha_{1}=\sqrt{-1}$ and $\alpha_{2}=-\sqrt{-1}.$ From this we can establish the following theorem.

Theorem** v.2**

The code $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-7})}}=\mathcal{C}(\mathbb{Q}(\sqrt{-7}),\sqrt{-1},-\sqrt{-1},-1)$ is an optimal $2\times 2$ space-time block code among all $2\times 2$ space-time block code over $\mathbb{Z}[\frac{1+\sqrt{-7}}{2}]$ with minimum determinant $1$.

Proof:

By according consideration of Section III, as $-7\equiv 1(mod\,4)$ it follows that $|det(M)|=\frac{\sqrt{7}}{2}.$

Thus, by (5),

[TABLE]

The value $|det(M)|^{2}$ is invariant because $\mathbb{F}=\mathbb{Q}(\sqrt{-7})$ is the base field, then we can remove it in our analysis.

In this case, by according Remark 1, $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-7})}}$ is optimal if $c_{\det}(\mathcal{C}(\mathbb{Q}(\sqrt{-7}),\beta_{1},\beta_{2},\gamma))\geq 4$, to any $2\times 2$ space-time block code $\mathcal{C}(\mathbb{Q}(\sqrt{-7}),\beta_{1},\beta_{2},\gamma).$

Suppose that there exists a $2\times 2$ space-time block code $\tilde{\mathcal{C}}=\mathcal{C}(\mathbb{Q}(\sqrt{-7}),\beta_{1},\beta_{2},\gamma)$ with minimum determinant 1 based on irreducible polynomial $x^{2}+px+q$, with $p,q\in\mathbb{Z}[\frac{1+\sqrt{-7}}{2}]$ whose roots are $\beta_{1}$ and $\beta_{2}$ such that

[TABLE]

By Proposition 2, without loss of generality, we can always assume that $|p|<|\frac{1+\sqrt{-7}}{2}|=\sqrt{2}.$

Since $p\in\mathbb{Z}[\frac{1+\sqrt{-7}}{2}]$ we have that $p=a+b\left(\frac{1+\sqrt{-7}}{2}\right),\,a,b\in\mathbb{Z}$ and then

[TABLE]

[TABLE]

Again, since $q\in\mathbb{Z}[\frac{1+\sqrt{-7}}{2}]$ we have that $q=c+d\left(\frac{1+\sqrt{-7}}{2}\right),\,c,d\in\mathbb{Z}$ and then to each pair $(a,b)$ of (8) we analyse when $|p^{2}-4q|=|(a+b\sqrt{-2})^{2}-4\left(c+d\left(\frac{1+\sqrt{-7}}{2}\right)\right)|<4$. Thus, we consider the following cases.

• (i)

$p^{2}=0.$ In this case, $|p^{2}-4q|<4\Leftrightarrow c=d=0,$ i.e., $q=0.$ When $q=0$ we have $x^{2}+px+q=x^{2}$, which is reducible over $\mathbb{Q}(\sqrt{-7}).$

• (ii)

$p^{2}=1$. In this case, $|p^{2}-4q|<3\Leftrightarrow c=d=0$ or $c=1,d=0,$ i.e., $q=0$ or $q=1.$ When $q=0$ we have $x^{2}+px+q=x^{2}\pm x$, which are reducible over $\mathbb{Q}(\sqrt{-7}).$ When $q=1$ we have $x^{2}+px+q=x^{2}\pm x+1$, which are irreducible over $\mathbb{Q}(\sqrt{-7}).$

Considering the irreducible polynomials obtained in (ii) we have that

[TABLE]

[TABLE]

However, if $\mathbb{K}=\mathbb{Q}(\sqrt{-7},\sqrt{-3})$ and $\mathbb{F}=\mathbb{Q}(\sqrt{-7})$, then

$N_{\mathbb{K}/\mathbb{F}}(0)=0,\,N_{\mathbb{K}/\mathbb{F}}(1)=1,N_{\mathbb{K}/\mathbb{F}}\left(\left(\frac{1+\sqrt{-7}}{2}-1\right)\left(\frac{1+\sqrt{-3}}{2}\right)-\frac{1+\sqrt{-7}}{2}\right)=-1.$

It means that $\gamma$ does not satisfy the conditions of Definition iii.1.

In conclusion, there is no $\tilde{\mathcal{C}}$ in the conditions above such that such that $c_{det}(\tilde{\mathcal{C}})<4$. Therefore $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-7})}}$ is an optimal $2\times 2$ space-time block code over $\mathbb{Z}[\frac{1+\sqrt{-7}}{2}].$ ∎

v-C Optimal $2\times 2$ Space-Time Block Codes over $\mathbb{Z}[\frac{1+\sqrt{-11}}{2}]$

We are interested in finding the optimal $2\times 2$ space-time block code $\mathcal{C}(\mathbb{Q}(\sqrt{-11}),\alpha_{1},\alpha_{2},\gamma)$ over $\mathbb{Z}[\frac{1+\sqrt{-11}}{2}]$. By Corollary 1, -1 is not an algebraic norm of $\mathbb{Q}(\sqrt{-11},\sqrt{-3})$ over $\mathbb{Q}(\sqrt{-11})$, so we can take $\gamma=-1.$ An irreducible polynomial of $\mathbb{Q}(\sqrt{-11},\sqrt{-3})$ over $\mathbb{Q}(\sqrt{-11})$ is $x^{2}-x+1,$ so $\alpha_{1}=\frac{1+\sqrt{-3}}{2}$ and $\alpha_{2}=\frac{1-\sqrt{-3}}{2}.$ From this we can establish the following theorem.

Theorem** v.3**

The code $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-11})}}=\mathcal{C}(\mathbb{Q}(\sqrt{-11}),\frac{1+\sqrt{-3}}{2},\frac{1-\sqrt{-3}}{2},-1)$ is an optimal $2\times 2$ space-time block code among all $2\times 2$ space-time block codes on $\mathbb{Z}[\frac{1+\sqrt{-11}}{2})$ with minimum determinant $1$.

Proof:

Analogous to proof of the Theorems v.1 and v.2, we obtain by (5),

[TABLE]

The value $|det(M)|^{2}$ is invariant because $\mathbb{F}=\mathbb{Q}(\sqrt{-11})$ is the base field, then we can remove it in our analysis.

In this case, by according Remark 1 $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-11})}}$ is optimal if $c_{\det}(\mathcal{C}(\mathbb{Q}(\sqrt{-11}),\beta_{1},\beta_{2},\gamma))\geq 3$, to any $2\times 2$ space-time block code $\mathcal{C}(\mathbb{Q}(\sqrt{-11}),\beta_{1},\beta_{2},\gamma).$

Suppose that there exists a $2\times 2$ space-time block code $\tilde{\mathcal{C}}=\mathcal{C}(\mathbb{Q}(\sqrt{-11}),\beta_{1},\beta_{2},\gamma)$ with minimum determinant 1 based on irreducible polynomial $x^{2}+px+q$, with $p,q\in\mathbb{Z}[\frac{1+\sqrt{-11}}{2}]$ whose roots are $\beta_{1}$ and $\beta_{2}$ such that

[TABLE]

By Proposition 2, without loss of generality, we can always assume that $|p|<|\frac{1+\sqrt{-11}}{2}|=\sqrt{3}.$ Since $p\in\mathbb{Z}[\frac{1+\sqrt{-11}}{2}]$ we have that $p=a+b\left(\frac{1+\sqrt{-11}}{2}\right),\,a,b\in\mathbb{Z}$ and then

[TABLE]

[TABLE]

Again, since $q\in\mathbb{Z}[\frac{1+\sqrt{-11}}{2}]$ we have that $q=c+d\left(\frac{1+\sqrt{-11}}{2}\right),\,c,d\in\mathbb{Z}$ and then to each pair $(a,b)$ of (10) we analyse when $|p^{2}-4q|=|\left(a+b\left(\frac{1+\sqrt{-11}}{2}\right)\right)^{2}-4\left(c+d\left(\frac{1+\sqrt{-11}}{2}\right)\right)|<3$. Thus, we consider the following cases.

• (i)

$p^{2}=0.$ In this case, $|p^{2}-4q|<3\Leftrightarrow c=d=0,$ i.e., $q=0.$ When $q=0$ we have $x^{2}+px+q=x^{2}$, which is reducible over $\mathbb{Q}(\sqrt{-11}).$

• (ii)

$p^{2}=1$. In this case, $|p^{2}-4q|<3\Leftrightarrow c=d=0,$ i.e., $q=0.$ When $q=0$ we have $x^{2}+px+q=x^{2}\pm x$, which are reducible over $\mathbb{Q}(\sqrt{-11}).$

Therefore there is no irreducible polynomials over $\mathbb{Q}(\sqrt{-11})$ that satisfy (9). In conclusion, there is no $\tilde{\mathcal{C}}$ in the conditions above such that $c_{det}(\tilde{\mathcal{C}})<3$. Therefore $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-11})}}$ is an optimal $2\times 2$ space-time block code over $\mathbb{Z}[\frac{1+\sqrt{-11}}{2}].$

∎

Thus, in this section we proved that the following theorem holds true.

Theorem** v.4**

The code $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-3})}}=\mathcal{C}(\mathbb{Q}(\zeta_{3}),\frac{-p+\sqrt{p^{2}-4q}}{2},\frac{-p-\sqrt{p^{2}-4q}}{2},\zeta_{6})$ is an optimal $2\times 2$ space-time block code among all $2\times 2$ space-time block codes with minimum determinant 1 over $\mathcal{O}_{\mathbb{F}}$, where $\mathbb{F}=\mathbb{Q}(\sqrt{-d})$, with $d$ a positive squarefree integer, where $p=-1-\zeta_{6}$ and $q=\sqrt{-3}$.

Remark 2

In order to improve the performance of space-time codes the requirement is that $|\gamma|=1,$ so that the average transmitted energy by each antenna in all time slots is equalized. Note that the space-time block code designs proposed here this requirement is satisfied.

However the designs proposed in [21], $|\gamma|\neq 1.$ This can be remedied by changing the codewords from

[TABLE]

to

[TABLE]

where $a,b,c,d\in\mathcal{O}_{\mathbb{F}},$ $\mathbb{F}$ is an imaginary quadratic field.

In the Table I we summarize what we have proved here.

The algebra $(-1,-1)_{\mathbb{Q}(\sqrt{-7})}$ is known as Silver Algebra. In [22] it is shown that an ideal in the Silver algebra generates the Silver code and its exact normalized minimum determinant is computed using the ideal structure of the code, whose value is $1/\sqrt{7}$. Here, since $\rho(\Lambda_{-7})\approx 0.0204\,(=1/7^{2})$ it follows that $\delta(\Lambda_{-d})=1/\sqrt{7}$, which is the same the one of the Silver code.

vi Conclusion

In this paper, we presented optimal $2\times 2$ space-time block codes designs over the ring of integers of $\mathbb{Q}(\sqrt{-d})$ with $d\in\{2,7,11\}$. This expands the design proposed in [21], where the authors only consider the informations symbols in $\mathbb{Z}[i]$ and $\mathbb{Z}[\zeta_{3}].$ We also proved that the optimal $2\times 2$ space-time block code over $\mathbb{Z}[\zeta_{3}]$ is an optimal $2\times 2$ space-time block code compared with any $2\times 2$ space-time block code over $\mathcal{O}_{\mathbb{F}},$ where $\mathbb{F}$ is an imaginary quadratic field. Although the optimal $2\times 2$ space-time block codes proposed here have normalized density smaller than the one of the codes proposed in [21], via the codes $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-2})}}$ and $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-7})}}$ we can construct the $E_{8}$-lattice. Moreover these codes are better suited for decoding using the method of algebraic reduction [33], that recently has also been generalized to a ring of imaginary quadratic integers [25] . In fact, we can define the codes $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-2})}}$ and $\mathcal{C}_{\mathcal{O}_{\mathbb{Q}(\sqrt{-7})}}$ in terms of the quaternion division algebra $\mathcal{A}=(-3,-1)_{\mathbb{Q}(\sqrt{-2})}$ and $\mathcal{A}=(-1,-1)_{\mathbb{Q}(\sqrt{-7})}$, respectively. In [34] the $E_{8}$-lattice was constructed via these algebras that give a group of units with smaller Tamagawa volume than the one corresponding to the Golden code, which is better to apply the method of algebraic reduction.