Jacobi forms over number fields from linear codes

1.   Introduction

There have been many developments in invariant theory and coding theory with various applications ^{[5,13,16,17,18,19,20,21,27]}. There are connections between invariants and weight enumerators for self-dual codes over various rings and fields ^{[1,2,3,4,7,8,9,10,11,12,14]}. Gleason ^[17] says the weight enumerators for binary Type Ⅱ codes are invariant by a certain finite group of order $192$. In ^[6], the authors prove that elliptic modular forms can be obtained from homogeneous weight enumerators of binary Type Ⅱ codes by using specific Jacobi theta series. For a finite ring $\Bbb Z_{2m}$, a Jacobi form of the full Jacobi group is suggested by complete weight enumerator of Type Ⅱ codes over the ring ^[10]. Recently, in ^[23], the authors determine some Jacobi forms from Type Ⅱ codes over $\Bbb Z_{2^m}$, and they also use shadow codes. Bannai et al. ^[2], construct Hermitian modular form by using Type Ⅱ codes over $\Bbb F_2+u \Bbb F_2$ with $u^2 = 0$. They suggest invariants concerning the symmetrized biweight enumerators, i.e., the genus is equal to $2$, of Type Ⅱ codes over the ring. In many works, a Jacobi form is studied over totally real fields (see ^[3,11,25]). For this reason, in this work, we figure out a Jacobi form which is not over a totally real field. We establish a connection between Jacobi forms and codes over $R = \Bbb F_4+u\Bbb F_4\; (u^2 = 0)$. Since the ring $R$ is described using the ring of integers in the quartic field $K = \Bbb Q(\sqrt 5, i)$, we should consider Jacobi forms over the field $K$, which is not totally real. The notion of Jacobi forms over arbitrary number fields was defined by Skogman in ^[24]. In this paper, we add examples of Jacobi forms over $K$ by modifying theta series of the lattices associated to codes over $R$.

We focus on the ring $\Bbb F_4+u\Bbb F_4$ with $u^2 = 0$ in this paper. This ring is a finite commutative local Frobenius ring of order $16$. A Frobenius ring is one of the most significant part in coding theory since we get the result that $|C||C^\perp| = |R^n|$ for a code $C$ of length $n$ over a Frobenius ring; this fact is connected to MacWilliams identity directly. Furthermore, for rings of order $16$, we already know about their generating characters from ^[13]. This means that we can adjust this information to our studies when finding MacWilliams identity. MacWilliams identity is one of significant results in coding theory that describes how the weight enumerator of a linear code and the weight enumerator of the dual code relate to each other. MacWilliams identity give various application in coding theory. For example, in ^[13], the author presents an upper bound for the minimum distances of divisible codes through their dual distances. In ^[21], the authors study Type Ⅱ codes over $\Bbb F_4+u\Bbb F_4$ and in particular the Gray map, the Lee weight, the construction of lattices and invariants. Actually, in ^[21], there are no results for invariants in higher genus.

In this paper, we suggest a Jacobi form from a linear code $C$ over $R: = \Bbb F_4+u\Bbb F_4$, where $u^2 = 0$ (Theorem 3.1). This Jacobi form is not over totally real field, and it is related to complete weight enumerator of the code $C$. We introduce MacWilliams identities for both, complete weight enumerator and symmetrized weight enumerator in higher genus $g\ge 1$ of a linear code over $R$. Finally, we give invariants via a self-dual code of even length over $R$ (Theorem 4.4).

2.   Preliminaries

Throughout this paper, we use the following notations.

Notation

${\rm{K}}$        $\mbox{an algebraic number field}$

$r_{1}$        $\mbox{the number of real embeddings of K}$

$r_{2}$        $\mbox{the number of conjugate pairs of complex embeddings of K}$

$\delta_K$        $\mbox{the different of}$ $K$

$\mathcal O_K$        $\mbox{the ring of integers of}$ $K$

$\mathcal Q$        $\mbox{the full ring of quaternions} \; \{x+y\kappa : x, y\in\mathbb{C}, \; \kappa^{2} = -1, \; a\kappa = \kappa\overline{a}, \; \forall a\in\mathbb{C}\}$

$\mathfrak{h}_{\mathcal{Q}}$        $\{x+y\kappa\in\mathcal{Q} : y\in\mathbb{R}_{ > 0}\}$

$\lVert u+v\kappa\rVert_{\mathbb{C}}$    $u+iv \; \mbox{ for} \; u+v\kappa\in\mathcal{Q}$

$\lVert u+v\kappa\rVert_{\overline{\mathbb{C}}}$    $\overline{u}+i\overline{v} \; \mbox{ for } \; u+v\kappa\in\mathcal{Q}$

$R$        $\mbox{a Frobenius ring } \; \Bbb F_4+u\Bbb F_4, \; \mbox{ where} \; u^2 = 0$

$\chi$        $\mbox{the generating character of $R$ }$

$wt_L$        $\mbox{the Lee weight in }\; \Bbb F_4$

$\hat wt_L$        $\mbox{the Lee weight in }\; R$

$N$        $\mbox{the cardinality of } \; R$

${}^tM$        $\mbox{the transpose of a matrix }$ $M$

$I_m$        $\mbox{an}$ $m\times m$ identity matrix

$diag(v_1, \ldots, v_m)$  $\mbox{an}$ $m\times m$ diagonal matrix, where an $(i, i)$ -th component is $v_i$ $(1 \le i \le m)$

$GL(m, F)$     $\mbox{the general linear group of degree}$ $m$ over $F$

$A \otimes B$     $\mbox{the Kronecker product of two matrices}$ $A$ and $B$

$e[z]$      $e^{2\pi iz}$

Let $K$ be an algebraic number field, and $\mathcal Q$ be the full ring of quaternions $\{x+y\kappa : x, y\in\mathbb{C}, \; \kappa^{2} = -1, \; a\kappa = \kappa\overline{a}, \; \forall a\in\mathbb{C}\}$; the set of all the elements of $\mathcal Q$ is equal to $\{a+bi+cj+d\kappa : a, b, c, d\in\mathbb{R}, \; i^{2} = j^{2} = \kappa^{2} = -1, \; ij = \kappa, \; j\kappa = i, \; ji = -\kappa\}$ (see [22,p. 220]). For an element $\alpha \in K$, we denote the real conjugates of $\alpha$ by $\alpha^{(1)}, \ldots, \alpha^{(r_{1})}$, and the complex conjugates of $\alpha$ by $\alpha^{(r_{1}+1)}, \ldots, \alpha^{(r_{1}+2r_{2})}$, where $\alpha^{(j+r_{2})} = \overline{\alpha^{(j)}}$ for $r_{1}+1\leq j\leq r_{1}+r_{2}$.

We set $\Gamma^{J}(K) = \Gamma\ltimes\mathcal{O}_{K}^{2}$, where $\Gamma = SL(2, \mathcal{O}_{K})$; the group $\Gamma^{J}(K)$ is called the Jacobi group of $K$. The group law of $\Gamma^{J}(K)$ is given by

Let $\mathfrak h$ be the upper half plane, and $\mathfrak h_{\mathcal Q} = \{x+y\kappa \in \mathcal Q\mid x \in \Bbb C, y \in \Bbb R^+\}$ be the quaternionic upper half plane. Let $\mathcal H$ be the space $\mathfrak{h}^{r_{1}}\times\mathfrak{h}_{\mathcal{Q}}^{r_{2}}\times\mathbb{C}^{r_{1}}\times\mathcal{Q}^{r_{2}}$; an element of $\mathcal H$ is written as $(\vec{\tau}, \vec{z}): = (\tau_{1}, \ldots, \tau_{r_{1}+r_{2}}, z_{1}, \ldots, z_{r_{1}+r_{2}})$.

The group $\Gamma^{J}(K)$ acts on $\mathcal H$, and the action is given as follows: For elements $\left(\begin{smallmatrix}\alpha & \beta \\ \gamma & \delta\end{smallmatrix}\right)\in SL(2, \mathcal{O}_{K}) \;\mbox{ and }\; [\lambda, \mu]\in\mathcal{O}_{K}^{2}$,

and

$[\lambda, \mu]\circ(\vec{\tau}, \vec{z}) = (\vec{\tau}, z_{1}+\tau_{1}\lambda^{(1)}+\mu^{(1)}, \ldots, z_{r_{1}+r_{2}}+\tau_{r_{1}+r_{2}}\lambda^{(r_{1}+r_{2})}+\mu^{(r_{1}+r_{2})}).$

For $\gamma, \delta, \lambda\in K$ and $(\vec{\tau}, \vec{z})\in\mathcal{H}$, set

where

$\bullet$ $z_{j} = u_{j}+v_{j}\kappa$ for $j = r_{1}+1, \ldots, r_{1}+r_{2}$.

$\bullet$ $\vec{m}^{(i)}$: vectors in $\mathbb{C}^{r_{1}+2r_{2}}$ $(1\le i \le r_1+2r_2)$ such that $\vec{m}^{(j+r_{2})} = \overline{\vec{m}^{(j)}}$ for $j = r_{1}+1, \ldots, r_{1}+r_{2}$.

$\bullet$ $M = (\vec{m}^{(1)}\quad\cdots\quad \vec{m}^{(r_{1}+2r_{2})})$: an $(r_1+2r_2)\times (r_1+2r_2)$-matrix over $\mathbb{C}$.

$\bullet$ $\widetilde{m}^{(j)} = {^{t}\vec{m}^{(j)}}\vec{m}^{(j)}$.

Next, we need to present a multiplier system for $SL(2, \mathcal O_K)$. For doing this, we set the following: For $A = \left(\begin{smallmatrix}\alpha & \beta \\ \gamma & \delta\end{smallmatrix}\right)\in SL(2, \mathcal{O}_{K})$ and $\vec{\tau} = (\tau_{1}, \ldots, \tau_{r_{1}+r_{2}})\in\mathfrak{h}^{r_{1}}\times\mathfrak{h}_{\mathcal{Q}}^{r_{2}}$,

and

where

and $\tau_{j} = x_{j}+y_{j}\kappa$ for $j = r_{1}+1, \ldots, r_{1}+r_{2}$. A multiplier system for $SL(2, \mathcal{O}_{K})$ is a function $\chi:SL(2, \mathcal{O}_{K})\rightarrow \mathbb{C}$ such that

for all $A, B\in SL(2, \mathcal{O}_{K})$ and $\vec{\tau}\in\mathfrak{h}^{r_{1}}\times\mathfrak{h}_{\mathcal{Q}}^{r_{2}}$.

The next definition is about a Jacobi form of weight $k$ and index $m$ with an index vector for a number field.

Definition 2.1. ^[22,24] Let $K$ be an algebraic number field, $k\in\frac{1}{2}\mathbb{Z}$, and $m\in\mathcal{O}_{K}$. Let $\chi$ be a multiplier system for $SL(2, \mathcal{O}_{K})$, and $\vec{m}$ be a vector in $\mathbb{C}^{n}$ such that ${}^{t}\vec{m}^{(j)}\vec{m}^{(j)} = m^{(j)}$ for $j = 1, \ldots, n$. A Jacobi form of weight $k$, index $m$, index vector $\vec{m}$ and the multiplier system $\chi$ for the number field $K$ is a function $\Phi:\mathcal{H}\rightarrow \mathbb{C}$ satisfying

and

for all $\left(\begin{smallmatrix}\alpha & \beta \\ \gamma & \delta\end{smallmatrix}\right)\in SL(2, \mathcal{O}_{K}), [\lambda, \mu]\in\mathcal{O}_{K}^{2}$, $\vec{\tau}\in\mathfrak{h}^{r_{1}}\times\mathfrak{h}_{\mathcal{Q}}^{r_{2}}$ and $\vec{z}\in\mathbb{C}^{r_{1}}\times\mathcal{Q}^{r_{2}}$.

A Frobenius ring is a finite commutative ring $R$ satisfying that the $R$-module $R$ is injective. We consider the number field $\Bbb Q(\sqrt 5, i)$ which has the ring of integers $\Bbb Z[\frac{1+\sqrt 5}{2}, i]$. Let $u$ be the residue class of $x+1$ in the quotient ring

i.e., $u^2 = 0$ in $R$. On the other side, the ring $R$ is isomorphic to $\Bbb F_2[u, v]/\langle v^2+v+1, u^2 \rangle$. This ring $R$ is a finite commutative local Frobenius ring of order $16$ (cf. ^[13]).

A code $C$ of length $n$ over $R$ is an $R$-submodule of $R^n$, and an element $c = (c_1, \ldots, c_n)$ in $C$ is called a codeword in $C$. The dual code $C^\perp$ of $C$ is $\{c \in R^n : c\cdot \hat c = 0 \;\mbox{ for all }\; \hat c\in C\}$ with respect to the Euclidean inner product. If $C\subseteq C^{\perp}$ (resp. $C = C^\perp$), then $C$ is a self-orthogonal (resp. self-dual) code.

The Lee weight $wt_L(a)$ of an element $a$ in $\Bbb F_4 = \{0, 1, \omega, \bar \omega\}$ is given as follows:

For a vector $w = (w_{1}, \ldots, w_{n})\in\mathbb{F}_{4}^{n}$, the Lee weight $wt_{L}(w)$ of $w$ is $\sum_{i = 1}^{n}wt_{L}(w_{i})$.

Definition 2.2 gives the Lee weight of an element in $R = \Bbb F_4+u\Bbb F_4$.

Definition 2.2. For an element $\alpha = a+bu$ in $R$ $(a, b \in \Bbb F_4)$, the Lee weight $\hat wt_L(\alpha)$ of $\alpha$ in $R$ is

where $wt_L$ is the Lee weight in $\Bbb F_4$. For a vector $v = (v_1, \ldots, v_n)$ in $R^n$, the Lee weight $\hat wt_L(v)$ of $v$ is $\sum_{i = 1}^n \hat wt_L(v_i)$.

In the following Table 1, we suggest the Lee weights of all the elements in $R$.

Proposition 2.3. ^[21]Let us define a map $\phi$ from $R^n$ to $\Bbb F_4^{2n}$ as follows:

where $a_i, b_i\in \Bbb F_4$ $(1\le i \le n)$. The map $\phi$ preserves the Lee weight from $R^n$ to $\Bbb F_4^{2n}$, it means that, the map $\phi$ is a Gray map.

For a self-dual code $C$ over $R$, $C$ is a Type II code if the Lee weight of every codeword is divisible by $4$. If not, the code $C$ is called a Type I code.

Lemma 2.4. ^[13,21] Let $C$ be a linear code of length $n$ over $R$.

(ⅰ) We have that $|C||C^{\perp}| = |R^n|$.

(ⅱ) There is a Type II code of length $n$ over $R$ if and only if $n$ is even.

From now on, we suggest some weight enumerators for a code $C$ over $R$: A complete weight enumerator and a symmetrized weight enumerator of $C$. Let $C$ be a linear code of length $n$ over $R$, and $v = (v_1, \ldots, v_n)$ be a codeword in $C$. Let $n_{a}(v)$ be the number of coordinates $v_i$ such that $v_i = a$, where $a\in R$ and $1\le i \le n$. We define the complete weight enumerator $cwe_C$ of $C$ as

where $x_j$ is an indeterminate and $a_j \in R$ for $1 \le j \le N$.

Let $U$ be a fixed subgroup of the unit group of $R$. We note that the group $U$ acts on $R$ by the multiplication. We denote $a\approx b$ if $a = ub$ for some $u\in U$; it is an equivalence relation in $R$. Moreover, $S = \{s_{1}, \ldots, s_{|S|}\}$ is a set of representatives of the distinct orbits of $U$. The symmetrized weight enumerator $swe_C$ of $C$ in $R^n$ is

3.   A Jacobi form from a linear code over $R$

We figure out a Jacobi form for a number field via a linear code over $R$. Before we do this, we first give a theta function as follows.

Let $K$ be a number field, and $\Lambda$ be a lattice in $K^n$, i.e., $\Lambda$ is a free $\mathcal O_K$-module of rank $n$. Now, for each $Y$ in $\Lambda$, we define a theta function $\Theta_{\Lambda, Y}:\mathcal{H}\rightarrow \mathbb{C}$ as

In this section, let $K = \Bbb Q(\sqrt 5, i)$, and then $\mathcal O_K = \Bbb Z[(1+\sqrt 5)/2, i]$; so $r_{1} = 0$ and $r_{2} = 2$. Here, we can check that $R\cong \Bbb F_2[u, v]/\langle u^2+u+1, v^2 \rangle \cong \mathcal O_K/2 \mathcal O_K$; the first equivalence is introduced in Section 2, and the second equivalence is from the ring isomorphism $\mathbb{F}_{2}[u, v]/\langle u^{2}+u+1, v^{2}\rangle \rightarrow \mathcal{O}_{K}/2\mathcal{O}_{K}$, where $u+\langle u^{2}+u+1, v^{2}\rangle \mapsto (1+\sqrt{5})/2+2\mathcal{O}_{K}$, and $v+\langle u^{2}+u+1, v^{2}\rangle \mapsto 1+i+2\mathcal{O}_{K}$.

Let $h:\mathcal O_K \rightarrow R$ be the reduction map by modulo $2$, and $\tilde h:\mathcal O_K^n \rightarrow R^n$ be defined as $(x_1, \ldots, x_n)\rightarrow (h(x_1), \ldots, h(x_n))$. For a linear code $C$ over $R$, the lattice $\Lambda(C)$ is $\frac{1}{\sqrt 2}\tilde h^{-1}(C)$.

In the following theorem, we obtain the relation between the theta function and complete weight enumerator for a linear code under the previous settings.

Theorem 3.1. Let $C$ be a linear code of length $n$ over $R$. Then we get

where

Proof. Let $v$ be a codeword $(v_{1}, \ldots, v_{n})$ in $C$, and $\widetilde{v}$ be a vector $(\widetilde{v}_{1}, \ldots, \widetilde{v}_{n})\in\widetilde{h}^{-1}(v)$. We easily check $\widetilde{h}$ is a ring homomorphism, thus $\widetilde{h}^{-1}(v) = \widetilde{h}^{-1}({\bf 0})+\widetilde{v}$. Then we obtain that

we only consider $\tau_1$ and $\tau_2$ because $r_1 = 0$ and $r_2 = 2$ as we mentioned before. And the second equation is from that

for all $u_{1}+v_{1}\kappa, u_{2}+v_{2}\kappa\in\mathcal{Q}$. Therefore, we have that

the fourth equation is from (3.1), where $n_{\mu}(v): = |\{j: v_{j} = \mu\}|$ for each $\mu\in\mathcal{O}_{K}/2\mathcal{O}_{K}$. The result is proved.

We fix an $\mathcal O_K$-basis $\{v_1, \ldots, v_n\}$ for a lattice $\Lambda$, and set $L$ to be an $n \times n$-matrix $(v_1, \ldots, v_n)$. The matrix $Q: = {}^{t}LL$ is a symmetric matrix, and $Q^{(j)}: = {}^{t}L_{j}L_{j}$, where $L_{j} = (v_{1}^{(j)}, \ldots, v_{n}^{(j)})$ for $1\leq j\leq n$. Morevoer, $Q^{(j)}$ is positive definite for $1\leq j\leq r_{1}$.

In ^[24], for $b\in \mathcal O_K^n$, the function $\theta_{Q, b}(\vec{\tau}, \vec{z})$ is defined as

The next theorem says that we can obtain a Jacobi form by using a linear code of length $n$ over $R$.

Theorem 3.2. We use the same notations as above.Let $C$ be a linear code of length $n$ over $R$, and $b$ an element of $\mathbb{Z}[\alpha, i]^{n}$ with $\alpha = (1+\sqrt 5)/2$.Then we get a Jacobi form of weight $\frac{n}{2}$, index $4n$ and index vector $(2, \ldots, 2)$ as follows:

where $Lb = (p_{1}+q_{1}\alpha, \ldots, p_{n}+q_{n}\alpha)\in\mathbb{Z}[\alpha]^{n}$ $($in particular, $\omega_{1, \mu}(\vec{\tau}, \vec{z})$ is introduced in Theorem 3.1$)$.

Proof. First, we claim that $Lb$ is in $\Bbb Z[\alpha]^n$. We set an $n$-tuple $Lb = (t_1, \ldots, t_n)$ satisfying ${}^{t}(Lb)Lb = t_{1}^{2}+\cdots+t_{n}^{2} = 4n$, and $\overline{{}^{t}(Lb)}Lb = |t_{1}|^{2}+\cdots+|t_{n}|^{2} = 4n$; the $\ell$-th component $t_{\ell}$ can be written as $p_{\ell}+q_{\ell}\alpha+r_{\ell}i+s_{\ell}\alpha i$, where $p_\ell, q_\ell, r_\ell, s_\ell\in \Bbb Z$ $(1 \le \ell \le n)$. It means that

Let $\sigma$ be the complex embedding of $K$ such that $\sigma:\sqrt 5 \mapsto -\sqrt 5$ and $i \mapsto i$. Applying $\sigma$ to (3.3), we have $(p_{1}+q_{1}\sigma(\alpha))^{2}+(r_{1}+s_{1}\sigma(\alpha))^{2}+\cdots+(p_{n}+q_{n}\sigma(\alpha))^{2}+(r_{n}+s_{n}\sigma(\alpha))^{2} = 4n$. It follows that

We get that combining (3.3) and (3.4), we see that

the first equality is obtained by expanding (3.2), and the second equality is from combining (3.3) and (3.4). Thus $r_{\ell} = s_{\ell} = 0$ since $r_{\ell}, s_{\ell} \in \Bbb Z$ for $\ell = 1, \ldots, n$. Hence we proved the first claim.

By using [24,p. 41], we can say that

is a Jacobi form of weight $\frac{n}{2}$, index $4n$ and index vector $(2, \ldots, 2)$.

Next, when $Lb = (2, \ldots, 2)$, $\theta_{Q, L^{-1}(2, \ldots, 2)}(\vec{\tau}, \vec{z})$ is obtained from the complete weight enumerator of the code $C$; in this case, the condition $p_{1}^{2}+q_{1}^{2}+\cdots+p_{n}^{2}+q_{n}^{2} = 4n$ is satisfied. In detail, we have that

where $b = L^{-1}(2, \ldots, 2)$; the first equation can be checked easily, and the second equation follows by the below reasoning

(here $Y = (2, \ldots, 2)$). This implies the second claim, thus this is the result here.

We close this section with an examlple.

Example 3.3. Let $C$ be the linear code over $R$ of length $2$ generated by the following $1\times 2$ matrix

(here $\alpha = (1+\sqrt{5})/2$).By using Table 1, we can check that the code $C$ is a Type II code over $R$. Meanwhile, $\tilde{h}^{-1}(C)$ is generated by $(\alpha, 1+\alpha+i), (2, 0)$ and $(0, 2)$ over $\mathcal{O}_{K}$. And then $\{(\alpha, 1+\alpha+i), (0, 2)\}$ is a basis for $\tilde{h}^{-1}(C)$; since $(\alpha, 1+\alpha+i)$ and $(0, 2)$ are linearly independent over $\mathcal{O}_{K}$, and $(2, 0) = (-2+2\alpha)(\alpha, 1+\alpha+i)+(-\alpha+i-\alpha i)(0, 2)$.Thus the matrix $L$ is obtained as

and the matrix $Q = {}^tLL$.We note that

As a result, by Theorem 3.2, we have a Jacobi form of weight 1, index 8 and index vector (2, 2) as follows:

where $b_1 = \left(\begin{smallmatrix}2\alpha-2 & 0 \\ -\alpha+i-\alpha i & 1\end{smallmatrix}\right)\left(\begin{smallmatrix}\pm 1\pm \alpha \\ 0\end{smallmatrix}\right)$, $b_2 = \left(\begin{smallmatrix}2\alpha-2 & 0 \\ -\alpha+i-\alpha i & 1\end{smallmatrix}\right)\left(\begin{smallmatrix}- 1 \\ - 1\end{smallmatrix}\right)$, $b_{3} = \pm\left(\begin{smallmatrix}2\alpha-2 & 0 \\ -\alpha+i-\alpha i & 1\end{smallmatrix}\right)\left(\begin{smallmatrix}1 \\ -1\end{smallmatrix}\right)$, $b_4 = \left(\begin{smallmatrix}2\alpha-2 & 0 \\ -\alpha+i-\alpha i & 1\end{smallmatrix}\right)\left(\begin{smallmatrix}\pm 1 \\ \pm\alpha\end{smallmatrix}\right)$, $b_5 = \left(\begin{smallmatrix}2\alpha-2 & 0 \\ -\alpha+i-\alpha i & 1\end{smallmatrix}\right)\left(\begin{smallmatrix}\pm \alpha \\ \pm 1\end{smallmatrix}\right)$, $b_6 = \left(\begin{smallmatrix}2\alpha-2 & 0 \\ -\alpha+i-\alpha i & 1\end{smallmatrix}\right)\left(\begin{smallmatrix}\pm \alpha \\ \pm\alpha\end{smallmatrix}\right)$, $b_7 = \left(\begin{smallmatrix}2\alpha-2 & 0 \\ -\alpha+i-\alpha i & 1\end{smallmatrix}\right)\left(\begin{smallmatrix}0 \\ \pm 1\pm\alpha\end{smallmatrix}\right)$

$(\mathit{\mbox{in detail }},\; \theta_{Q, \frac{1}{2}\left(\begin{smallmatrix}2\alpha-2 & 0 \\ -\alpha+i-\alpha i & 1\end{smallmatrix}\right)\left(\begin{smallmatrix}2 \\ 2\end{smallmatrix}\right)}(\vec{\tau}, \vec{z})$ $\mathit{\mbox{ term is equal to }} \;cwe_{C}\left(\omega_{1, \mu}(4\vec{\tau}, 4\vec{z})\mid\mu\in \mathcal{O}_{K}/2\mathcal{O}_{K}\right))$.

4.   Invariants via a self-dual code over $R$

In this section, we study invariants by using self-dual codes over $R$. First, we give definition for some weight enumerators in higher genus $g \ge 1$.

Definition 4.1. Let $C$ be a code of length $n$ over $R$.

(ⅰ) The complete weight enumerator in genus $g$ is

where $n_a(c_1, \ldots, c_g)$ is the number of $i$ such that $a = ^t(c_{1i}, \ldots, c_{gi})$.

(ⅱ) The symmetrized weight enumerator in genus $g$ is

where $S$ is the set introduced in Section 2, and $n_{[a]}(c_1, \ldots, c_g)$ is the number of $i$ satisfying $[a] = [^t(c_{1i, }, \ldots, c_{gi})]$.

The following lemma says the MacWilliams identity for linear codes over $R$. For this, we give that an $n\times n$ matrix $M = (m_{ij})$ over $\Bbb C$ acts on $\Bbb C[x_1, \ldots, x_n]$ as follows:

where $h(x_1, \ldots, x_n)\in \Bbb C[x_1, \ldots, x_n]$.

As we mentioned before, the ring $R$ is a finite commutative local Frobenius ring of order $16$. We note that the generating character $\chi$ of $R\cong \Bbb F_2[u, v]/\langle v^2+v+1, u^2 \rangle$ is

where $a, b, c, d \in \Bbb F_2$.

Lemma 4.2 is about the MacWilliams identity for a linear code over $R$ with weight enumerators in genus $g$. We recall that $S$ is a set of representatives of the distinct orbits of a fixed unit subgroup $U$ in $R$.

Lemma 4.2. Let $C$ be a linear code of length $n$ over $R$, and $\chi$ be the generating character of $R$.Let $a_i$ (resp. $b_j$) be an element of $R$ (resp. $S$) with any ordering for $1\le i \le N$ (resp. $1 \le j \le |S|$).

(ⅰ) Let $T_1$ be a $N \times N$-matrix such that$(i, j)$-th component of $T_1$ is $\chi(a_ia_j)$ with $1\le i, j \le N$.For complete weight enumerator, we have

(ⅱ) Let $T_2$ be a $|S|\times |S|$-matrix such that $(i, j)$-th component of $T_2$ is $\sum_{a'\in\tau}T_{a, a'}$, where $[a] = b_{i}$ and $\tau = \{a' \in R : a' \approx b_j\}$.For symmetrized weight enumerator, we get

Proof. We give an equivalence relation as $a\approx b$ if $a = bu$, where $u \in U$, and $U$ is a unit subgroup in $R$. By the equivalence relation, the generating character for the ring $R$, and [28,Theorem 8.4], we obtain the result.

For an arbitrary unit subgroup for the ring $R$, we can suggest a symmetrized weight enumerator for a code over $R$. In the following remark, we show a symmterized weight enumerator and MacWilliams identity for a certain unit group of $R$.

Remark 4.3. Let $U$ be a subgroup of the unit group of $R$.There are eight subgroups of the unit group of $R$.For example, let $U = \langle \omega \rangle$.Then we get the set $S = \{0, 1, u, 1+u, 1+\omega u, u+ \omega\}$ which is the set of representatives of the distinct orbits of $U$. We note that the equivalence classes are $[0] = \{0\}$, $[1]= \{1, \omega, 1+\omega\}$, $[u] = \{u, \omega u, u+\omega u\}$, $[1+u] = \{1+u, \omega+\omega u, 1+\omega+u+\omega u\}$, $[1+\omega u] = \{1+\omega u, \omega+u+\omega u, 1+\omega+u\}$ and $[u+\omega] = \{u+\omega, 1+\omega+\omega u, 1+u+\omega u\}$.The ordering in $S$ is given as $0, 1, u, 1+u, 1+\omega u, u+ \omega$.In this case, we obtain the following matrix $T_2$ as follows:

For the genus $g = 1$, the MacWilliams identity for a linear code $C$ of length $n$ over $R$ is

where

$x_{[i]}$ and $y_j$ are indeterminates.

Finally, we will show invariants by the complete weight enumerator and symmetrized weight enumerator for a self-dual code over $R$. First, we define a subgroup $G_g$ of $GL(N^g, \Bbb C)$ as

where $M_g = \left(\frac{-1}{\sqrt N}\right)^g { \otimes ^g}\; T_1$ and $M_J = diag((-1)^{{}^taJa} \;\mbox{ with }\; a\in R^g)$. Similarly, let us a subgroup $\hat G_g$ of $GL(|S|^g, \Bbb C)$ as

where $\hat M_g = \left(\frac{-1}{\sqrt N}\right)^g { \otimes ^g}\; T_2$ and $\hat M_J = diag((-1)^{{}^taJa}\; \mbox{ with }\; a\in S^g)$.

Especially, for a self-dual code of even length over $R$, we obtain the result of invariant.

Theorem 4.4. Let $C$ be a self-dual code of even length $n$ over $R$.The complete weight enumerator $cwe_{C, g}$ of $C$ in genus $g$ is invariant under the action of the group $G_g$.Similarly, the symmetrized weight enumerator $swe_{C, g}$ of $C$ in genus $g$ is invariant under the action of the group $\hat G_g$.

Proof. We prove the statement for the complete weight enumerator for the code $C$. We can easily check that $-I_{n^g}$ and $M_g$ invariant for $cwe_C(x_i)$; first, $-I_{N^g}\cdot cwe_C(x_i) = cwe_C(x_i)$ since $cwe_C(x_i) = \sum_{c\in C}x_i^{n_{x_i}(c)} = \sum_{c\in C}((-1)x_i)^{n_{x_i}(c)}$ by $2\mid n$. Second, $M_g\cdot cwe_C(x_i) = cwe_C(x_i)$ is from the MacWilliams identity Lemma 4.2. Now, we claim that $M_J$ is in $G_g$. We note that

In detail,

the right hand side value is divisible by $2$ since the code $C$ is self-dual over $R$.

Thus, we have

by the above equations. We proved the statement. For symmetrized weight enumerator for $C$, the proof is similar with the previous case.

If a linear code $C$ is Type Ⅱ of length $n$ over $R$, then the length $n$ is automatically even. Thus, we obtain the following corollary.

Corollary 4.5. For a Type II code $C$ of length $n$ over $R$, $cwe_{C, g}$ and $swe_{C, g}$ are invariant under the action of the group $G_g$ and $\hat G_g$, respectively.

Proof. By Lemma 2.4 (ⅱ), for a Type Ⅱ code of length $n$ over $R$, $n$ is even. So by using Theorem 4.4, the result follows.

5.   Conclusions

We suggested a Jacobi form from a linear code $C$ over $R: = \Bbb F_4+u\Bbb F_4$, where $u^2 = 0$. This Jacobi form is not over totally real field, and it is related to complete weight enumerator of the code $C$. We introduced MacWilliams identities for both, complete weight enumerator and symmetrized weight enumerator in an arbitrary genus $g\ge 1$ of a linear code over $R$. Moreover, we presented invariants via a self-dual code of even length over $R$. In the future work, we can consider another finite ring for linear codes and their various weight enumerators. From these results, we can also establish a new Jacobi form.

Acknowledgments

Boran Kim is a corresponding author and supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2021R1C1C2012517 and NRF-2019R1I1A1A01060467). Chang Heon Kim is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2018R1D1A1B07045618 and 2016R1A5A1008055). Soonhak Kwon is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2016R1A5A1008055). Yeong-Wook Kwon is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2018R1D1A1B07045618 and 2020R1A4A1016649).

Conflict of interest

The authors declare no conflict of interest.