# Some restrictions on the weight enumerators of near-extremal ternary self-
dual codes and quaternary Hermitian self-dual codes

Makoto Araya and Masaaki Harada Department of Computer Science, Shizuoka
University, Hamamatsu 432–8011, Japan. email<EMAIL_ADDRESS>Research Center for Pure and Applied Mathematics, Graduate School of
Information Sciences, Tohoku University, Sendai 980–8579, Japan. email:
<EMAIL_ADDRESS>
###### Abstract

We give restrictions on the weight enumerators of ternary near-extremal self-
dual codes of length divisible by $12$ and quaternary near-extremal Hermitian
self-dual codes of length divisible by $6$. We consider the weight enumerators
for which there is a ternary near-extremal self-dual code of length $12m$ for
$m=3,4,5,6$. Also we consider the weight enumerators for which there is a
quaternary near-extremal Hermitian self-dual code of length $6m$ for
$m=4,5,6$.

##  1 Introduction

Self-dual codes are one of the most interesting classes of codes. This
interest is justified by many combinatorial objects and algebraic objects
related to self-dual codes (see e.g., [6] and [25]).

Let $\mathbb{F}_{q}$ denote the finite field of order $q$, where $q$ is a
prime power. A code $C$ over $\mathbb{F}_{q}$ of length $n$ is said to be
_self-dual_ if $C=C^{\perp}$, where the dual code $C^{\perp}$ of $C$ is
defined as $C^{\perp}=\\{x\in\mathbb{F}_{q}^{n}\mid\langle
x,y\rangle_{E}=0\text{ for all }y\in C\\}$ under the standard inner product
$\langle x,y\rangle_{E}$. A code $C$ over $\mathbb{F}_{q^{2}}$ of length $n$
is said to be _Hermitian self-dual_ if $C=C^{\perp_{H}}$, where the Hermitian
dual code $C^{\perp_{H}}$ of $C$ is defined as
$C^{\perp_{H}}=\\{x\in\mathbb{F}_{q^{2}}^{n}\mid\langle x,y\rangle_{H}=0\text{
for all }y\in C\\}$ under the Hermitian inner product $\langle
x,y\rangle_{H}$. By the Gleason–Pierce theorem, there are nontrivial divisible
self-dual codes over $\mathbb{F}_{q}$ for $q=2$ and $3$ only, and there are
nontrivial divisible Hermitian self-dual codes over $\mathbb{F}_{q^{2}}$ for
$q=2$ only. This is one of the reasons why much work has been done concerning
these self-dual codes. Codes over $\mathbb{F}_{3}$ and $\mathbb{F}_{4}$ are
called _ternary_ and _quaternary_ , respectively. In this paper, we
concentrate on ternary self-dual codes and quaternary Hermitian self-dual
codes.

The minimum weight $d$ of a ternary (resp. quaternary Hermitian) self-dual
code of length $n$ is bounded by $d\leq 3\lfloor n/12\rfloor+3$ [21] (resp.
$d\leq 2\lfloor n/6\rfloor+2$ [19]). A ternary (resp. quaternary Hermitian)
self-dual code of length $n$ and minimum weight $3\lfloor n/12\rfloor+3$
(resp. $2\lfloor n/6\rfloor+2$) is called _extremal_. By the Assmus–Mattson
theorem [2], the supports of codewords of weight $i$ in a ternary extremal
(resp. quaternary extremal Hermitian) self-dual code of length $12m$ (resp.
$6m$) form a $5$-design for $i\leq 6m+3$ [21] (resp. $i\leq
2\lfloor(3m+2)/2\rfloor$ [19, Theorem 18]).

A ternary (resp. quaternary Hermitian) self-dual code of length $n$ and
minimum weight $3\lfloor n/12\rfloor$ (resp. $2\lfloor n/6\rfloor$) is called
_near-extremal_. Recently, Miezaki, Munemasa and Nakasora [22] gave a new
Assmus–Mattson type theorem for ternary near-extremal self-dual codes of
length $12m$ and quaternary near-extremal Hermitian self-dual codes of length
$6m$. The supports of codewords of weight $i$ in a ternary near-extremal
(resp. quaternary near-extremal Hermitian) self-dual code of length $12m$
(resp. $6m$) form a $1$-design for $i\leq 6m-3$ (resp. $i\leq 3m-1$). This
motivates our study of ternary near-extremal self-dual codes of length $12m$
and quaternary near-extremal Hermitian self-dual codes of length $6m$. In this
paper, we give restrictions on the weight enumerators of these codes.
Furthermore, we give a divisibility property of the coefficients of the weight
enumerators of these codes. This divisibility property is the main result of
this paper.

This paper is organized as follows. In Section 2, we give some definitions,
notations and basic results. In Section 3, we show that if there is a ternary
near-extremal self-dual code $C$ of length $12m$ then $A_{3i}(C)\equiv
0\pmod{8}$ for $i=m,m+1,\ldots,4m$, where $A_{i}(C)$ denotes the number of
codewords of weight $i$ in $C$ (Theorem 3.2). Furthermore, we show that if
there is a quaternary near-extremal Hermitian self-dual code $C$ of length
$6m$ then ${A_{2i}(C)\equiv 0\pmod{9}}$ for ${i=m,m+1,\ldots,3m}$ (Theorem
3.5). Theorems 3.2 and 3.5 are the main results of this paper. As a
consequence, it is shown that if there is a ternary extremal (resp. quaternary
extremal Hermitian) self-dual code $C$ of length $12m$ (resp. $6m$) then
$A_{3i}(C)\equiv 0\pmod{8}$ (resp. $A_{2i}(C)\equiv 0\pmod{9}$) for
$i=m+1,m+2,\ldots,4m$ (resp. $3m$). In Section 4, we consider the weight
enumerators for which there is a ternary near-extremal self-dual code of
length $12m$ for $m=3,4,5,6$. In Section 5, we consider the weight enumerators
for which there is a quaternary near-extremal Hermitian self-dual code of
length $6m$ for $m=4,5,6$.

All computer calculations in this paper were done using programs in Magma [3]
and Mathematica [29].

##  2 Preliminaries

In this section, we prepare some definitions, notations and basic results.

###  2.1 Self-dual codes

From now on, we suppose that $q=3$ or $4$. We denote the finite fields of
orders $3$ and $4$ by $\mathbb{F}_{3}=\\{0,1,2\\}$ and
$\mathbb{F}_{4}=\\{0,1,\omega,\omega^{2}\\}$, respectively, where
$\omega^{2}=\omega+1$. An $[n,k]$ _code_ $C$ over $\mathbb{F}_{q}$ is a
$k$-dimensional vector subspace of $\mathbb{F}_{q}^{n}$. A code $C$ over
$\mathbb{F}_{3}$ (resp. $\mathbb{F}_{4}$) is called _ternary_ (resp.
_quaternary_). The parameter $n$ is called the _length_ of $C$. A generator
matrix of an $[n,k]$ code $C$ over $\mathbb{F}_{q}$ is a $k\times n$ matrix
such that the rows of the matrix generate $C$. Two codes $C$ and $C^{\prime}$
over $\mathbb{F}_{q}$ are _equivalent_ if there is a monomial matrix $P$ over
$\mathbb{F}_{q}$ with $C^{\prime}=C\cdot P$, where $C\cdot P=\\{xP\mid x\in
C\\}$.

The _support_ $\operatorname{supp}(x)$ of a vector
$x=(x_{1},x_{2},\ldots,x_{n})\in\mathbb{F}_{q}^{n}$ is a subset $\\{i\mid
x_{i}\neq 0\\}$ of $\\{1,2,\ldots,n\\}$. The _weight_ $\operatorname{wt}(x)$
of a vector $x$ is defined as $|\operatorname{supp}(x)|$. A vector of a code
$C$ is called a _codeword_ of $C$. The minimum non-zero weight of all
codewords in $C$ is called the _minimum weight_ of $C$. An $[n,k,d]$ code over
$\mathbb{F}_{q}$ is an $[n,k]$ code over $\mathbb{F}_{q}$ with minimum weight
$d$. The _weight enumerator_ of $C$ is given by $\sum_{c\in
C}y^{\operatorname{wt}(c)}$.

The _dual_ code $C^{\perp}$ of a ternary code $C$ of length $n$ is defined as

$C^{\perp}=\\{x\in\mathbb{F}_{3}^{n}\mid\langle x,y\rangle_{E}=0\text{ for all
}y\in C\\},$

where $\langle x,y\rangle_{E}=\sum_{i=1}^{n}x_{i}{y_{i}}$ for
$x=(x_{1},x_{2},\ldots,x_{n}),y=(y_{1},y_{2},\ldots,y_{n})\in\mathbb{F}_{3}^{n}$.
A ternary code $C$ is said to be _self-dual_ if $C=C^{\perp}$. Every codeword
of a ternary self-dual code has weight a multiple of $3$. A ternary self-dual
code of length $n$ exists if and only if $n>0$ with $n\equiv 0\pmod{4}$ [21].
All ternary self-dual codes were classified in [5], [14], [20] and [24] for
lengths up to $24$.

The _Hermitian dual_ code $C^{\perp_{H}}$ of a quaternary code $C$ of length
$n$ is defined as

$C^{\perp_{H}}=\\{x\in\mathbb{F}_{4}^{n}\mid\langle x,y\rangle_{H}=0\text{ for
all }y\in C\\},$

where $\langle x,y\rangle_{H}=\sum_{i=1}^{n}x_{i}y_{i}^{2}$ for
$x=(x_{1},x_{2},\ldots,x_{n}),y=(y_{1},y_{2},\ldots,y_{n})\in\mathbb{F}_{4}^{n}$.
A quaternary code $C$ is said to be _Hermitian self-dual_ if
$C=C^{\perp_{H}}$. All codewords of a quaternary Hermitian self-dual code have
even weights. A quaternary Hermitian self-dual code of length $n$ exists if
and only if $n>0$ with $n\equiv 0\pmod{2}$ [19]. All quaternary Hermitian
self-dual codes were classified in [5], [13], [15] and [19] for lengths up to
$20$.

###  2.2 Near-extremal self-dual codes

The minimum weight $d$ of a ternary (resp. quaternary Hermitian) self-dual
code of length $n$ is bounded by $d\leq 3\lfloor n/12\rfloor+3$ [21] (resp.
$d\leq 2\lfloor n/6\rfloor+2$ [19]). A ternary (resp. quaternary Hermitian)
self-dual code of length $n$ and minimum weight $3\lfloor n/12\rfloor+3$
(resp. $2\lfloor n/6\rfloor+2$) is called _extremal_. A ternary (resp.
quaternary Hermitian) self-dual code of length $n$ and minimum weight
$3\lfloor n/12\rfloor$ (resp. $2\lfloor n/6\rfloor$) is called _near-
extremal_.

In this paper, we are interested in ternary near-extremal self-dual codes of
length $12m$ and quaternary near-extremal Hermitian self-dual codes of length
$6m$.

######  Theorem 2.1 (Han and Kim [11]).

  * (i)

There is no ternary near-extremal self-dual code of length $12m$ for $m\geq
147$.

  * (ii)

There is no quaternary near-extremal Hermitian self-dual code of length $6m$
for $m\geq 38$.

The above theorem is used in the proofs of Theorems 3.2 and 3.5.

###  2.3 Designs in self-dual codes

A _$t$ -$(v,k,\lambda)$ design_ $\mathcal{D}$ is a pair $(X,\mathcal{B})$,
where $X$ is a set of $v$ _points_ and a collection $\mathcal{B}$ of
$k$-subsets of $X$ called _blocks_ such that every $t$-subset of $X$ is
contained in exactly $\lambda$ blocks. A $t$-$(v,k,\lambda)$ design is simply
called a $t$-design. The number of blocks that contain a given point is
traditionally denoted by $r$, and the total number of blocks is $b$. If
$\mathcal{D}$ is a $1$-design, then it holds that $bk=vr$.

The Assmus–Mattson theorem shows that the supports of codewords of a fixed
weight in a given code are the blocks of a $t$-design under certain conditions
[2]. We often identify a $t$-design with its set of blocks. By the
Assmus–Mattson theorem, the supports of codewords of weight $i$ in a ternary
extremal self-dual code of length $12m$ form a $5$-design for
$i=3m+3,3m+6,\ldots,6m+3$ (see [21]). By the Assmus–Mattson theorem, the
supports of codewords of weight $i$ in a quaternary extremal Hermitian self-
dual code of length $6m$ form a $5$-design for
$i=2m+2,2m+4,\ldots,2\lfloor(3m+2)/2\rfloor$ (see [19, Theorem 18]).

Recently, Miezaki, Munemasa and Nakasora [22] gave an analogue of the result
for ternary near-extremal self-dual codes of length $12m$ and quaternary near-
extremal Hermitian self-dual codes of length $6m$.

######  Theorem 2.2 (Miezaki, Munemasa and Nakasora [22]).

  * (i)

The supports of codewords of weight $i$ in a ternary near-extremal self-dual
code of length $12m$ form a $1$-design for $i=3m,3m+3,\ldots,6m-3$.

  * (ii)

The supports of codewords of weight $i$ in a quaternary near-extremal
Hermitian self-dual code of length $6m$ form a $1$-design for
$i=2m,2m+2,\ldots,3m-1$.

The above theorem motivates our study of ternary near-extremal self-dual codes
of length $12m$ and quaternary near-extremal Hermitian self-dual codes of
length $6m$. The theorem is also used in the proofs of Lemmas 3.1 and 3.4.

###  2.4 Gleason type theorems

It is well known that the possible weight enumerators of ternary self-dual
codes can be determined by the Gleason type theorem (see [21]). The weight
enumerator $W$ of a ternary self-dual code of length $n$ is written as

$W=\sum_{j=0}^{\lfloor\frac{n}{12}\rfloor}a_{j}(1+8y^{3})^{\frac{n}{4}-3j}(y^{3}(1-y^{3})^{3})^{j},$
(1)

using some integers $a_{j}$. As an example, we give the weight enumerator
$W_{3,12m}$ of a ternary near-extremal self-dual code of length $12m$ for
$m=1,2,3$:

$\begin{split}W_{3,12}=&1+\alpha
y^{3}+(264-3\alpha)y^{6}+(440+3\alpha)y^{9}+(24-\alpha)y^{12},\\\
W_{3,24}=&1+\alpha y^{6}+(4048-6\alpha)y^{9}+(61824+15\alpha)y^{12}\\\
&+(242880-20\alpha)y^{15}+(198352+15\alpha)y^{18}\\\
&+(24288-6\alpha)y^{21}+(48+\alpha)y^{24},\\\ W_{3,36}=&1+\alpha
y^{9}+(42840-9\alpha)y^{12}+(1400256+36\alpha)y^{15}\\\
&+(18452280-84\alpha)y^{18}+(90370368+126\alpha)y^{21}\\\
&+(162663480-126\alpha)y^{24}+(97808480+84\alpha)y^{27}\\\
&+(16210656-36\alpha)y^{30}+(471240+9\alpha)y^{33}+(888-\alpha)y^{36},\end{split}$
(2)

where $\alpha$ is a positive integer. This calculation was done by Mathematica
[29].

Also, it is well known that the possible weight enumerators of quaternary
Hermitian self-dual codes can be determined by the Gleason type theorem (see
[18]). The weight enumerator $W$ of a quaternary Hermitian self-dual code of
length $n$ is written as

${W=\sum_{j=0}^{\lfloor\frac{n}{6}\rfloor}a_{j}(1+3y^{2})^{\frac{n}{2}-3j}(y^{2}(1-y^{2})^{2})^{j},}$
(3)

using some integers $a_{j}$. As an example, we give the weight enumerator
$W_{4,6m}$ of a quaternary near-extremal Hermitian self-dual code of length
$6m$ for $m=1,2,3$:

$\begin{split}W_{4,6}=&1+\alpha y^{2}+(45-2\alpha)y^{4}+(18+\alpha)y^{6},\\\
W_{4,12}=&1+\alpha y^{4}+(396-4\alpha)y^{6}+(1485+6\alpha)y^{8}\\\
&+(1980-4\alpha)y^{10}+(234+\alpha)y^{12},\\\ W_{4,18}=&1+\alpha
y^{6}+(2754-6\alpha y^{8}+(18360+15\alpha)y^{10}\\\
&+(77112-20\alpha)y^{12}+(110160+15\alpha)y^{14}\\\
&+(50949-6\alpha)y^{16}+(2808+\alpha)y^{18},\end{split}$ (4)

where $\alpha$ is a positive integer. This calculation was done by
Mathematica.

###  2.5 Methods for constructing self-dual codes

We give methods for constructing self-dual codes used in this paper.

Throughout this paper, let $I_{n}$ denote the identity matrix of order $n$ and
let $A^{T}$ denote the transpose of a matrix $A$. An $n\times n$ _$\mu$
-circulant_ matrix has the following form:

$\left(\begin{array}[]{cccccc}r_{0}&r_{1}&r_{2}&\cdots&r_{n-2}&r_{n-1}\\\ \mu
r_{n-1}&r_{0}&r_{1}&\cdots&r_{n-3}&r_{n-2}\\\ \mu r_{n-2}&\mu
r_{n-1}&r_{0}&\cdots&r_{n-4}&r_{n-3}\\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\\
\mu r_{1}&\mu r_{2}&\mu r_{3}&\cdots&\mu r_{n-1}&r_{0}\end{array}\right),$

where $\mu\neq 0$. If $\mu=1$ and $-1$, then the matrix is called _circulant_
and _negacirculant_ , respectively.

If $A$ and $B$ are $n\times n$ circulant (resp. negacirculant) matrices, then
a $[4n,2n]$ code over $\mathbb{F}_{q}$ having the following generator matrix:

$\left(\begin{array}[]{ccc@{}c}&{\Large I_{2n}}&&\begin{array}[]{cc}A&B\\\
-B^{T}&A^{T}\end{array}\end{array}\right)$ (5)

is called a _four-circulant_ (resp.  _four-negacirculant_) code. If $A$ is an
$(n-1)\times(n-1)$ circulant matrix, then a $[2n,n]$ code over
$\mathbb{F}_{q}$ having the following generator matrix:

$\left(\begin{array}[]{ccccccccc}{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&0&1&\cdots&1\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&1&{}\hfil&{}\hfil&{}\hfil\\\
{}\hfil&{}\hfil&I_{n}&{}\hfil&{}\hfil&\vdots&{}\hfil&A&{}\hfil\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&1&{}\hfil&{}\hfil&{}\hfil\\\
\end{array}\right),$ (6)

is called a _bordered double circulant_ code. If $A$ is an $n\times n$
$\mu$-circulant matrix, then a $[2n,n]$ code over $\mathbb{F}_{q}$ having the
following generator matrix:

$\left(\begin{array}[]{cc}I_{n}&A\\\ \end{array}\right)$ (7)

is called a _$\mu$ -circulant_ code or a quasi-twisted code. Many four-
circulant self-dual codes, four-negacirculant self-dual codes, bordered double
circulant codes and $\mu$-circulant codes having large minimum weights are
known (see e.g., [8], [9], [10] and [12]).

For a code $C$ of length $n$ over $\mathbb{F}_{q}$ and a vector
$x\in\mathbb{F}_{q}^{n}$, we denote by $\langle x\rangle$ the code generated
by $x$, and we denote by $\langle C,x\rangle$ the code generated by $C$ and
$x$. Let $C$ be a ternary self-dual code of length $n$ and let $x$ be a vector
of $\mathbb{F}_{3}^{n}$. If $\langle x,x\rangle_{E}=0$ and $x\not\in C$, then

$\operatorname{Nei}_{E}(C,x)=\langle C\cap\langle x\rangle^{\perp},x\rangle$
(8)

is a ternary self-dual code. Let $D$ be a quaternary Hermitian self-dual code
of length $n$ and let $y$ be a vector of $\mathbb{F}_{4}^{n}$. If $\langle
y,y\rangle_{H}=0$ and $y\not\in D$, then

$\operatorname{Nei}_{H}(D,y)=\langle D\cap\langle
y\rangle^{\perp_{H}},y\rangle$ (9)

is a quaternary Hermitian self-dual code. These codes are often called
_neighbors_ of $C$ and $D$, respectively. For all ternary self-dual codes
$\operatorname{Nei}_{E}(C,x)$ and quaternary Hermitian self-dual codes
$\operatorname{Nei}_{H}(D,y)$ constructed in this paper, the vectors $x$ and
$y$ have forms $(0,0,\ldots,0,\hat{x})$ and $(0,0,\ldots,0,\hat{y})$,
respectively, where $\hat{x}\in\mathbb{F}_{3}^{n/2}$ and
$\hat{y}\in\mathbb{F}_{4}^{n/2}$. Throughout this paper, we simply write
$\operatorname{Nei}_{E}(C,x)$ and $\operatorname{Nei}_{H}(D,y)$ by
$\operatorname{Nei}_{E}(C,\hat{x})$ and $\operatorname{Nei}_{H}(D,\hat{y})$,
respectively.

###  2.6 Largest minimum weights $d_{3}(12m)$ and $d^{H}_{4}(6m)$

Let $d_{3}(n)$ denote the largest minimum weight among ternary self-dual codes
of length $n$. For $m=3,4,\ldots,10$, we list the current information on
$d_{3}(12m)$ in Table 1 along with the references. Remark that there is a
ternary extremal self-dual code of length $12m$ for $m=1,2,3,4,5$. Throughout
this paper, $QR_{12m}$ and $P_{12m}$ denote the extended quadratic residue
code and the Pless symmetry code of length $12m$, respectively.

Table 1: Largest minimum weights $d_{3}(12m)$
---
$m$ | $d_{3}(12m)$ | Reference | $m$ | $d_{3}(12m)$ | Reference
3 | $12$ | $P_{36}$ | 7 |  $21$ or $24$ | $QR_{84},P_{84}$
4 | $15$ | $QR_{48},P_{48}$ | 8 | $24$ | $P_{96}$
5 | $18$ |  $QR_{60},P_{60}$, [23] | 9 |  $27$ or $30$ |  [7, Table 3], [23]
6 | $18$ |  $QR_{72}$, Section 4 | 10 |  $24,27$ or $30$ | [7, Table 3]
Table 2: Largest minimum weights $d^{H}_{4}(6m)$
---
$m$ | $d^{H}_{4}(6m)$ | Reference | $m$ | $d^{H}_{4}(6m)$ | Reference
4 | $8$ |  [17], Section 5 | 8 |  $14,16$ or $18$ | [7, Table 5]
5 | $12$ | [19] | 9 |  $16,18$ or $20$ | [7, Table 5]
6 |  $12$ or $14$ |  [8], Section 5 | 10 |  $16,18,20$ or $22$ | [7, Table 5]
7 |  $12,14$ or $16$ | [7, Table 5] |  |  |

Let $d^{H}_{4}(n)$ denote the largest minimum weight among quaternary
Hermitian self-dual codes of length $n$. For $m=4,5,\ldots,10$, we list the
current information on $d^{H}_{4}(6m)$ in Table 2 along with the references.
It is currently not known whether there is a quaternary near-extremal
Hermitian self-dual code of length $6m$ or not for $m=7,8,9,10$ [7, Table 5].

##  3 Restrictions on weight enumerators of self-dual codes

In this section, we give restrictions on the weight enumerators of ternary
near-extremal self-dual codes of length $12m$ and quaternary near-extremal
Hermitian self-dual codes of length $6m$. Furthermore, we give a divisibility
property of the coefficients of the weight enumerators of ternary near-
extremal self-dual codes of length $12m$ (Theorem 3.2) and quaternary near-
extremal Hermitian self-dual codes of length $6m$ (Theorem 3.5). Theorems 3.2
and 3.5 are the main results of this paper.

###  3.1 Ternary near-extremal self-dual codes of length $12m$

######  Lemma 3.1.

Let $C$ be a ternary near-extremal self-dual code of length $12m$. Let
$A_{3m}$ denote the number of codewords of weight $3m$ in $C$. Then
$A_{3m}\equiv 0\pmod{8}$.

###### Proof.

Let $C(3m)$ be the set of codewords of weight $3m$ in $C$. There is a set $X$
such that

$C(3m)=X\cup Y\text{ and }X\cap Y=\emptyset,$

where $Y=\\{2x\mid x\in X\\}$. Let $x$ and $y$ be codewords of $C(3m)$ such
that $\operatorname{supp}(x)=\operatorname{supp}(y)$. If $x\not\in\\{y,2y\\}$,
then $0<\min\\{\operatorname{wt}(x+y),\operatorname{wt}(x+2y)\\}<3m$. This
implies that

$|X|=|\\{\operatorname{supp}(x)\mid x\in X\\}|=\frac{A_{3m}}{2}.$

By Theorem 2.2, the supports of codewords of weight $3m$ in $C$ form a
$1$-design. Thus, the supports of codewords in $X$ form a $1$-design. Hence,
it holds that $\frac{A_{3m}}{2}3m=12mr$, where $r$ is a positive integer.
Therefore, $A_{3m}\equiv 0\pmod{8}$. ∎

The above lemma gives a restriction on the weight enumerators of ternary near-
extremal self-dual codes of length $12m$. Furthermore, we have the following
divisibility property of the coefficients of the weight enumerators of these
codes.

######  Theorem 3.2.

If there is a ternary near-extremal self-dual code of length $12m$, then
$A_{3i}\equiv 0\pmod{8}$ for $i=m,m+1,\ldots,4m$, where $A_{3i}$ denotes the
number of codewords of weight $3i$ in the code.

###### Proof.

By Theorem 2.1, it is sufficient to consider for $m\leq 146$. Our proof is
numerical. Let $W_{3,12m}$ be the weight enumerator of a ternary near-extremal
self-dual code of length $12m$. We numerically calculated the weight
enumerator $W_{3,12m}$ using $A_{3m}$, say $\alpha=A_{3m}$ by the Gleason type
theorem (1) for each $m=1,2,\ldots,146$. This calculation was done by
Mathematica [29]. For $m=1,2,3$, the weight enumerators $W_{3,12m}$ are given
in (2). For $m=4,5,\ldots,10$, the weight enumerators $W_{3,12m}$ are given in
Tables 5, 9, 27 and 28. For the remaining $m$, the weight enumerators
$W_{3,12m}$ can be obtained electronically from [1]. For each
$m=1,2,\ldots,146$, $A_{3i}$ can be written as $s_{3i}+t_{3i}\alpha$, using
some integers $s_{3i},t_{3i}$ $(i=m+1,m+2,\ldots,4m)$. For each
$m=1,2,\ldots,146$, we numerically verified by Mathematica that $s_{3i}\equiv
0\pmod{8}$ $(i=m+1,m+2,\ldots,4m)$. By Lemma 3.1, $\alpha\equiv 0\pmod{8}$.
This completes the proof. ∎

All ternary self-dual codes were classified in [5], [14], [20] and [24] for
lengths up to $24$. For $m=1,2$, we list in Table 3 the numbers $N_{12m}$ of
inequivalent ternary near-extremal self-dual codes of length $12m$ along with
the references. We also list the values $\alpha$ in $W_{3,12m}$ for which
there is a ternary near-extremal self-dual code of length $12m$.

Table 3: Small ternary near-extremal self-dual codes of length $12m$
---
$m$ | Reference | $N_{12m}$ |  $\alpha$ in $W_{3,12m}$
$1$ | [20] | $2$ | $8,24$
$2$ | [14] | $166$ | $8\beta\ (\beta=2,3,\ldots,16,18,21,24,25,30,36,66)$

As a consequence, Theorem 3.2 gives a divisibility property of the
coefficients of the weight enumerators of ternary extremal self-dual codes of
length $12m$.

######  Corollary 3.3.

If there is a ternary extremal self-dual code of length $12m$, then
$B_{3i}\equiv 0\pmod{8}$ for $i=m+1,m+2,\ldots,4m$, where $B_{3i}$ denotes the
number of codewords of weight $3i$ in the code.

###### Proof.

Note that there is no ternary extremal self-dual code of length $12m$ for
$m\geq 70$ [30]. Let $A_{3i}$ denote the number of codewords of weight $3i$ in
a ternary near-extremal self-dual code of length $12m$. As described in the
proof of Theorem 3.2, $A_{3i}$ can be written as $s_{3i}+t_{3i}A_{3m}$, using
some integers $s_{3i},t_{3i}$, and $s_{3i}\equiv 0\pmod{8}$
$(i=m+1,m+2,\ldots,4m)$. It is trivial that $B_{3i}=s_{3i}$
$(i=m+1,m+2,\ldots,4m)$. ∎

###  3.2 Quaternary near-extremal Hermitian self-dual codes of length $6m$

######  Lemma 3.4.

Let $D$ be a quaternary near-extremal Hermitian self-dual code of length $6m$.
Let $A_{2m}$ denote the number of codewords of weight $2m$ in $D$. Then
${A_{2m}\equiv 0\pmod{9}}$.

###### Proof.

Let $D(2m)$ be the set of codewords of weight $2m$ in $D$. There is a set $X$
such that

$D(2m)=X\cup Y\cup Z\text{ and }X\cap Y=X\cap Z=Y\cap Z=\emptyset,$

where $Y=\\{\omega x\mid x\in X\\}$ and $Z=\\{\omega^{2}x\mid x\in X\\}$. Let
$x$ and $y$ be codewords of $D(2m)$ such that
$\operatorname{supp}(x)=\operatorname{supp}(y)$. If $x\not\in\\{y,\omega
y,\omega^{2}y\\}$, then
$0<\min\\{\operatorname{wt}(x+y),\operatorname{wt}(x+\omega
y),\operatorname{wt}(x+\omega^{2}y)\\}<2m$. This implies that

$|X|=|\\{\operatorname{supp}(x)\mid x\in X\\}|=\frac{A_{2m}}{3}.$

By Theorem 2.2, the supports of codewords of weight $2m$ in $D$ form a
$1$-design. Thus, the supports of codewords in $X$ form a $1$-design. Hence,
it holds that $\frac{A_{2m}}{3}2m=6mr$, where $r$ is a positive integer.
Therefore, ${A_{2m}\equiv 0\pmod{9}}$. ∎

The above lemma gives a restriction on the weight enumerators of quaternary
near-extremal Hermitian self-dual codes of length $6m$. Furthermore, we have
the following divisibility property of the coefficients of the weight
enumerators of these codes.

######  Theorem 3.5.

If there is a quaternary near-extremal Hermitian self-dual code of length
$6m$, then ${A_{2i}\equiv 0\pmod{9}}$ for ${i=m,m+1,\ldots,3m}$, where
$A_{2i}$ denotes the number of codewords of weight $2i$ in the code.

###### Proof.

By Theorem 2.1, it is sufficient to consider for $m\leq 37$. Our proof is
numerical. Let $W_{4,6m}$ be the weight enumerator of a quaternary near-
extremal Hermitian self-dual code of length $6m$. We numerically calculated by
Mathematica the weight enumerator $W_{4,6m}$ using $A_{2m}$, say
$\alpha=A_{2m}$ by the Gleason type theorem (3) for each $m=1,2,\ldots,37$.
For $m=1,2,3$, the weight enumerators $W_{4,6m}$ are given in (4). For
$m=4,5,\ldots,10$, the weight enumerators $W_{4,6m}$ are given in Tables 19,
21 and 29. For the remaining $m$, the weight enumerators $W_{4,6m}$ can be
obtained electronically from [1]. For each $m=1,2,\ldots,37$, $A_{2i}$ can be
written as $s_{2i}+t_{2i}\alpha$, using some integers $s_{2i},t_{2i}$
$(i=m+1,m+2,\ldots,3m)$. For each $m=1,2,\ldots,37$, we numerically verified
by Mathematica that $s_{2i}\equiv 0\pmod{9}$ $(i=m+1,m+2,\ldots,3m)$. By Lemma
3.4, $\alpha\equiv 0\pmod{9}$. This completes the proof. ∎

All quaternary Hermitian self-dual codes were classified in [5], [13], [15]
and [19] for lengths up to $20$. For $m=1,2,3$, we list in Table 4 the numbers
$N_{6m}$ of inequivalent quaternary near-extremal Hermitian self-dual codes of
length $6m$ along with the references. We also list the values $\alpha$ in
$W_{4,6m}$ for which there is a quaternary near-extremal Hermitian self-dual
code of length $6m$.

Table 4: Small quaternary near-extremal Hermitian self-dual codes of length
$6m$
---
$m$ | Reference | $N_{6m}$ |  $\alpha$ in $W_{4,6m}$
$1$ | [19] | $1$ | $9$
$2$ | [19] | $5$ | $9,18,36,45,90$
$3$ | [13] | $30$ | $27,45,72,81,99,108$

As a consequence, Theorem 3.5 gives a divisibility property of the
coefficients of the weight enumerators of quaternary extremal Hermitian self-
dual codes of length $6m$.

######  Corollary 3.6.

If there is a quaternary extremal Hermitian self-dual code of length $6m$,
then $B_{2i}\equiv 0\pmod{9}$ for $i=m+1,m+2,\ldots,3m$, where $B_{2i}$
denotes the number of codewords of weight $2i$ in the code.

###### Proof.

Note that there is no quaternary extremal Hermitian self-dual code of length
$6m$ for $m\geq 17$ [30]. Let $A_{2i}$ denote the number of codewords of
weight $2i$ in a quaternary near-extremal Hermitian self-dual code of length
$6m$. As described in the proof of Theorem 3.5, $A_{2i}$ can be written as
$s_{2i}+t_{2i}A_{2m}$, using some integers $s_{2i},t_{2i}$, and $s_{2i}\equiv
0\pmod{9}$ $(i=m+1,m+2,\ldots,3m)$. It is trivial that $B_{2i}=s_{2i}$
$(i=m+1,m+2,\ldots,3m)$. ∎

##  4 Existence of ternary near-extremal self-dual codes of length $12m$

In this section, we consider the weight enumerators for which there is a
ternary near-extremal self-dual code of length $12m$ for $m=3,4,5,6$. All
computer calculations in this section and the next section were done by Magma
[3].

###  4.1 Length 72

The smallest length for which $3m$ is the largest minimum weight among ternary
self-dual codes of length $12m$ is $72$ (see Table 1). From the viewpoint, the
most interesting length is $72$. Here we consider the weight enumerators for
which there is a ternary near-extremal self-dual code of length $72$.

For the weight enumerator $W_{3,72}=\sum_{i=0}^{72}A_{i}y^{i}$ of a ternary
near-extremal self-dual code of length $72$, $A_{i}$ are listed in Table 5.

######  Fact 4.1.

For the weight enumerator $W_{3,72}$, $\alpha=8\beta$ and

$\beta\in\\{14466,14467,\ldots,251482\\}.$

###### Proof.

Follows from Lemma 3.1, $A_{21}\geq 0$ and $A_{72}\geq 0$. ∎

Table 5: Weight enumerator $W_{3,72}$
---
$i$ | $A_{i}$ | $i$ | $A_{i}$
$0$ | $1$ | $45$ | $33060336837654336-48620\alpha$
$18$ | $\alpha$ | $48$ | $44727764434320240+43758\alpha$
$21$ | $36213408-18\alpha$ | $51$ | $34777061847747648-31824\alpha$
$24$ | $2634060240+153\alpha$ | $54$ | $14918066750540800+18564\alpha$
$27$ | $126284566912-816\alpha$ | $57$ | $3328272012763584-8568\alpha$
$30$ | $3525613242624+3060\alpha$ | $60$ | $354029045223168+3060\alpha$
$33$ | $59358705673680-8568\alpha$ | $63$ | $15690890307840-816\alpha$
$36$ | $607797076070496+18564\alpha$ | $66$ | $230384763504+153\alpha$
$39$ | $3798847469410560-31824\alpha$ | $69$ | $707807520-18\alpha$
$42$ | $14443524566748288+43758\alpha$ | $72$ | $-115728+\alpha$

The ternary extended quadratic residue code $QR_{72}$ of length $72$ is a
ternary near-extremal self-dual code of length $72$ having weight enumerator
with $\alpha=357840$. A ternary near-extremal self-dual code of length $72$
was found in [7, Table 3]. We verified that the code has weight enumerator
with $\alpha=213936$.

By considering four-negacirculant codes, we found $700$ ternary near-extremal
self-dual codes of length $72$ with distinct weight enumerators. The values
$\alpha$ of the $700$ weight enumerators are given by:

$\alpha\in\Gamma_{72}=\\{24\beta\mid\beta\in\\{8500,8501,\ldots,9142\\}\cup\Gamma_{72,1}\cup\Gamma_{72,2}\setminus\Gamma_{72,3}\\},$
(10)

where $\Gamma_{72,1}$, $\Gamma_{72,2}$ and $\Gamma_{72,3}$ are listed in Table
6. The codes have generator matrices of the form (5), and the first rows
$r_{A}$ and $r_{B}$ of the negacirculant matrices $A$ and $B$ can be obtained
electronically from [1].

Table 6: $\Gamma_{72,1}$, $\Gamma_{72,2}$ and $\Gamma_{72,3}$
---
$\Gamma_{72,1}$ | 8244, 8316, 8326, 8366, 8376, 8401, 8403, 8415, 8421, 8426, 8439,
| 8445, 8454, 8456, 8458, 8470, 8472, 8475, 8478, 8479, 8481, 8489,
| 8490, 8492, 8493, 8494, 8498
$\Gamma_{72,2}$ | 9144, 9146, 9150, 9151, 9153, 9154, 9156, 9157, 9158, 9159, 9160,
| 9161, 9162, 9165, 9166, 9168, 9171, 9172, 9174, 9179, 9180, 9181,
| 9182, 9184, 9185, 9186, 9190, 9192, 9193, 9194, 9195, 9198, 9199,
| 9202, 9204, 9207, 9208, 9210, 9211, 9212, 9216, 9220, 9224, 9237,
| 9242, 9244, 9247, 9258, 9264, 9274, 9280, 9284, 9286, 9306, 9328,
| 9330, 9342, 9372, 9374, 9424, 9442, 9458, 9811
$\Gamma_{72,3}$ | 8502, 8503, 8504, 8505, 8506, 8507, 8518, 8527, 8529, 8534, 8537,
| 8543, 8546, 8552, 8555, 8558, 8567, 8571, 8573, 8577, 8583, 8585,
| 8594, 8914, 9077, 9083, 9104, 9110, 9113, 9119, 9121, 9122, 9139

In order to construct more ternary near-extremal self-dual codes of length
$72$, we consider ternary codes having the following generator matrices:

$\left(\begin{array}[]{ccc@{}c}&{\Large
I_{36}}&&\begin{array}[]{cccc}A&B&C&D\\\ B&-A&D&-C\\\
C^{T}&-D^{T}&-A^{T}&B^{T}\\\ D^{T}&C^{T}&-B^{T}&-A^{T}\\\
\end{array}\end{array}\right),$

where $A,B,C$ and $D$ are $9\times 9$ negacirculant matrices. Remark that the
right halves of the above generator matrices are known as the Ito array in the
context of Hadamard matrices (see [28, Section 3.3]). Using this construction
method, we found $40$ more ternary near-extremal self-dual codes $C_{72,i}$
$(i=1,2,\ldots,40)$ of length $72$ with distinct weight enumerators. The
values $\alpha$ of the $40$ weight enumerators are given by:

$\alpha\in\\{24\beta\mid\beta\in\Gamma^{\prime}_{72}\\},$

where $\Gamma^{\prime}_{72}$ is listed in Table 7. For the $40$ codes, the
values $\alpha$ and the first rows $r_{A},r_{B},r_{C}$ and $r_{D}$ of the
negacirculant matrices $A,B,C$ and $D$ are listed in Tables 22 and 23.

Table 7: $\Gamma^{\prime}_{72}$
---
8350, 8431, 8442, 8448, 8460, 8465, 8484, 8486, 8502, 8507, 8527, 8529,
8534, 8546, 8552, 8558, 8573, 8577, 8583, 8594, 9077, 9104, 9110, 9122,
9139, 9145, 9148, 9152, 9155, 9176, 9178, 9217, 9218, 9232, 9234, 9272,
9273, 9300, 9419, 9542

In order to construct ternary near-extremal self-dual codes of length $72$
having weight enumerator $W_{3,72}$ with $\alpha\equiv 8,16\pmod{24}$, we
consider bordered double circulant codes. Then we found $25$ more ternary
near-extremal self-dual codes $C_{72,i}$ $(i=41,42,\ldots,65)$ of length $72$
with distinct weight enumerators. The values $\alpha$ of the $25$ weight
enumerators are given by:

$\alpha\in\left\\{8\beta\mid\beta\in\Delta_{72}\right\\},$

where $\Delta_{72}$ is listed in Table 8. The codes have generator matrices of
the form (6), and the values $\alpha$ and the first rows $r_{A}$ of the
circulant matrices $A$ are listed in Table 24.

Table 8: $\Delta_{72}$
---
25550, 25795, 25970, 26005, 26075, 26110, 26215, 26285, 26320, 26390,
26425, 26495, 26530, 26600, 26635, 26705, 26740, 26810, 26845, 26950,
27020, 27160, 27265, 27580, 27755

In summary, we have the following:

######  Proposition 4.2.

There is a ternary near-extremal self-dual code of length $72$ having weight
enumerator $W_{3,72}$ listed in Table 5 for

$\alpha\in\\{24\beta\mid\beta\in\\{8914,14910\\}\cup\Gamma_{72}\cup\Gamma^{\prime}_{72}\\}\cup\\{8\beta\mid\beta\in\Delta_{72}\\}$

where $\Gamma_{72}$, $\Gamma^{\prime}_{72}$ and $\Delta_{72}$ are listed in
(10), Tables 7 and 8, respectively.

###  4.2 Length 36

There is a ternary extremal self-dual code of length $36$. Thus, the largest
minimum weight among ternary self-dual codes of this length is $12$ not $9$.
However, the smallest length for which the classification of ternary self-dual
codes of length $12m$ has not been completed is $36$. From the viewpoint, here
we consider the weight enumerators for which there is a ternary near-extremal
self-dual code of length $36$.

For the weight enumerator $W_{3,36}=\sum_{i=0}^{36}A_{i}y^{i}$ of a ternary
near-extremal self-dual code of length $36$, $A_{i}$ are listed in Table 9.

######  Fact 4.3.

For the weight enumerator $W_{3,36}$, $\alpha=8\beta$ and

$\beta\in\\{1,2,\ldots,111\\}.$

###### Proof.

Follows from Lemma 3.1, $A_{9}>0$ and $A_{36}\geq 0$. ∎

Table 9: Weight enumerator $W_{3,36}$
---
$i$ | $A_{i}$ | $i$ | $A_{i}$ | $i$ | $A_{i}$
$0$ | $1$ | $18$ | $18452280-84\alpha$ | $30$ | $16210656-36\alpha$
$9$ | $\alpha$ | $21$ | $90370368+126\alpha$ | $33$ | $471240+9\alpha$
$12$ | $42840-9\alpha$ | $24$ | $162663480-126\alpha$ | $36$ | $888-\alpha$
$15$ | $1400256+36\alpha$ | $27$ | $97808480+84\alpha$ |  |

By considering four-negacirculant codes, we found $19$ ternary near-extremal
self-dual codes $C_{36,i}$ $(i=1,2,\ldots,19)$ of length $36$ with distinct
weight enumerators. These codes have generator matrices of the form (5), and
the first rows $r_{A}$ and $r_{B}$ of the negacirculant matrices $A$ and $B$
are given in Table 10. The values $\alpha$ of the weight enumerators are also
given in Table 10.

Table 10: Ternary four-negacirculant self-dual codes of length $36$
---
Code | $\alpha$ | $r_{A}$ | $r_{B}$
$C_{36,1}$ | $72$ | $(0,1,2,0,0,0,0,1,2)$ | $(1,2,2,1,1,1,1,0,0)$
$C_{36,2}$ | $96$ | $(0,2,1,1,0,0,0,1,2)$ | $(1,0,2,1,1,1,0,1,0)$
$C_{36,3}$ | $144$ | $(0,1,1,1,0,0,0,0,2)$ | $(2,1,0,2,1,1,1,1,0)$
$C_{36,4}$ | $168$ | $(0,2,2,1,0,0,0,1,1)$ | $(1,2,1,1,0,2,0,1,0)$
$C_{36,5}$ | $216$ | $(0,2,0,1,0,0,0,0,2)$ | $(1,2,1,2,0,0,1,0,0)$
$C_{36,6}$ | $240$ | $(0,2,2,0,0,0,0,2,1)$ | $(2,1,1,2,1,1,1,0,0)$
$C_{36,7}$ | $288$ | $(0,2,1,0,0,0,0,0,0)$ | $(2,1,2,1,0,1,1,0,0)$
$C_{36,8}$ | $312$ | $(0,0,1,1,0,0,0,0,2)$ | $(1,1,2,2,1,2,1,1,0)$
$C_{36,9}$ | $360$ | $(0,0,2,0,0,0,0,1,0)$ | $(2,1,2,2,2,0,1,0,0)$
$C_{36,10}$ | $384$ | $(0,2,0,0,0,0,0,2,0)$ | $(1,2,2,1,2,1,0,0,0)$
$C_{36,11}$ | $432$ | $(0,0,0,0,0,0,0,1,2)$ | $(1,1,2,1,2,0,1,0,0)$
$C_{36,12}$ | $456$ | $(0,1,0,0,0,0,0,2,2)$ | $(2,2,1,2,0,1,0,0,0)$
$C_{36,13}$ | $504$ | $(0,1,0,1,0,0,0,1,2)$ | $(1,2,1,0,0,1,0,0,0)$
$C_{36,14}$ | $528$ | $(0,1,2,1,0,0,0,2,1)$ | $(2,1,1,0,2,1,1,0,0)$
$C_{36,15}$ | $576$ | $(0,0,0,2,0,0,0,2,2)$ | $(1,2,2,2,1,2,2,1,0)$
$C_{36,16}$ | $600$ | $(0,2,1,2,0,0,0,1,2)$ | $(1,1,1,2,2,1,1,1,1)$
$C_{36,17}$ | $648$ | $(0,0,0,1,0,0,0,2,1)$ | $(2,2,1,0,1,1,0,0,0)$
$C_{36,18}$ | $720$ | $(0,1,1,0,0,0,0,1,0)$ | $(1,2,1,2,1,0,0,0,0)$
$C_{36,19}$ | $744$ | $(0,0,1,0,0,0,0,1,2)$ | $(1,1,1,1,1,0,0,0,0)$

By considering bordered double circulant codes, we found three ternary near-
extremal self-dual codes $C_{36,i}$ $(i=20,21,22)$ of length $36$ with
distinct weight enumerators. These codes have generator matrices of the form
(6), and the first rows of the circulant matrices $A$ are

$\displaystyle(2,1,0,0,0,0,2,1,2,1,2,0,2,0,1,1,0),$
$\displaystyle(2,0,1,0,0,0,1,1,1,1,2,0,1,1,0,0,1),$
$\displaystyle(2,0,0,0,0,0,0,0,2,2,0,1,2,0,2,1,0),$

respectively. We verified that $C_{36,i}$ $(i=20,21,22)$ have weight
enumerators with $\alpha=136$, $408$, $544$, respectively.

By considering neighbors, we found $53$ more ternary near-extremal self-dual
codes
$N_{36,i}=\operatorname{Nei}_{E}(C,{x})=\operatorname{Nei}_{E}(C,\hat{x})$
$(i=1,2,\ldots,52)$ (see (8) for $\operatorname{Nei}_{E}(C,x)$), where $C$ and
$\hat{x}=(x_{17},x_{18},\ldots,x_{36})$ are listed in Table 25. In the table,
$P_{36}$ denotes the Pless symmetry code of length $36$ having generator
matrix of form (6), where the first row of the circulant matrix $A$ is

$(0,1,1,2,1,2,2,2,1,1,2,2,2,1,2,1,1).$

The values $\alpha$ of the weight enumerators are also given in Table 25.

In summary, we have the following:

######  Proposition 4.4.

There is a ternary near-extremal self-dual code of length $36$ having weight
enumerator $W_{3,36}$ listed in Table 9 for

$\displaystyle\alpha\in\\{8\beta\mid\beta\in\\{9,12,14,16,17,\ldots,83,85,90,93\\}\\}.$

###  4.3 Lengths 48 and 60

For the weight enumerators $W_{3,48}=\sum_{i=0}^{48}A_{i}y^{i}$ and
$W_{3,60}=\sum_{i=0}^{60}A_{i}y^{i}$ of ternary near-extremal self-dual codes
of lengths $48$ and $60$, $A_{i}$ are listed in Tables 11 and 12,
respectively.

######  Fact 4.5.

For the weight enumerators $W_{3,48}$ and $W_{3,60}$, $\alpha=8\beta$ and

$\beta\in\\{1,2,\ldots,4324\\}\text{ and }\beta\in\\{1,2,\ldots,5148\\},$

respectively.

###### Proof.

Follows from Lemma 3.1, $A_{12}>0$ and $A_{15}\geq 0$ (resp. $A_{15}>0$ and
$A_{60}\geq 0$) for $W_{3,48}$ (resp. $W_{3,60}$). ∎

Table 11: Weight enumerator $W_{3,48}$
---
$i$ | $A_{i}$ | $i$ | $A_{i}$ | $i$ | $A_{i}$
$0$ | $1$ | $24$ | $5745355200+495\alpha$ | $39$ | $9794378880-220\alpha$
$12$ | $\alpha$ | $27$ | $31815369344-792\alpha$ | $42$ | $573051072+66\alpha$
$15$ | $415104-12\alpha$ | $30$ | $83368657152+924\alpha$ | $45$ | $6503296-12\alpha$
$18$ | $20167136+66\alpha$ | $33$ | $99755406432-792\alpha$ | $48$ | $96+\alpha$
$21$ | $497709696-220\alpha$ | $36$ | $50852523072+495\alpha$ |  |
Table 12: Weight enumerator $W_{3,60}$
---
$i$ | $A_{i}$ | $i$ | $A_{i}$
$0$ | $1$ | $39$ | $63958467767040+6435\alpha$
$15$ | $\alpha$ | $42$ | $59278900150800-5005\alpha$
$18$ | $3901080-15\alpha$ | $45$ | $27270640178880+3003\alpha$
$21$ | $241456320+105\alpha$ | $48$ | $5739257192760-1365\alpha$
$24$ | $8824242960-455\alpha$ | $51$ | $485029078560+455\alpha$
$27$ | $172074038080+1365\alpha$ | $54$ | $13144038880-105\alpha$
$30$ | $1850359081824-3003\alpha$ | $57$ | $71451360+15\alpha$
$33$ | $11014750094040+5005\alpha$ | $60$ | $41184-\alpha$
$36$ | $36099369380880-6435\alpha$ |  |
Table 13: $\Gamma_{60,1}$
---
821, 845, 865, 870, 871, 875, 876, 880, 885, 886, 890, 891, 895, 896, 900,
901, 905, 906, 910, 911, 915, 916, 920, 921, 925, 926, 930, 931, 935, 936,
940, 941, 945, 946, 950, 951, 955, 956, 960, 961, 965, 966, 970, 971, 975,
976, 980, 981, 985, 986, 990, 991, 995, 996, 1000, 1001, 1005, 1006, 1010,
1011, 1015, 1016, 1020, 1021, 1025, 1026, 1030, 1031, 1035, 1036, 1040,
1041, 1045, 1046, 1050, 1051, 1055, 1056, 1060, 1061, 1065, 1066, 1070,
1071, 1075, 1076, 1080, 1081, 1085, 1086, 1090, 1091, 1095, 1096, 1100,
1101, 1105, 1106, 1110, 1111, 1115, 1116, 1120, 1121, 1125, 1126, 1130,
1131, 1135, 1136, 1140, 1141, 1145, 1146, 1150, 1151, 1155, 1156, 1160,
1161, 1165, 1166, 1170, 1171, 1175, 1176

By considering four-negacirculant codes, we found $86$ and $126$ ternary near-
extremal self-dual codes of lengths $48$ and $60$ with distinct weight
enumerators, respectively. The values $\alpha$ of the $86$ weight enumerators
are given by:

$\alpha\in\left\\{48\beta\mid\beta\in\Gamma_{48,1}\right\\},$

where

$\Gamma_{48,1}=\left\\{\begin{array}[]{l}33,34,36,37,\ldots,107,110,113,115,116,\\\
117,118,123,126,132,142,166,246\end{array}\right\\}.$ (11)

The values $\alpha$ of the $126$ weight enumerators are given by:

$\displaystyle\alpha\in\\{24\beta\mid\beta\in\Gamma_{60,1}\\},$

where $\Gamma_{60,1}$ is listed in Table 13. These codes have generator
matrices of the form (5), and the first rows $r_{A}$ and $r_{B}$ of the
negacirculant matrices $A$ and $B$ can be obtained electronically from [1].

By considering neighbors, we found $61$ more ternary near-extremal self-dual
codes $N_{48,i}=\operatorname{Nei}_{E}(P^{\prime}_{48},\hat{x})$
$(i=1,2,\ldots,61)$ of length $48$, where the vectors
$\hat{x}=(x_{25},x_{26},\ldots,x_{48})$ are listed in Table 26. Here
$P^{\prime}_{48}$ is a bordered double circulant code having generator matrix
of form (6), where the first row of the circulant matrix $A$ is

$(0,1,1,1,1,2,1,2,1,1,2,2,1,1,2,2,1,2,1,2,2,2,2).$

Note that $P^{\prime}_{48}$ is equivalent to $P_{48}$. The values $\alpha$ of
the weight enumerators are also given in Table 26, where

$\alpha\in\\{8\beta\mid\beta\in\Gamma_{48,2}\\},$

and $\Gamma_{48,2}$ is listed in Table 14.

Table 14: $\Gamma_{48,2}$
---
180, 181, 182, 184, 188, 210, 212, 215, 217, 218, 219, 220, 221, 223, 224,
225, 226, 227, 229, 230, 231, 232, 233, 235, 236, 237, 238, 239, 241, 242,
243, 244, 245, 247, 248, 249, 250, 251, 253, 254, 255, 256, 257, 259, 260,
261, 262, 263, 265, 266, 267, 268, 269, 271, 272, 273, 274, 275, 277, 278,
281

Again, by considering neighbors, we found $60$ more ternary near-extremal
self-dual codes $N_{60,i}=\operatorname{Nei}_{E}(P_{60},\hat{x})$
$(i=1,2,\ldots,60)$ of length $60$, where the vectors
$\hat{x}=(x_{31},x_{32},\ldots,x_{60})$ are listed in Tables 16 and 17. Here
$P_{60}$ denotes the Pless symmetry code of length $60$ having generator
matrix of form (6), where the first row of the circulant matrix $A$ is

$(0,1,2,2,1,1,1,1,2,1,2,2,2,1,2,2,1,2,2,2,1,2,1,1,1,1,2,2,1).$

The values $\alpha$ of the weight enumerators are also given in Tables 16 and
17, where

$\alpha\in\\{8\beta\mid\beta\in\Gamma_{60,2}\\},$

and $\Gamma_{60,2}$ is listed in Table 15.

Table 15: $\Gamma_{60,2}$
---
1869, 1871, 1872, 1874, 1879, 1884, 1886, 1887, 1888, 1889, 1890, 1892,
1894, 1895, 1896, 1897, 1898, 1899, 1900, 1901, 1902, 1903, 1904, 1906,
1907, 1908, 1909, 1910, 1911, 1912, 1913, 1914, 1915, 1916, 1917, 1918,
1919, 1920, 1921, 1922, 1923, 1924, 1925, 1926, 1928, 1930, 1931, 1933,
1934, 1936, 1937, 1938, 1939, 1940, 1943, 1944, 1947, 1948, 1952, 1960
Table 16: Ternary near-extremal self-dual codes $N_{60,i}$ of length $60$
---
$i$ | $\alpha$ | $\hat{x}$
1 | $14952$ | $(2,2,1,0,1,0,0,1,2,0,1,0,0,1,0,2,0,0,0,1,2,2,2,1,0,2,2,2,0,1)$
2 | $14968$ | $(2,2,2,1,1,0,0,1,2,0,1,0,0,1,0,2,0,0,0,1,0,2,2,1,0,2,2,1,0,1)$
3 | $14976$ | $(0,2,2,0,1,0,0,1,2,0,1,0,0,1,0,2,0,0,0,1,0,2,2,1,0,2,2,2,0,0)$
4 | $14992$ | $(2,2,2,0,1,1,0,1,2,0,1,0,0,1,0,2,0,0,0,1,0,2,2,1,0,2,1,2,0,1)$
5 | $15032$ | $(2,1,2,0,1,0,0,1,2,0,1,0,0,1,0,2,0,0,0,1,2,2,2,1,0,2,2,2,0,1)$
6 | $15072$ | $(1,0,2,2,0,2,0,1,1,0,2,2,2,2,0,1,1,2,1,1,0,2,1,0,2,1,1,2,0,0)$
7 | $15088$ | $(2,1,0,2,1,0,0,1,0,0,2,2,1,1,2,2,2,1,0,1,0,1,2,2,2,0,0,2,1,1)$
8 | $15096$ | $(2,1,1,1,0,1,1,2,2,1,1,1,2,0,0,1,1,2,2,2,2,1,2,1,1,1,2,1,1,2)$
9 | $15104$ | $(0,0,1,2,1,2,1,0,2,2,2,2,0,0,0,0,0,0,1,0,2,1,2,1,2,0,0,0,0,0)$
10 | $15112$ | $(2,0,0,0,1,0,0,1,2,0,1,0,0,1,0,2,0,0,0,1,0,2,2,1,0,2,2,2,0,1)$
11 | $15120$ | $(2,2,2,2,1,2,0,1,1,0,2,2,2,2,0,1,1,2,1,1,1,0,0,0,2,2,2,0,0,0)$
12 | $15136$ | $(0,2,0,1,1,0,1,2,2,1,1,1,2,0,0,1,1,2,2,2,2,0,1,1,2,1,0,1,0,0)$
13 | $15152$ | $(2,1,2,1,1,0,0,1,2,0,1,0,0,1,0,2,0,0,0,1,0,2,2,1,0,2,2,2,0,1)$
14 | $15160$ | $(2,1,0,1,2,1,1,2,2,1,1,1,2,0,0,1,1,2,2,2,2,1,2,1,1,1,2,1,1,2)$
15 | $15168$ | $(2,2,0,0,2,0,0,1,0,0,2,2,1,1,2,2,2,1,0,2,0,1,1,2,2,0,0,1,0,0)$
16 | $15176$ | $(2,1,1,0,2,1,1,2,2,1,1,1,2,0,0,1,1,2,2,2,2,1,2,1,1,1,2,1,1,2)$
17 | $15184$ | $(0,1,1,1,2,2,1,2,2,1,1,1,2,0,0,1,1,2,2,1,0,0,0,1,0,1,0,0,2,1)$
18 | $15192$ | $(0,1,2,1,1,1,0,0,1,2,0,2,1,1,2,0,1,0,0,2,1,2,1,1,1,2,0,1,0,2)$
19 | $15200$ | $(1,0,1,2,2,0,1,0,0,2,0,0,0,1,0,1,0,2,2,2,1,2,1,0,2,2,0,1,1,0)$
20 | $15208$ | $(2,0,0,1,0,2,0,1,1,0,2,2,2,2,0,1,1,2,1,0,0,1,1,0,1,0,0,1,2,0)$
Table 17: Ternary near-extremal self-dual codes $N_{60,i}$
---
$i$ | $\alpha$ | $\hat{x}$
21 | $15216$ | $(0,0,2,0,1,0,0,1,2,0,1,0,0,1,0,2,0,0,0,1,0,2,2,1,0,2,2,2,0,1)$
22 | $15224$ | $(2,1,2,1,1,1,0,0,1,2,0,2,1,1,2,0,1,0,0,2,1,2,1,0,1,2,0,1,0,2)$
23 | $15232$ | $(0,2,2,2,0,2,0,1,1,0,2,2,2,2,0,1,1,2,1,1,0,2,1,0,2,1,1,2,0,0)$
24 | $15248$ | $(0,2,2,0,0,0,0,1,2,0,1,0,0,1,0,2,0,0,0,1,0,2,2,1,0,2,2,2,0,1)$
25 | $15256$ | $(1,1,2,1,1,1,0,0,1,2,0,2,1,1,2,0,1,0,0,2,1,2,1,0,1,2,0,1,0,2)$
26 | $15264$ | $(2,2,2,0,1,0,0,1,2,0,1,0,0,1,0,2,0,0,0,1,0,2,2,1,0,2,2,0,0,0)$
27 | $15272$ | $(0,1,1,1,2,1,1,2,2,1,1,1,2,0,0,1,1,2,2,2,2,1,2,1,1,1,2,1,1,2)$
28 | $15280$ | $(1,0,0,0,0,0,1,2,2,1,1,1,2,0,0,1,1,2,2,0,2,0,1,1,2,2,1,0,0,0)$
29 | $15288$ | $(1,1,0,2,2,0,0,0,1,2,0,2,1,1,2,0,1,0,0,2,0,0,0,2,0,2,0,0,0,1)$
30 | $15296$ | $(2,1,1,1,2,1,1,2,2,1,1,1,2,0,0,1,1,2,2,2,2,1,2,1,1,1,2,1,1,0)$
31 | $15304$ | $(2,0,1,1,2,1,1,2,2,1,1,1,2,0,0,1,1,2,2,2,2,1,2,1,1,1,2,1,1,2)$
32 | $15312$ | $(1,1,0,1,2,0,2,0,2,0,0,2,0,0,0,2,0,1,1,1,0,0,0,1,1,1,0,0,0,1)$
33 | $15320$ | $(2,0,1,1,2,0,1,2,2,1,1,1,2,0,0,1,1,2,2,0,0,2,1,0,1,1,0,2,0,1)$
34 | $15328$ | $(0,0,0,1,0,1,2,0,2,0,0,2,0,0,0,2,0,1,1,2,1,0,0,2,1,2,2,0,2,0)$
35 | $15336$ | $(1,2,2,2,0,2,2,1,0,1,2,0,2,1,1,2,2,2,1,1,0,0,2,1,1,1,2,0,1,1)$
36 | $15344$ | $(2,2,2,0,2,0,0,1,2,0,1,0,0,1,0,2,0,0,0,1,2,2,2,1,0,2,2,2,0,1)$
37 | $15352$ | $(0,0,0,1,0,0,1,1,0,1,1,2,0,0,0,2,2,1,0,2,1,1,1,1,2,0,2,2,0,2)$
38 | $15360$ | $(0,0,2,2,0,2,0,1,1,0,2,2,2,2,0,1,1,2,1,1,2,2,1,0,2,1,1,2,0,0)$
39 | $15368$ | $(0,1,1,1,0,2,0,0,1,2,0,2,1,1,2,0,1,0,0,1,2,2,1,2,2,1,2,0,2,1)$
40 | $15376$ | $(1,0,1,2,2,2,0,1,1,0,2,2,2,2,0,1,1,2,1,2,2,0,1,2,0,0,2,0,2,0)$
41 | $15384$ | $(1,0,0,2,1,2,2,0,2,0,0,2,0,0,0,2,0,1,1,1,2,0,0,1,1,1,0,2,1,2)$
42 | $15392$ | $(1,2,1,0,0,0,0,1,0,0,2,2,1,1,2,2,2,1,0,2,2,1,2,1,2,1,1,2,0,0)$
43 | $15400$ | $(1,2,1,0,1,1,0,1,1,0,2,2,2,2,0,1,1,2,1,1,2,1,2,1,1,0,2,0,1,1)$
44 | $15408$ | $(0,2,2,0,2,2,0,1,1,0,2,2,2,2,0,1,1,2,1,2,0,2,0,1,2,1,0,0,1,2)$
45 | $15424$ | $(0,2,2,0,2,1,0,1,1,0,2,2,2,2,0,1,1,2,1,2,0,2,0,1,2,1,0,0,1,2)$
46 | $15440$ | $(2,2,1,1,0,0,0,1,1,0,2,2,2,2,0,1,1,2,1,1,0,1,1,1,2,0,0,0,2,2)$
47 | $15448$ | $(0,0,2,2,2,2,0,1,1,0,2,2,2,2,0,1,1,2,1,1,0,2,1,0,2,1,1,2,0,0)$
48 | $15464$ | $(1,2,1,1,1,1,0,1,1,0,2,2,2,2,0,1,1,2,1,1,0,1,2,1,1,0,2,0,1,1)$
49 | $15472$ | $(1,2,2,1,0,1,0,1,1,0,2,2,2,2,0,1,1,2,1,0,0,0,1,1,2,1,0,2,0,2)$
50 | $15488$ | $(1,0,0,1,2,1,0,1,2,0,1,0,0,1,0,2,0,0,0,0,0,2,2,1,2,2,2,1,2,1)$
51 | $15496$ | $(0,1,2,2,0,2,0,1,1,0,2,2,2,2,0,1,1,2,1,2,2,2,2,1,2,1,2,0,2,1)$
52 | $15504$ | $(1,0,0,1,2,1,0,1,1,0,2,2,2,2,0,1,1,2,1,0,0,0,0,1,1,1,2,2,1,1)$
53 | $15512$ | $(0,0,1,0,2,2,1,0,0,2,0,0,0,1,0,1,0,2,2,1,0,2,1,0,0,2,1,2,0,0)$
54 | $15520$ | $(1,0,2,0,2,1,2,2,2,1,1,1,2,0,0,1,1,2,2,2,1,2,1,1,0,2,1,2,0,2)$
55 | $15544$ | $(1,0,1,2,2,0,0,1,0,0,2,2,1,1,2,2,2,1,0,1,1,1,2,0,2,1,2,0,0,2)$
56 | $15552$ | $(2,1,2,2,1,0,0,1,2,0,1,0,0,1,0,2,0,0,0,1,0,2,2,1,0,2,2,2,0,1)$
57 | $15576$ | $(2,0,1,1,0,0,2,0,2,0,0,2,0,0,0,2,0,1,1,0,2,1,1,0,0,1,2,0,0,1)$
58 | $15584$ | $(1,1,0,0,2,2,0,1,0,0,2,2,1,1,2,2,2,1,0,2,0,0,0,2,2,2,1,2,2,2)$
59 | $15616$ | $(1,1,0,1,0,2,0,1,0,0,2,2,1,1,2,2,2,1,0,2,0,0,0,2,2,2,1,2,2,2)$
60 | $15680$ | $(1,2,2,2,1,0,0,1,2,0,1,0,0,1,0,2,0,0,0,1,0,2,2,1,0,2,2,2,0,1)$

In summary, we have the following:

######  Proposition 4.6.

  * (1)

There is a ternary near-extremal self-dual code of length $48$ having weight
enumerator $W_{3,48}$ listed in Table 11 for

$\alpha\in\\{48\beta\mid\beta\in\Gamma_{48,1}\\}\cup\\{8\beta\mid\beta\in\Gamma_{48,2}\\},$

where $\Gamma_{48,1}$ and $\Gamma_{48,2}$ are listed in (11) and Table 14,
respectively.

  * (2)

There is a ternary near-extremal self-dual code of length $60$ having weight
enumerator $W_{3,60}$ listed in Table 12 for

$\alpha\in\\{24\beta\mid\beta\in\Gamma_{60,1}\\}\cup\\{8\beta\mid\beta\in\Gamma_{60,2}\\},$

where $\Gamma_{60,1}$ and $\Gamma_{60,2}$ are listed in Tables 13 and 15,
respectively.

###  4.4 Remarks

In this section, we constructed ternary near-extremal self-dual codes having
distinct weight enumerators for lengths $12m$ $(m=3,4,5,6)$ (see Propositions
4.2, 4.4 and 4.6). The results give rise to the following natural question.

######  Question 1.

Determine if there is a ternary near-extremal self-dual code of length $12m$
by constructing or proving the nonexistence for the remaining cases $\alpha$
in the weight enumerators $W_{3,12m}$ when $m=3,4,5,6$.

For the weight enumerator $W_{3,12m}=\sum_{i=0}^{12m}A_{i}y^{i}$ of a ternary
near-extremal self-dual code of length $12m$ $(m=7,8,9,10)$, $A_{i}$ are
listed in Tables 27 and 28. For $m\geq 7$, currently known ternary near-
extremal self-dual codes of length $12m$ are $QR_{84}$, $P_{84}$, $P_{96}$ and
two ternary near-extremal self-dual codes of length $108$ in [7, Table 3] and
[23] (see Table 1).

We verified that $QR_{84}$ and $P_{84}$ have weight enumerators $W_{3,84}$
with

$\alpha=2368488\text{ and }\alpha=1259520,$

respectively. We verified that $P_{96}$ has weight enumerator $W_{3,96}$ with

$\alpha=15358848.$

Due to the computational complexity, it seems infeasible to calculate the
number of codewords of minimum weight of the two codes of length $108$ in [7,
Table 3] and [23].

##  5 Existence of quaternary near-extremal Hermitian self-dual codes of
length $6m$

In this section, we consider the weight enumerators for which there is a
quaternary near-extremal Hermitian self-dual code of length $6m$ for
$m=4,5,6$.

###  5.1 Length 24

The smallest length for which $2m$ is the largest minimum weight among
quaternary Hermitian self-dual codes of length $6m$ is $24$. The smallest
length for which the classification of quaternary (near-extremal) Hermitian
self-dual codes of length $6m$ has not been completed is also $24$. From these
viewpoints, the most interesting length is $24$. Here we consider the weight
enumerators for which there is a quaternary near-extremal Hermitian self-dual
code of length $24$.

For the weight enumerator $W_{4,24}=\sum_{i=0}^{24}A_{i}y^{i}$ of a quaternary
near-extremal Hermitian self-dual code of length $24$, $A_{i}$ are listed in
Table 18.

######  Fact 5.1.

For the weight enumerator $W_{4,24}$, $\alpha=9\beta$ and

$\beta\in\\{1,2,\ldots,253\\}.$

###### Proof.

Follows from Lemma 3.4, $A_{8}>0$ and $A_{10}\geq 0$. ∎

Table 18: Weight enumerator $W_{4,24}$
---
$i$ | $A_{i}$ | $i$ | $A_{i}$ | $i$ | $A_{i}$
$0$ | $1$ | $14$ | $1147608-56\alpha$ | $22$ | $1038312-8\alpha$
$8$ | $\alpha$ | $16$ | $3736557+70\alpha$ | $24$ | $32778+\alpha$
$10$ | $18216-8\alpha$ | $18$ | $6248088-56\alpha$ |  |
$12$ | $156492+28\alpha$ | $20$ | $4399164+28\alpha$ |  |

Quaternary near-extremal Hermitian self-dual codes of length $24$ having
weight enumerators $W_{4,24}$ are known, where

$\displaystyle\alpha=$ $\displaystyle 756\text{
(see~{}\cite[cite]{[\@@bibref{}{CP}{}{}]})},$ $\displaystyle\alpha=$
$\displaystyle 522,1089,594,2277\text{
(see~{}\cite[cite]{[\@@bibref{}{G00}{}{}]})},$ $\displaystyle\alpha=$
$\displaystyle 513,540,549,576,621,630,657,684,702,738,765,792\text{
(see~{}\cite[cite]{[\@@bibref{}{K01}{}{}]})},$ $\displaystyle\alpha=$
$\displaystyle 513,522,594,630,657,684,702,738,756,765,792,837,846,900,$
$\displaystyle 954,981,1089,1197,1242,1413,2277\text{
(see~{}\cite[cite]{[\@@bibref{}{R02}{}{}]})}.$

######  Remark 5.2.

It is claimed in [16] that the code $C_{24,11}$ in [16, Table IV] has weight
enumerator $W_{4,24}$ with $\alpha=627$ ($\beta=114$ in the notation of [16]).
However, this value is incorrect, as it contradicts Lemma 3.4. We verified
that the correct value is $657$ ($\beta=144$ in the notation of [16]).
Moreover, we point out that $\overline{(l_{i}|r_{i})}$ in Theorem 1 of [16] is
${(l_{i}|r_{i})}$.

All quaternary near-extremal Hermitian self-dual double circulant codes of
length $24$ are known [8, Table I]. The code $C_{24,4}$ in [8, Table I] is a
bordered double circulant code with generator matrix of form (6), where the
circulant matrix $A$ has with first row $(1,\omega,1,1,\omega,1,0,0,0,0,0)$.
By considering neighbors, we found two quaternary near-extremal Hermitian
self-dual codes
$N_{24,i}=\operatorname{Nei}_{H}(C_{24,4},y)=\operatorname{Nei}_{H}(C_{24,4},\hat{y})$
$(i=1,2)$ (see (9) for $\operatorname{Nei}_{H}(C,y)$), where

$\displaystyle\hat{y}$
$\displaystyle=(0,0,0,0,1,\omega^{2},1,1,\omega,\omega^{2},1,1),$
$\displaystyle\hat{y}$
$\displaystyle=(1,0,0,0,\omega^{2},\omega^{2},1,\omega,\omega,0,1,1),$

respectively. These codes $N_{24,1}$ and $N_{24,2}$ have weight enumerators
$W_{4,24}$ with $\alpha=864$ and $1026$, respectively.

In summary, we have the following:

######  Proposition 5.3.

There is a quaternary near-extremal Hermitian self-dual code of length $24$
having weight enumerator $W_{4,24}$ listed in Table 18 for

$\displaystyle\alpha\in\left\\{9\beta~{}\middle|~{}\beta\in\left\\{\begin{array}[]{l}57,58,60,61,64,66,69,70,73,76,78,82,84,85,88,\\\
93,94,96,100,106,109,114,121,133,138,157,253\end{array}\right\\}\right\\}.$

###  5.2 Length 30

There is a quaternary extremal Hermitian self-dual code of length $30$ [19]
(see also [8]). Here we consider the weight enumerators for which there is a
quaternary near-extremal Hermitian self-dual code of length $30$.

For the weight enumerator $W_{4,30}=\sum_{i=0}^{30}A_{i}y^{i}$ of a quaternary
near-extremal Hermitian self-dual code of length $30$, $A_{i}$ are listed in
Table 19.

######  Fact 5.4.

For the weight enumerator $W_{4,30}$, $\alpha=9\beta$ and

$\beta\in\\{1,2,\ldots,1319\\}.$

###### Proof.

Follows from Lemma 3.4, $A_{10}>0$ and $A_{12}\geq 0$. ∎

Table 19: Weight enumerator $W_{4,30}$
---
$i$ | $A_{i}$ | $i$ | $A_{i}$ | $i$ | $A_{i}$
$0$ | $1$ | $16$ | $12038625-120\alpha$ | $24$ | $312800670-120\alpha$
$10$ | $\alpha$ | $18$ | $61752600+210\alpha$ | $26$ | $129570840+45\alpha$
$12$ | $118755-10\alpha$ | $20$ | $195945750-252\alpha$ | $28$ | $18581895-10\alpha$
$14$ | $1151010+45\alpha$ | $22$ | $341403660+210\alpha$ | $30$ | $378018+\alpha$

All quaternary extremal Hermitian self-dual double circulant codes of length
$30$ are known [8, Table I]. The code $C_{30}$ in [8, Table I] has the
following generator matrix
$\left(\begin{array}[]{ccccccccc}I_{15}&R\end{array}\right)$, where $R$ is the
circulant matrix with first row
$(1,1,1,\omega,0,\omega,\omega^{2},\omega,0,\omega,1,1,1,0,0)$. By considering
neighbors, we found $19$ quaternary near-extremal Hermitian self-dual codes
$N_{30,i}=\operatorname{Nei}_{H}(C_{30},\hat{y})$ of length $30$, where
$\hat{y}$ and the values $\alpha$ of the weight enumerators are listed in
Table 20 $(i=1,2,\ldots,19)$.

Table 20: Quaternary near-extremal Hermitian self-dual codes $N_{30,i}$
---
$i$ | $C$ | $\alpha$ | $\hat{y}$
${1}$ | $C_{30}$ | $1917$ | $(0,1,0,0,0,\omega^{2},0,1,\omega^{2},1,\omega^{2},\omega^{2},1,\omega^{2},1)$
${2}$ | $C_{30}$ | $2088$ | $(0,1,0,0,0,1,0,\omega,1,\omega,1,1,\omega,1,1)$
${3}$ | $C_{30}$ | $2268$ | $(1,\omega^{2},0,0,0,\omega^{2},1,0,0,\omega^{2},1,1,1,1,1)$
${4}$ | $C_{30}$ | $2349$ | $(1,0,0,0,0,\omega,0,1,\omega,1,1,\omega^{2},\omega,1,1)$
${5}$ | $C_{30}$ | $2430$ | $(1,1,0,0,0,\omega^{2},1,\omega,0,\omega,\omega^{2},\omega,0,\omega,1)$
${6}$ | $C_{30}$ | $2520$ | $(0,0,0,0,0,\omega,1,\omega,1,1,\omega,1,\omega,1,1)$
${7}$ | $C_{30}$ | $2592$ | $(0,1,0,0,0,\omega,\omega,1,1,\omega,\omega,\omega^{2},1,0,1)$
${8}$ | $C_{30}$ | $2673$ | $(1,0,0,0,0,1,1,\omega,\omega,\omega,\omega^{2},\omega,0,\omega,1)$
${9}$ | $C_{30}$ | $2700$ | $(0,1,0,0,0,\omega,\omega,1,1,\omega,\omega,\omega^{2},\omega^{2},0,1)$
${10}$ | $C_{30}$ | $2781$ | $(1,0,0,0,0,1,\omega,1,\omega,\omega^{2},\omega,0,\omega,1,1)$
${11}$ | $C_{30}$ | $2808$ | $(0,0,0,0,0,\omega^{2},\omega,1,\omega^{2},1,\omega^{2},\omega^{2},1,\omega^{2},1)$
${12}$ | $C_{30}$ | $2862$ | $(1,0,0,0,0,1,\omega^{2},\omega,1,1,\omega^{2},\omega^{2},\omega^{2},0,1)$
${13}$ | $C_{30}$ | $2898$ | $(1,0,0,0,0,\omega^{2},\omega,\omega,1,1,\omega,\omega,\omega^{2},0,1)$
${14}$ | $C_{30}$ | $2925$ | $(1,0,0,0,0,\omega,1,\omega,1,1,\omega,1,\omega,0,1)$
${15}$ | $C_{30}$ | $2970$ | $(0,0,0,0,0,\omega,1,\omega,1,1,\omega,1,\omega,\omega,1)$
${16}$ | $C_{30}$ | $3024$ | $(1,0,0,0,0,1,\omega,\omega^{2},\omega^{2},1,\omega,0,1,\omega^{2},1)$
${17}$ | $C_{30}$ | $3060$ | $(0,1,0,0,0,\omega,1,1,1,\omega,\omega,\omega^{2},\omega^{2},0,1)$
${18}$ | $C_{30}$ | $3105$ | $(1,0,0,0,0,1,\omega^{2},\omega,1,\omega^{2},\omega^{2},\omega^{2},\omega^{2},0,1)$
${19}$ | $C_{30}$ | $3168$ | $(1,0,0,0,0,1,\omega,\omega^{2},\omega,\omega^{2},\omega,0,\omega,1,1)$
${20}$ | $D_{30,1}$ | $3213$ | $(1,\omega,\omega,0,1,\omega,0,\omega,0,0,\omega^{2},1,\omega^{2},0,1)$
${21}$ | $D_{30,1}$ | $3240$ | $(1,\omega,1,\omega,1,\omega^{2},0,1,\omega,0,1,\omega^{2},0,\omega^{2},1)$
${22}$ | $N_{30,20}$ | $3186$ | $(1,\omega,1,\omega,1,\omega,0,\omega,1,\omega^{2},1,\omega,1,\omega^{2},1)$
${23}$ | $N_{30,20}$ | $3222$ | $(1,0,1,\omega,\omega,\omega,1,1,1,\omega^{2},1,\omega^{2},\omega^{2},\omega^{2},1)$

By considering $\omega$-circulant codes, we found a quaternary near-extremal
Hermitian self-dual code $D_{30,1}$ with generator matrix of form (7), where
the $\omega$-circulant matrix $A$ has the following first row:

$(0,0,0,\omega,\omega,1,1,1,\omega^{2},\omega,\omega,0,\omega,\omega^{2},\omega^{2}).$

The code $D_{30,1}$ has weight enumerator $W_{4,30}$ with $\alpha=3249$.
Moreover, by considering neighbors of $D_{30,1}$ and its neighbors, we found
four more quaternary near-extremal Hermitian self-dual codes
$N_{30,i}=\operatorname{Nei}_{H}(C,\hat{y})$ of length $30$, where $C$,
$\hat{y}$ and the values $\alpha$ of the weight enumerators are listed in
Table 20 $(i=20,21,22,23)$.

In summary, we have the following:

######  Proposition 5.5.

There is a quaternary near-extremal Hermitian self-dual code of length $30$
having weight enumerator $W_{4,30}$ listed in Table 19 for

$\displaystyle\alpha\in\left\\{9\beta~{}\middle|~{}\beta\in\left\\{\begin{array}[]{l}213,232,252,261,270,280,288,297,\\\
300,309,312,318,322,325,330,336,\\\
340,345,352,354,357,358,360,361\end{array}\right\\}\right\\}.$

###  5.3 Length 36

It is not known whether there is a quaternary extremal Hermitian self-dual
code of length $36$ (see [8]). Here we consider the weight enumerators for
which there is a quaternary near-extremal Hermitian self-dual code of length
$36$.

For the weight enumerator $W_{4,36}=\sum_{i=0}^{36}A_{i}y^{i}$ of a quaternary
near-extremal Hermitian self-dual code of length $36$, $A_{i}$ are listed in
Table 21.

######  Fact 5.6.

For the weight enumerator $W_{4,36}$, $\alpha=9\beta$ and

$\beta\in\\{1,2,\ldots,7140\\}.$

###### Proof.

Follows from Lemma 3.4, $A_{12}>0$ and $A_{14}\geq 0$. ∎

Table 21: Weight enumerator $W_{4,36}$
---
$i$ | $A_{i}$ | $i$ | $A_{i}$
$0$ | $1$ | $24$ | $10270161720+924\alpha$
$12$ | $\alpha$ | $26$ | $18820571616-792\alpha$
$14$ | $771120-12\alpha$ | $28$ | $20136623760+495\alpha$
$16$ | $7846146+66\alpha$ | $30$ | $11676761712-220\alpha$
$18$ | $106611120-220\alpha$ | $32$ | $3175038405+66\alpha$
$20$ | $732024216+495\alpha$ | $34$ | $306134640-12\alpha$
$22$ | $3482598240-792\alpha$ | $36$ | $4334040+\alpha$

There are two inequivalent quaternary near-extremal Hermitian self-dual double
circulant codes of length $36$ [10]. The two codes are denoted by $P_{36,1}$
and $P_{36,2}$ in [10]. The codes $P_{36,1}$ and $P_{36,2}$ in [10] have
weight enumerators $W_{4,36}$ with $\alpha=20844$ and $28764$, respectively
[10]. Recently, quaternary near-extremal Hermitian self-dual codes of length
$36$ having weight enumerator $W_{4,36}$ with $\alpha=19548$ and $22149$ have
been found in [26].

Our extensive search failed to discover a quaternary near-extremal Hermitian
self-dual code of length $36$ having weight enumerator $W_{4,36}$ with
$\alpha\not\in\\{19548,20844,22149,28764\\}$ by considering four-circulant
codes and $\omega$-circulant codes.

In summary, we have the following:

######  Proposition 5.7.

There is a quaternary near-extremal Hermitian self-dual code of length $36$
having weight enumerator $W_{4,36}$ listed in Table 21 for

$\alpha\in\\{19548,20844,22149,28764\\}.$

###  5.4 Remarks

In this section, we constructed quaternary near-extremal Hermitian self-dual
codes having distinct weight enumerators for lengths $6m$ $(m=4,5)$ (see
Propositions 5.3 and 5.5). We also discussed quaternary near-extremal
Hermitian self-dual codes having distinct weight enumerators for length $36$
(see Proposition 5.7). The results give rise to the following natural
question.

######  Question 2.

Determine if there is a quaternary near-extremal Hermitian self-dual code of
length $6m$ by constructing or proving the nonexistence for the remaining
cases $\alpha$ in the weight enumerators $W_{4,6m}$ when $m=4,5,6$.

For the weight enumerator $W_{4,6m}=\sum_{i=0}^{6m}A_{i}y^{i}$ of a quaternary
near-extremal Hermitian self-dual code of length $6m$ ($m=7,8,9,10$), $A_{i}$
are listed in Table 29. As described above, it is currently not known whether
there is a quaternary near-extremal Hermitian self-dual code of length $6m$ or
not $(m=7,8,9,10)$ [7, Table 5] (see Table 2).

Acknowledgments. This work was supported by JSPS KAKENHI Grant Numbers
19H01802 and 21K03350.

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Table 22: Ternary near-extremal self-dual codes $C_{72,i}$ of length $72$
---
$i$ | $\alpha$ | $r_{A}$ | $r_{B}$ | $r_{C}$ | $r_{D}$
1 | $200400$ | $(2,2,0,1,2,0,2,1,0)$ | $(0,2,0,2,1,0,2,0,0)$ | $(2,0,0,0,2,0,2,1,1)$ | $(2,0,2,0,0,0,2,2,2)$
2 | $202344$ | $(2,2,0,1,0,0,0,0,0)$ | $(0,0,1,0,1,2,0,1,2)$ | $(0,1,2,1,2,1,2,0,0)$ | $(2,2,1,0,0,1,2,2,0)$
3 | $202608$ | $(1,2,1,0,0,1,0,1,0)$ | $(0,2,2,1,1,1,2,0,1)$ | $(2,0,0,1,1,1,2,2,2)$ | $(2,2,0,0,1,2,2,2,1)$
4 | $202752$ | $(2,2,0,0,1,0,0,2,2)$ | $(2,2,0,0,1,0,0,2,2)$ | $(0,0,1,2,0,2,0,1,1)$ | $(0,2,2,2,1,0,0,0,1)$
5 | $203040$ | $(2,1,2,2,2,0,0,0,0)$ | $(1,2,1,0,0,0,0,2,2)$ | $(2,1,2,2,0,0,2,1,0)$ | $(2,0,2,2,2,1,0,1,2)$
6 | $203160$ | $(0,0,0,1,1,2,1,1,2)$ | $(1,2,0,1,1,2,2,1,1)$ | $(0,0,0,1,1,1,0,2,1)$ | $(1,0,2,2,1,2,0,2,2)$
7 | $203616$ | $(2,0,1,0,0,0,2,0,1)$ | $(2,1,1,1,1,2,2,0,1)$ | $(1,0,1,0,2,0,1,1,1)$ | $(2,1,0,1,0,1,0,0,2)$
8 | $203664$ | $(0,0,0,0,1,0,1,0,1)$ | $(0,2,0,0,0,0,1,0,1)$ | $(0,1,0,2,2,1,1,2,1)$ | $(2,0,0,0,2,1,0,0,2)$
9 | $204048$ | $(1,2,0,0,2,0,2,1,1)$ | $(0,0,1,2,1,1,2,0,2)$ | $(2,0,2,2,0,1,1,0,0)$ | $(2,0,0,0,2,1,2,2,2)$
10 | $204168$ | $(2,2,0,1,2,1,0,2,1)$ | $(1,0,2,2,0,1,1,2,1)$ | $(2,2,2,2,2,1,0,2,0)$ | $(2,1,2,2,1,2,2,2,0)$
11 | $204648$ | $(0,0,2,2,0,1,0,0,1)$ | $(0,2,0,2,0,0,1,2,2)$ | $(1,2,1,0,0,0,0,1,2)$ | $(0,2,1,1,2,0,1,2,0)$
12 | $204696$ | $(0,2,0,2,0,1,0,2,1)$ | $(0,1,1,2,2,1,1,1,0)$ | $(2,0,2,0,2,2,0,0,1)$ | $(1,1,0,2,2,0,2,2,0)$
13 | $204816$ | $(2,2,0,0,0,0,2,1,0)$ | $(1,2,0,1,0,2,2,0,0)$ | $(1,1,2,0,0,2,0,0,0)$ | $(2,2,0,0,0,2,0,0,2)$
14 | $205104$ | $(2,1,2,0,2,2,0,2,2)$ | $(0,0,2,1,1,0,2,0,2)$ | $(2,1,0,0,1,0,0,1,1)$ | $(2,1,1,1,1,2,1,1,1)$
15 | $205248$ | $(1,1,0,0,2,2,1,1,1)$ | $(0,1,0,2,0,1,1,0,0)$ | $(2,2,2,1,1,1,0,0,0)$ | $(0,2,1,1,2,0,2,2,0)$
16 | $205392$ | $(2,2,1,0,1,1,2,1,1)$ | $(1,1,1,1,2,1,1,0,1)$ | $(2,2,1,1,2,0,2,2,0)$ | $(0,2,1,2,2,0,0,1,1)$
17 | $205752$ | $(0,0,2,2,1,0,1,2,0)$ | $(1,1,0,2,0,1,1,0,0)$ | $(2,2,2,1,1,0,0,1,2)$ | $(2,0,1,0,1,1,2,0,2)$
18 | $205848$ | $(0,0,0,2,1,2,2,1,1)$ | $(2,2,0,1,1,1,0,1,1)$ | $(1,0,1,1,0,2,2,0,2)$ | $(0,0,0,2,0,1,2,0,1)$
19 | $205992$ | $(2,0,1,0,1,2,2,2,2)$ | $(2,0,1,2,2,1,1,0,1)$ | $(0,2,1,2,0,2,2,1,1)$ | $(1,0,2,1,2,2,1,2,2)$
20 | $206256$ | $(0,2,0,0,0,0,2,1,1)$ | $(0,1,0,1,1,2,0,0,0)$ | $(1,2,2,1,1,2,1,0,1)$ | $(1,1,0,2,0,1,1,1,1)$
Table 23: Ternary near-extremal self-dual codes $C_{72,i}$ of length $72$
---
$i$ | $\alpha$ | $r_{A}$ | $r_{B}$ | $r_{C}$ | $r_{D}$
21 | $217848$ | $(0,2,2,0,2,2,1,2,1)$ | $(1,2,0,0,2,0,0,2,2)$ | $(2,0,1,0,0,1,1,1,2)$ | $(1,2,1,2,1,1,1,2,0)$
22 | $218496$ | $(2,2,2,2,1,1,0,2,2)$ | $(0,1,2,0,0,1,0,1,0)$ | $(0,0,2,2,2,2,1,0,1)$ | $(0,0,0,1,2,0,2,1,1)$
23 | $218640$ | $(0,0,1,2,0,2,0,1,1)$ | $(2,1,0,1,1,2,0,0,0)$ | $(1,2,2,1,1,0,0,1,2)$ | $(0,1,1,1,1,1,0,0,1)$
24 | $218928$ | $(2,2,1,2,1,2,1,1,0)$ | $(2,2,0,1,1,1,0,1,1)$ | $(1,0,1,1,0,1,1,2,1)$ | $(2,0,2,1,2,1,1,0,1)$
25 | $219336$ | $(2,2,1,1,1,2,0,2,0)$ | $(1,0,1,2,0,0,2,1,1)$ | $(0,2,0,2,0,1,0,1,0)$ | $(1,2,2,2,1,2,1,2,1)$
26 | $219480$ | $(1,0,1,2,0,2,2,0,2)$ | $(0,0,1,1,0,0,0,0,0)$ | $(2,0,0,2,2,2,2,2,0)$ | $(1,2,0,0,1,1,0,1,1)$
27 | $219552$ | $(0,0,0,2,0,1,2,1,0)$ | $(1,0,0,2,1,2,0,0,0)$ | $(0,2,2,1,2,2,1,0,0)$ | $(1,1,0,0,0,1,2,2,1)$
28 | $219648$ | $(1,0,2,0,1,1,0,1,2)$ | $(1,0,1,2,0,0,2,1,1)$ | $(0,2,0,2,0,1,2,0,0)$ | $(0,0,0,2,1,0,2,0,2)$
29 | $219720$ | $(2,1,1,1,2,1,1,2,2)$ | $(2,2,2,0,2,1,2,2,2)$ | $(2,1,2,2,2,0,2,2,0)$ | $(0,2,0,2,0,2,1,0,1)$
30 | $220224$ | $(1,2,2,2,2,1,0,2,2)$ | $(2,0,1,2,2,2,2,1,1)$ | $(1,0,1,2,1,1,1,1,2)$ | $(1,2,2,1,1,2,2,0,1)$
31 | $220272$ | $(2,0,0,1,0,2,2,2,2)$ | $(0,0,1,0,0,1,1,1,0)$ | $(0,2,1,0,1,2,2,1,0)$ | $(0,0,2,1,0,0,1,2,0)$
32 | $221208$ | $(1,0,2,2,1,1,1,1,2)$ | $(2,0,2,2,0,0,0,0,1)$ | $(1,2,1,1,1,0,1,0,1)$ | $(0,2,1,2,2,0,2,1,1)$
33 | $221232$ | $(0,0,1,0,2,2,1,1,0)$ | $(0,1,2,0,0,2,0,2,2)$ | $(2,1,2,2,0,2,2,2,1)$ | $(2,1,1,1,0,1,1,2,1)$
34 | $221568$ | $(1,2,1,1,1,2,0,2,2)$ | $(1,2,2,0,2,0,2,2,0)$ | $(0,0,2,2,2,0,0,1,0)$ | $(0,0,1,0,2,2,1,2,0)$
35 | $221616$ | $(1,2,2,2,1,1,0,2,1)$ | $(1,2,2,0,0,1,1,1,1)$ | $(2,0,1,2,1,0,1,1,1)$ | $(0,2,2,2,2,0,2,1,2)$
36 | $222528$ | $(0,0,0,1,1,0,2,0,1)$ | $(0,1,0,2,0,1,1,0,0)$ | $(2,2,2,1,1,0,2,0,0)$ | $(2,2,1,1,1,2,1,1,2)$
37 | $222552$ | $(0,2,2,0,2,2,0,0,1)$ | $(2,2,0,1,1,1,0,1,1)$ | $(1,0,1,1,0,0,0,0,2)$ | $(2,0,1,0,2,0,0,2,0)$
38 | $223200$ | $(2,1,1,2,2,1,1,1,1)$ | $(0,1,0,1,1,2,0,0,0)$ | $(1,2,2,1,1,0,0,0,2)$ | $(0,2,1,0,0,2,0,2,0)$
39 | $226056$ | $(0,1,1,2,2,0,1,0,1)$ | $(2,1,0,0,2,1,1,1,1)$ | $(0,0,0,1,0,0,0,0,2)$ | $(2,2,2,1,0,0,0,0,2)$
40 | $229008$ | $(1,1,1,0,1,0,1,1,1)$ | $(2,2,2,0,2,0,2,2,2)$ | $(2,1,1,2,2,2,1,1,2)$ | $(1,1,1,1,1,1,1,1,1)$
Table 24: Ternary near-extremal self-dual codes $C_{72,i}$ of length $72$
---
$i$ | $\alpha$ | $r_{A}$
41 | 204400 | $(2,1,1,1,1,2,0,1,1,1,2,0,2,0,1,1,0,2,2,1,1,2,0,1,2,0,1,0,0,0,0,0,1,0,0)$
42 | 206360 | $(0,0,0,1,0,2,2,2,2,2,2,1,1,1,2,1,1,2,2,1,2,1,0,0,1,1,0,0,1,0,2,0,1,1,1)$
43 | 207760 | $(1,1,0,0,0,2,2,1,1,2,2,1,2,1,2,2,0,0,1,0,1,2,2,2,1,1,1,2,1,0,0,1,0,1,0)$
44 | 208040 | $(1,1,0,1,1,2,0,0,0,0,1,0,2,0,1,2,1,2,2,0,0,0,1,1,1,1,0,1,1,0,1,2,0,2,2)$
45 | 208600 | $(1,0,2,2,0,0,2,0,1,0,2,0,0,1,2,1,0,2,0,2,1,0,0,1,0,1,1,1,0,1,0,1,0,0,2)$
46 | 208880 | $(0,0,1,2,0,1,1,0,2,0,1,2,2,2,0,2,0,0,0,1,2,0,2,1,1,1,2,1,0,0,1,1,2,0,2)$
47 | 209720 | $(2,0,2,1,0,1,0,0,1,0,1,1,0,2,0,0,2,2,0,0,2,2,0,0,2,1,0,1,2,2,1,2,1,1,1)$
48 | 210280 | $(2,2,2,1,1,0,0,2,2,1,0,0,2,1,0,1,0,0,2,2,2,0,2,0,0,2,2,1,2,2,2,1,1,2,2)$
49 | 210560 | $(2,0,0,1,2,1,2,0,1,0,2,0,2,1,2,2,0,1,1,2,0,0,0,0,0,1,2,1,2,0,0,0,0,2,0)$
50 | 211120 | $(2,2,1,0,0,1,2,0,0,0,2,1,0,0,2,1,0,1,0,0,0,0,2,1,0,2,1,0,1,1,0,2,0,1,1)$
51 | 211400 | $(1,0,1,2,0,1,0,1,1,0,0,0,0,0,1,1,1,2,2,2,2,0,2,2,2,2,0,1,0,2,2,1,0,0,1)$
52 | 211960 | $(2,2,0,2,2,1,2,1,0,0,1,1,0,2,0,0,0,0,1,2,0,2,2,1,0,1,0,1,2,0,2,0,1,1,1)$
53 | 212240 | $(1,0,0,1,2,2,2,0,1,0,1,1,2,0,1,0,0,2,2,2,1,0,2,2,1,0,2,0,1,1,2,1,1,1,1)$
54 | 212800 | $(2,2,2,2,1,0,0,2,2,0,0,0,0,0,2,0,0,2,0,0,1,0,2,0,1,1,0,1,2,2,1,0,1,0,1)$
55 | 213080 | $(1,0,1,2,2,1,1,1,1,1,1,1,1,2,2,0,1,2,2,1,2,0,2,1,2,2,2,1,2,0,0,2,0,2,0)$
56 | 213640 | $(0,2,0,0,0,2,2,2,1,1,2,2,0,2,0,1,2,2,1,2,2,2,2,1,2,1,2,2,1,2,2,0,2,2,1)$
57 | 213920 | $(1,2,2,1,2,1,1,0,2,2,0,1,2,2,0,0,2,2,2,1,0,0,0,2,2,0,1,0,2,2,2,1,2,2,0)$
58 | 214480 | $(2,2,2,1,1,1,2,2,0,0,1,1,2,2,1,2,2,0,1,1,1,2,1,1,0,0,0,0,1,1,1,1,2,1,1)$
59 | 214760 | $(2,2,0,2,2,0,1,2,1,0,0,1,2,2,2,2,2,0,1,2,0,2,0,0,1,0,0,2,2,2,1,1,2,2,1)$
60 | 215600 | $(1,1,2,0,2,1,1,1,2,1,2,1,2,0,2,2,2,0,2,0,2,0,0,0,2,1,2,2,1,0,1,2,0,0,1)$
61 | 216160 | $(0,1,2,0,1,0,2,2,2,1,0,1,0,2,1,0,1,1,0,1,1,2,0,0,2,1,1,1,1,0,2,2,2,1,2)$
62 | 217280 | $(2,2,1,2,0,2,2,2,1,1,2,0,1,2,0,2,0,1,0,1,0,0,1,2,1,2,0,2,1,0,1,0,1,2,2)$
63 | 218120 | $(2,2,2,1,1,1,1,0,1,1,0,2,0,0,1,2,2,0,1,1,0,1,0,1,0,2,0,0,2,2,1,2,1,1,2)$
64 | 220640 | $(0,1,2,0,0,0,2,1,0,1,2,1,2,1,2,2,0,2,0,2,2,2,2,1,1,2,1,1,0,1,2,1,2,0,0)$
65 | 222040 | $(0,1,2,2,0,2,1,0,2,0,1,1,1,1,0,0,2,0,2,1,0,2,1,2,0,2,0,0,0,2,0,0,0,2,0)$
Table 25: Ternary near-extremal self-dual codes $N_{36,i}$ of length $36$
---
$i$ | $C$ | $\alpha$ | $\hat{x}$ | $i$ | $C$ | $\alpha$ | $\hat{x}$
${1}$ | $P_{36}$ | $128$ | $(0,0,2,0,0,0,1,2,2,1,2,1,1,2,2,1,0,1)$ | ${27}$ | $C_{36,2}$ | $400$ | $(2,1,0,0,0,0,1,0,0,0,1,1,1,2,1,0,1,0)$
${2}$ | $P_{36}$ | $152$ | $(2,1,1,0,0,0,2,0,1,1,2,1,1,0,0,0,0,0)$ | ${28}$ | $C_{36,3}$ | $416$ | $(1,0,0,0,0,0,2,1,0,0,2,1,1,1,1,2,0,0)$
${3}$ | $P_{36}$ | $160$ | $(2,2,1,0,0,0,0,1,1,1,0,2,1,1,0,0,0,0)$ | ${29}$ | $C_{36,3}$ | $424$ | $(1,0,0,1,0,0,2,1,2,1,2,1,1,2,1,0,2,0)$
${4}$ | $P_{36}$ | $176$ | $(0,2,1,0,0,0,2,0,1,1,2,1,1,1,0,0,0,0)$ | ${30}$ | $C_{36,5}$ | $440$ | $(2,1,0,0,0,0,1,2,1,0,0,2,1,0,2,1,0,0)$
${5}$ | $P_{36}$ | $184$ | $(0,0,1,0,0,0,1,1,1,1,1,1,1,1,0,0,0,0)$ | ${31}$ | $C_{36,5}$ | $448$ | $(0,1,0,0,0,0,1,2,2,0,0,0,0,2,1,2,2,2)$
${6}$ | $P_{36}$ | $192$ | $(0,1,1,0,0,0,2,2,1,1,0,1,1,1,0,0,0,0)$ | ${32}$ | $C_{36,6}$ | $464$ | $(0,0,1,0,0,0,2,2,2,1,0,1,0,0,2,0,1,1)$
${7}$ | $P_{36}$ | $200$ | $(0,2,0,0,0,0,1,2,2,2,1,1,1,1,0,0,0,0)$ | ${33}$ | $C_{36,7}$ | $472$ | $(0,1,0,0,0,0,2,1,0,1,2,0,2,1,2,0,0,1)$
${8}$ | $P_{36}$ | $208$ | $(2,0,1,0,0,0,2,2,2,2,1,1,1,0,0,0,0,0)$ | ${34}$ | $C_{36,7}$ | $480$ | $(2,2,1,0,0,0,1,0,0,0,1,1,2,0,1,0,0,1)$
${9}$ | $P_{36}$ | $224$ | $(2,1,0,0,0,0,2,2,1,1,1,1,1,0,0,0,0,0)$ | ${35}$ | $C_{36,8}$ | $488$ | $(0,1,0,0,0,0,2,0,1,1,0,2,2,1,1,0,0,2)$
${10}$ | $P_{36}$ | $232$ | $(1,0,0,0,0,0,2,2,1,1,1,1,1,1,0,0,0,0)$ | ${36}$ | $C_{36,8}$ | $496$ | $(2,0,1,0,0,0,1,1,0,2,0,2,2,2,2,2,2,2)$
${11}$ | $P_{36}$ | $248$ | $(1,2,0,0,0,0,2,1,1,1,1,1,1,0,0,0,0,0)$ | ${37}$ | $C_{36,8}$ | $512$ | $(2,0,0,1,0,0,2,0,0,1,2,2,2,1,1,1,2,2)$
${12}$ | $P_{36}$ | $256$ | $(0,0,0,0,0,0,1,2,1,1,1,1,1,1,1,0,0,0)$ | ${38}$ | $C_{36,12}$ | $520$ | $(1,1,0,0,0,0,2,1,0,1,2,0,1,2,0,1,0,0)$
${13}$ | $P_{36}$ | $264$ | $(2,0,0,0,0,0,1,2,1,1,1,1,1,1,0,0,0,0)$ | ${39}$ | $C_{36,12}$ | $536$ | $(2,0,2,0,0,0,2,0,0,0,2,0,2,0,1,2,2,1)$
${14}$ | $P_{36}$ | $272$ | $(0,1,0,0,0,0,1,2,1,1,1,1,1,1,0,0,0,0)$ | ${40}$ | $C_{36,13}$ | $552$ | $(0,1,1,0,0,0,2,2,2,2,1,0,2,2,2,0,2,1)$
${15}$ | $P_{36}$ | $280$ | $(1,1,0,0,0,0,1,2,2,2,1,1,1,0,0,0,0,0)$ | ${41}$ | $C_{36,14}$ | $560$ | $(2,2,2,0,1,0,2,2,1,2,2,2,2,2,1,0,1,1)$
${16}$ | $P_{36}$ | $296$ | $(1,0,1,0,0,0,2,1,1,2,2,1,2,0,0,0,0,0)$ | ${42}$ | $C_{36,15}$ | $568$ | $(0,2,2,0,0,0,1,2,0,2,1,0,1,1,1,2,2,1)$
${17}$ | $P_{36}$ | $304$ | $(2,2,0,0,0,0,1,2,1,1,1,1,1,0,0,0,0,0)$ | ${43}$ | $C_{36,16}$ | $584$ | $(0,1,1,0,0,0,1,0,2,1,0,1,0,2,0,0,1,1)$
${18}$ | $P_{36}$ | $320$ | $(1,1,1,0,0,0,1,0,2,1,2,0,2,1,0,0,0,0)$ | ${44}$ | $C_{36,16}$ | $592$ | $(2,2,0,0,0,0,2,1,0,0,2,0,0,2,0,2,2,1)$
${19}$ | $P_{36}$ | $328$ | $(1,2,1,0,0,0,1,2,2,1,2,2,2,1,1,0,0,0)$ | ${45}$ | $C_{36,18}$ | $608$ | $(2,0,0,0,0,0,2,2,1,0,1,2,0,0,0,2,1,1)$
${20}$ | $C_{36,1}$ | $112$ | $(2,2,0,0,0,0,1,0,1,2,1,1,1,0,2,1,2,1)$ | ${46}$ | $C_{36,18}$ | $616$ | $(1,0,0,0,0,0,2,2,2,1,2,1,0,0,0,0,2,1)$
${21}$ | $C_{36,1}$ | $336$ | $(0,0,0,0,0,0,2,2,2,2,2,2,1,1,1,0,0,0)$ | ${47}$ | $C_{36,18}$ | $624$ | $(1,1,1,0,0,0,0,2,1,2,1,1,0,1,0,0,0,0)$
${22}$ | $C_{36,1}$ | $344$ | $(1,0,0,0,0,0,1,2,1,1,1,2,2,1,0,0,0,0)$ | ${48}$ | $C_{36,18}$ | $632$ | $(1,1,0,0,0,0,2,1,2,2,1,1,0,0,0,0,0,1)$
${23}$ | $C_{36,1}$ | $352$ | $(2,0,0,0,0,0,1,0,2,2,1,2,1,2,1,0,0,0)$ | ${49}$ | $C_{36,18}$ | $664$ | $(1,2,0,0,0,0,2,1,0,2,1,1,2,0,0,0,0,1)$
${24}$ | $C_{36,1}$ | $368$ | $(1,1,0,0,0,0,0,1,1,1,1,1,2,0,2,0,0,0)$ | ${50}$ | $C_{36,19}$ | $640$ | $(1,2,0,0,0,0,2,1,2,0,2,2,2,0,0,0,0,1)$
${25}$ | $C_{36,1}$ | $376$ | $(2,1,0,0,0,0,2,1,2,2,0,1,2,0,1,0,0,0)$ | ${51}$ | $N_{36,46}$ | $656$ | $(0,0,0,0,1,0,2,2,2,2,1,0,0,0,0,2,2,2)$
${26}$ | $C_{36,1}$ | $392$ | $(0,1,0,0,0,0,1,2,0,1,0,1,2,1,1,0,1,0)$ | ${52}$ | $N_{36,47}$ | $680$ | $(2,1,0,2,1,1,0,1,0,1,0,1,1,2,0,2,1,0)$
Table 26: Ternary near-extremal self-dual codes $N_{48,i}$ of length $48$
---
$i$ | $\alpha$ | $\hat{x}$ | $i$ | $\alpha$ | $\hat{x}$
1 | $1440$ | $(1,1,2,0,1,2,0,2,1,2,2,2,0,1,0,0,2,0,1,1,1,0,0,0)$ | 32 | $1952$ | $(0,2,2,1,1,2,0,2,1,1,2,2,1,1,0,0,0,0,0,0,0,0,0,0)$
2 | $1448$ | $(1,2,2,0,1,2,0,2,1,2,0,1,2,2,1,2,1,0,2,2,1,0,1,0)$ | 33 | $1960$ | $(0,0,2,1,1,2,0,2,1,1,2,2,0,1,1,1,0,0,0,0,0,0,0,0)$
3 | $1456$ | $(2,0,1,2,1,2,0,2,1,0,2,1,0,1,2,1,2,0,0,1,0,0,1,0)$ | 34 | $1976$ | $(2,2,2,1,1,2,0,2,1,2,1,2,1,0,0,0,0,0,0,0,0,0,0,0)$
4 | $1472$ | $(1,0,0,2,1,2,0,2,1,1,0,2,1,1,0,1,2,0,0,0,0,0,0,0)$ | 35 | $1984$ | $(0,2,0,1,1,2,0,2,1,1,2,1,1,2,1,0,0,0,0,0,0,0,0,0)$
5 | $1504$ | $(1,0,2,1,1,0,0,1,0,0,2,1,0,0,2,0,0,2,0,1,0,1,0,2)$ | 36 | $1992$ | $(1,0,0,1,1,2,0,2,1,2,1,2,2,0,1,1,0,0,0,0,0,0,0,0)$
6 | $1680$ | $(2,2,2,0,1,2,0,2,1,1,1,2,0,2,2,1,2,1,2,1,2,2,1,1)$ | 37 | $2000$ | $(1,2,0,1,1,2,0,2,1,1,0,1,1,2,1,0,0,0,0,0,0,0,0,0)$
7 | $1696$ | $(0,0,1,2,1,2,0,2,1,2,2,2,2,2,1,0,1,1,0,0,0,0,1,0)$ | 38 | $2008$ | $(1,2,2,1,1,2,0,2,1,1,0,1,2,1,0,0,0,0,0,0,0,0,0,0)$
8 | $1720$ | $(0,2,2,0,1,2,0,2,1,0,1,0,1,2,2,2,2,1,2,2,0,0,0,0)$ | 39 | $2024$ | $(2,0,2,1,1,2,0,2,1,1,1,2,2,1,0,0,0,0,0,0,0,0,0,0)$
9 | $1736$ | $(1,2,1,2,1,2,0,2,1,1,1,2,1,1,2,2,1,0,1,1,0,0,0,0)$ | 40 | $2032$ | $(2,1,1,1,1,2,0,2,1,2,1,1,2,0,0,0,0,0,0,0,0,0,0,0)$
10 | $1744$ | $(1,2,0,2,1,2,0,2,1,1,0,2,1,1,2,1,2,1,0,0,0,0,0,0)$ | 41 | $2040$ | $(2,2,1,1,1,2,0,2,1,0,2,0,2,2,1,0,0,0,0,0,0,0,0,0)$
11 | $1752$ | $(0,1,2,0,1,2,0,2,1,0,0,1,0,1,2,2,2,1,2,1,0,1,0,0)$ | 42 | $2048$ | $(1,1,2,2,1,2,0,2,1,1,1,0,1,0,2,0,0,0,0,0,0,0,0,0)$
12 | $1760$ | $(2,1,1,2,1,2,0,2,1,0,0,0,2,0,1,0,2,0,1,0,0,0,0,0)$ | 43 | $2056$ | $(2,1,2,1,1,2,0,2,1,1,1,1,2,0,0,0,0,0,0,0,0,0,0,0)$
13 | $1768$ | $(0,0,2,2,1,2,0,2,1,2,0,2,2,2,1,0,0,1,0,0,0,0,0,0)$ | 44 | $2072$ | $(1,0,2,1,1,2,0,2,1,1,2,1,1,1,0,0,0,0,0,0,0,0,0,0)$
14 | $1784$ | $(2,0,0,2,1,2,0,2,1,0,2,1,2,0,1,2,2,2,1,2,0,0,0,0)$ | 45 | $2080$ | $(1,1,1,1,1,2,0,2,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0)$
15 | $1792$ | $(1,1,1,2,1,2,0,2,1,1,0,2,1,1,0,1,2,1,0,0,0,0,0,0)$ | 46 | $2088$ | $(0,0,0,1,1,2,0,2,1,0,2,2,1,2,2,1,1,0,0,0,0,0,0,0)$
16 | $1800$ | $(0,0,1,1,1,2,0,2,1,0,2,1,1,1,1,1,0,0,0,0,0,0,0,0)$ | 47 | $2096$ | $(1,1,2,1,1,2,0,2,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0)$
17 | $1808$ | $(2,1,0,2,1,2,0,2,1,2,1,1,0,2,0,1,0,0,0,0,0,0,0,0)$ | 48 | $2104$ | $(2,0,1,1,1,2,0,2,1,0,2,0,1,2,1,2,0,0,0,0,0,0,0,0)$
18 | $1816$ | $(1,1,0,2,1,2,0,2,1,2,2,1,0,0,0,1,0,0,1,0,0,0,0,0)$ | 49 | $2120$ | $(0,1,2,2,1,2,0,2,1,1,0,2,2,0,1,1,0,0,0,0,0,0,0,0)$
19 | $1832$ | $(0,2,2,2,1,2,0,2,1,0,0,2,1,1,1,0,0,2,0,0,0,0,0,0)$ | 50 | $2128$ | $(2,2,2,2,1,2,0,2,1,1,1,1,0,2,2,0,2,1,0,0,0,0,0,0)$
20 | $1840$ | $(1,0,2,2,1,2,0,2,1,2,0,2,2,1,0,0,0,1,0,0,0,0,0,0)$ | 51 | $2136$ | $(1,0,1,1,1,2,0,2,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0)$
21 | $1848$ | $(1,2,2,2,1,2,0,2,1,2,2,2,2,2,2,0,1,0,0,0,0,0,0,0)$ | 52 | $2144$ | $(0,1,0,2,1,2,0,2,1,0,2,1,1,2,1,2,2,1,0,1,0,0,0,0)$
22 | $1856$ | $(0,2,1,1,1,2,0,2,1,2,0,1,1,1,1,0,0,0,0,0,0,0,0,0)$ | 53 | $2152$ | $(2,0,2,2,1,2,0,2,1,1,0,0,1,2,0,0,1,2,0,0,0,0,0,0)$
23 | $1864$ | $(0,1,2,1,1,2,0,2,1,1,0,1,1,2,1,0,0,0,0,0,0,0,0,0)$ | 54 | $2168$ | $(2,2,0,2,1,2,0,2,1,0,0,2,1,0,0,1,0,2,2,0,0,0,0,0)$
24 | $1880$ | $(0,1,0,1,1,2,0,2,1,0,1,2,2,2,1,1,0,0,0,0,0,0,0,0)$ | 55 | $2176$ | $(0,2,1,2,1,2,0,2,1,2,0,1,2,0,0,2,1,1,1,2,0,0,0,0)$
25 | $1888$ | $(2,1,2,2,1,2,0,2,1,2,0,2,2,2,1,2,1,0,0,0,0,0,0,0)$ | 56 | $2184$ | $(0,1,1,2,1,2,0,2,1,2,1,2,2,0,0,2,0,0,1,1,1,0,0,0)$
26 | $1896$ | $(1,2,1,1,1,2,0,2,1,2,1,0,2,0,1,0,0,0,0,0,0,0,0,0)$ | 57 | $2192$ | $(2,2,1,2,1,2,0,2,1,2,2,2,0,2,2,1,0,1,1,1,2,0,0,0)$
27 | $1904$ | $(0,1,1,1,1,2,0,2,1,1,2,1,0,2,1,0,0,0,0,0,0,0,0,0)$ | 58 | $2200$ | $(1,0,1,2,1,2,0,2,1,2,1,0,1,1,0,2,1,0,0,0,1,2,0,0)$
28 | $1912$ | $(2,0,0,1,1,2,0,2,1,1,0,2,1,1,1,1,0,0,0,0,0,0,0,0)$ | 59 | $2216$ | $(0,2,0,2,1,2,0,2,1,1,2,1,2,1,0,0,2,1,1,2,0,0,0,0)$
29 | $1928$ | $(2,2,0,1,1,2,0,2,1,0,1,1,2,2,2,0,0,0,0,0,0,0,0,0)$ | 60 | $2224$ | $(0,0,0,2,1,2,0,2,1,2,2,0,0,1,2,2,1,1,0,0,0,0,0,0)$
30 | $1936$ | $(2,1,0,1,1,2,0,2,1,2,2,1,1,0,1,0,0,0,0,0,0,0,0,0)$ | 61 | $2248$ | $(2,1,2,0,1,2,0,2,1,1,0,2,1,0,1,2,2,2,2,1,1,2,0,0)$
31 | $1944$ | $(1,1,0,1,1,2,0,2,1,1,2,2,2,1,0,0,0,0,0,0,0,0,0,0)$ |  |  |
Table 27: Weight enumerators $W_{3,84}$ and $W_{3,96}$
---
$i$ | $A_{i}$ | $i$ | $A_{i}$
$0$ | $1$ | $0$ | $1$
$21$ | $\alpha$ | $27$ | $\alpha$
24 | $334446840-21\alpha$ | $27$ | $3082778880-24\alpha$
27 | $27251986272+210\alpha$ | $30$ | $272857821696+276\alpha$
30 | $1595856832416-1330\alpha$ | $33$ | $18642386018880-2024\alpha$
33 | $57941510353272+5985\alpha$ | $36$ | $827849897536896+10626\alpha$
36 | $1352260881447840-20349\alpha$ | $39$ | $24804181974320640-42504\alpha$
39 | $20473083660798720+54264\alpha$ | $42$ | $505747055590698240+134596\alpha$
42 | $202449354096116160-116280\alpha$ | $45$ | $7072369265609436672-346104\alpha$
45 | $1310283470243301120+203490\alpha$ | $48$ | $68123007669831753600+735471\alpha$
48 | $5538707497101706200-293930\alpha$ | $51$ | $452631286236320964864-1307504\alpha$
51 | $15191881032059825184+352716\alpha$ | $54$ | $2071548949601690296320+1961256\alpha$
54 | $26733400854085789248-352716\alpha$ | $57$ | $6502086811906783649280-2496144\alpha$
57 | $29675352547018793664+293930\alpha$ | $60$ | $13891892631756684300288+2704156\alpha$
60 | $20292322648045462560-203490\alpha$ | $63$ | $19981992697305182069760-2496144\alpha$
63 | $8274112209487238400+116280\alpha$ | $66$ | $19059746794624115235648+1961256\alpha$
66 | $1923875932140643320-54264\alpha$ | $69$ | $11815486097051591516160-1307504\alpha$
69 | $239704140659316480+20349\alpha$ | $72$ | $4635854680301645299200+735471\alpha$
72 | $14629846917558600-5985\alpha$ | $75$ | $1111643979567788848896-346104\alpha$
75 | $381315149927712+1330\alpha$ | $78$ | $155474678628222036480+134596\alpha$
78 | $3368733804576-210\alpha$ | $81$ | $11895672707972542592-42504\alpha$
81 | $6274641968+21\alpha$ | $84$ | $454433325623896320+10626\alpha$
84 | $2368656-\alpha$ | $87$ | $7545707021698560-2024\alpha$
|  | $90$ | $43157012131584+276\alpha$
|  | $93$ | $53638295040-24\alpha$
|  | $96$ | $-13283136+\alpha$
Table 28: Weight enumerators $W_{3,108}$ and $W_{3,120}$
---
$i$ | $A_{i}$ | $i$ | $A_{i}$
$0$ | $1$ | $0$ | $1$
$27$ | $\alpha$ | $30$ | $\alpha$
$30$ | $28403815968-27\alpha$ | $33$ | $261792869520-30\alpha$
$33$ | $2673811124712+351\alpha$ | $36$ | $25817030152160+435\alpha$
$36$ | $206590654211760-2925\alpha$ | $39$ | $2206243011598080-4060\alpha$
$39$ | $10760877549676800+17550\alpha$ | $42$ | $130825652486394240+27405\alpha$
$42$ | $392999730265933920-80730\alpha$ | $45$ | $5612809762417673088-142506\alpha$
$45$ | $10138440079202647680+296010\alpha$ | $48$ | $175296412571121223200+593775\alpha$
$48$ | $186220899739005047280-888030\alpha$ | $51$ | $4016223213579257466240-2035800\alpha$
$51$ | $2448009164124666552384+2220075\alpha$ | $54$ | $67868355146310115461120+5852925\alpha$
$54$ | $23102325452246741611680-4686825\alpha$ | $57$ | $849119988307524499949760-14307150\alpha$
$57$ | $156672606424451974380576+8436285\alpha$ | $60$ | $7882969705335235952872704+30045015\alpha$
$60$ | $762760272369004823178720-13037895\alpha$ | $63$ | $54343678095173262461802240-54627300\alpha$
$63$ | $2657742513817704808435200+17383860\alpha$ | $66$ | $277988814357219566717271120+86493225\alpha$
$66$ | $6593245862016933524881800-20058300\alpha$ | $69$ | $1052828118567489363251967360-119759850\alpha$
$69$ | $11557119125796366858624000+20058300\alpha$ | $72$ | $2940998734258869894606360000+145422675\alpha$
$72$ | $14167741347435074934974400-17383860\alpha$ | $75$ | $6026510372574451248316638720-155117520\alpha$
$75$ | $11984618814237321888400128+13037895\alpha$ | $78$ | $8992710676827639928253905920+145422675\alpha$
$78$ | $6876079739002410705532512-8436285\alpha$ | $81$ | $9679917353459122424429865440-119759850\alpha$
$81$ | $2617616851829250108509168+4686825\alpha$ | $84$ | $7427460198130322024755753920+86493225\alpha$
$84$ | $642838612913585065972800-2220075\alpha$ | $87$ | $4002608863954458573348798720-54627300\alpha$
$87$ | $98201262285146563994880+888030\alpha$ | $90$ | $1487111607797231811094207872+30045015\alpha$
$90$ | $8893951642438004207520-296010\alpha$ | $93$ | $372219109982756445972447360-14307150\alpha$
$93$ | $447418538339167578240+80730\alpha$ | $96$ | $60959736643469146369614000+5852925\alpha$
$96$ | $11398404115890038520-17550\alpha$ | $99$ | $6293059289046900207864000-2035800\alpha$
$99$ | $127900763897680800+2925\alpha$ | $102$ | $389971756726008705480960+593775\alpha$
$102$ | $500637943155168-351\alpha$ | $105$ | $13580154047777800897728-142506\alpha$
$105$ | $423068395680+27\alpha$ | $108$ | $242127348895859481600+27405\alpha$
$108$ | $161827872-\alpha$ | $111$ | $1921175681643429120-4060\alpha$
|  | $114$ | $5368260223959360+435\alpha$
|  | $117$ | $3343797869440-30\alpha$
|  | $120$ | $-1208841792+\alpha$
Table 29: Weight enumerators $W_{4,42},W_{4,48},W_{4,54}$ and $W_{4,60}$
---
$i$ | $A_{i}$ | $i$ | $A_{i}$ | $i$ | $A_{i}$ | $i$ | $A_{i}$
$0$ | $1$ | $0$ | $1$ | $0$ | $1$ | $0$ | $1$
$14$ | $\alpha$ | $16$ | $\alpha$ | $18$ | $\alpha$ | $20$ | $\alpha$
$16$ | $5005854-14\alpha$ | $18$ | $32533776-16\alpha$ | $20$ | $211795155-18\alpha$ | $22$ | $1381290300-20\alpha$
$18$ | $51040080+91\alpha$ | $20$ | $321057000+120\alpha$ | $22$ | $1963918710+153\alpha$ | $24$ | $11695925475+190\alpha$
$20$ | $859085136-364\alpha$ | $22$ | $6538354992-560\alpha$ | $24$ | $47987926740-816\alpha$ | $26$ | $343952371980-1140\alpha$
$22$ | $7211157408+1001\alpha$ | $24$ | $63324394380+1820\alpha$ | $26$ | $514922408208+3060\alpha$ | $28$ | $3965025528090+4845\alpha$
$24$ | $45668111580-2002\alpha$ | $26$ | $497930686992-4368\alpha$ | $28$ | $4809711507372-8568\alpha$ | $30$ | $42720147944496-15504\alpha$
$26$ | $192104653104+3003\alpha$ | $28$ | $2714283514008+8008\alpha$ | $30$ | $31973165850600+18564\alpha$ | $32$ | $332209447756065+38760\alpha$
$28$ | $550338821280-3432\alpha$ | $30$ | $10702668824496-11440\alpha$ | $32$ | $160726125267045-31824\alpha$ | $34$ | $2024219799388080-77520\alpha$
$30$ | $1034970326208+3003\alpha$ | $32$ | $29678410515501+12870\alpha$ | $34$ | $594847410234480+43758\alpha$ | $36$ | $9383151491321400+125970\alpha$
$32$ | $1240187813865-2002\alpha$ | $34$ | $57163456332720-11440\alpha$ | $36$ | $1615407895545210-48620\alpha$ | $38$ | $33173750202208200-167960\alpha$
$34$ | $894987802800+1001\alpha$ | $36$ | $74293883912664+8008\alpha$ | $38$ | $3163481567347140+43758\alpha$ | $40$ | $88400914224548310+184756\alpha$
$36$ | $358107409296-364\alpha$ | $38$ | $62784083316240-4368\alpha$ | $40$ | $4380680515423860-31824\alpha$ | $42$ | $175586877143389800-167960\alpha$
$38$ | $68742928800+91\alpha$ | $40$ | $32595760120572+1820\alpha$ | $42$ | $4166728491884400+18564\alpha$ | $44$ | $255572564219099100+125970\alpha$
$40$ | $4762814364-14\alpha$ | $42$ | $9541274664240-560\alpha$ | $44$ | $2616430701397500-8568\alpha$ | $46$ | $266691475753110000-77520\alpha$
$42$ | $49541328+\alpha$ | $44$ | $1361387920680+120\alpha$ | $46$ | $1023782853997800+3060\alpha$ | $48$ | $193631697367494750+38760\alpha$
|  | $46$ | $71055167760-16\alpha$ | $48$ | $228726938682450-816\alpha$ | $50$ | $93892876873052400-15504\alpha$
|  | $48$ | $565394634+\alpha$ | $50$ | $25204966879248+153\alpha$ | $52$ | $28677315561812280+4845\alpha$
|  |  |  | $52$ | $1026632343051-18\alpha$ | $54$ | $5050176407767260-1140\alpha$
|  |  |  | $54$ | $6447073014+\alpha$ | $56$ | $442697147254215+190\alpha$
|  |  |  |  |  | $58$ | $14463285616620-20\alpha$
|  |  |  |  |  | $60$ | $73479968154+\alpha$