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\begin{document}
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\title[Formally self-dual codes from block $\lambda$-circulant matrices]{%
Constructing formally self-dual codes from block $\lambda$-circulant matrices%
}
\author[A.\,Kaya, B.\,Yildiz]{Abidin Kaya\affil{1}\comma\corrauth and Bahattin Yildiz\affil{2}}
\markboth{A.~Kaya and B.~Yildiz}{Formally self-dual codes from block
$\lambda$-circulant matrices}

% multiple authors:
% Please mark \corrauth after the name of the corresponding author.
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\address{\affilnum{1}\ Sampoerna Academy, L'Avenue Campus, 12\,780, Jakarta,
Indonesia\\   \affilnum{2}\ Department of Mathematics and Statistics,
Northern Arizona University, Flagstaff, AZ 86\,011, USA}


\emails{{\texttt{abidin.kaya@sampoernaacademy.sch.id} (A.\,Kaya)}, {\texttt{%
bahattin.yildiz@nau.edu} (B.\,Yildiz)}}


\begin{abstract}
In this work, construction methods for formally self-dual codes are
generalized in the form of block $\lambda$-circulant matrices. The
constructions are applied over the rings $\mathbb{F}_{2}$, $R_{1}=\mathbb{F}%
_{2}+u\mathbb{F}_{2}$ and $S=\mathbb{F}_{2}[u]/(u^{3}-1)$. Using $n$-block $%
\lambda $-circulant matrices for suitable integers $n$ and units $\lambda $,
many binary FSD codes (as Gray images) with a higher minimum distance than
best known self-dual codes of lengths 34, 40, 44, 54, 58, 70, 72 and 74 were
obtained. In particular, ten new even FSD $\left[ 72,36,14\right]$ codes
were constructed together with eight new near-extremal FSD even codes of
length 44 and twenty-five new near-extremal FSD even codes of length 36.
\end{abstract}
\ams{94B05, 94B99.}
\keywords{formally self-dual codes,
near-extremal codes, circulant codes } \maketitle

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\ams{Primary 94B05, 94B99; Secondary 11T71, 13M99 }

\section{Introduction}

Formally self-dual codes are those that have the same weight enumerators as
their duals. Iso-dual codes are codes that are equivalent to their duals. Of
course self-dual codes are both iso-dual and formally self-dual but there
are formally self-dual codes that are not self-dual. Formally self-dual
codes can have better minimum distances than self-dual codes of the same
lengths, as will be showcased throughout the paper. Formally self-dual codes
work in the exact same way for the Assmus-Mattson theorem as self-dual
codes and they have been studied quite extensively in the literature. For
shortness of the notation, throughout the paper, we will use the
abbreviation FSD to describe formally self-dual codes.

Betsumiya and Harada studied binary optimal odd FSD codes in \cite{betsumiya}%
, whereas Dougherty et al. considered optimal FSD codes over $\mathbb{F}_{5}$
and $\mathbb{F}_{7}$ in \cite{dougherty}. In \cite{betsumiya2}, a
classification of FSD even codes of lengths up to 16 was done. In \cite%
{fields}, a classification of extremal even FSD codes of lengths 20 and 22
was done. Extremal double circulant FSD even codes were classified in \cite%
{gulliver}. Betsumiya and Harada considered FSD codes related to Type II
codes in \cite{betsumiya3}. Near extremal FSD even codes of lengths 24 and
32 and later FSD even codes of lengths divisible by 8 were studied in \cite%
{gulliver2} and \cite{kim}, respectively. In \cite{bouyuklieva}, a
classification of extremal even FSD codes of length 30 was made. In \cite%
{freibert}, optimal subcodes of FSD codes of lengths 16-22 were analysed by
using their distance profiles.

Parallel to our main methods, some recent studies have shown that good FSD
codes can be obtained from FSD codes over certain rings. A family of rings
of characteristic 2 was used quite effectively in \cite{karadeniz} to obtain
FSD codes with better minimum distances than the self-dual ones as well as
some near-extremal FSD codes. In \cite{batoul}, constructions were applied
to  the rings $\mathbb{F}_{3}+v\mathbb{F}_{3}$ and $\mathbb{F}_{5}+v\mathbb{F}%
_{5}$ to find some good FSD codes over the fields $\mathbb{F}_{3}$ and $%
\mathbb{F}_{5}$ with better minimum distances than the self-dual codes.

Two main construction methods for FSD codes use double circulant and
bordered double circulant matrices over the binary field as well as over
appropriate rings. In this work, we will generalize these methods by
introducing a general block-matrix construction as well as $\lambda $%
-circulant matrices. Our construction methods generalize all the existing
methods of constructions and their effectiveness is demonstrated by the
highly substantial number of good FSD codes (near-extremal or better than
self-dual) that we have obtained. We will apply our methods to three main
alphabets, all of characteristic 2, namely the binary field $\mathbb{F}_{2}$%
, the ring $R_{1}=\mathbb{F}_{2}+u\mathbb{F}_{2}$ and $S=\mathbb{F}%
_{2}[u]/(u^{3}-1)$. In the case of $S$, since the Gray map is not
duality-preserving, we will prove the existence of the MacWilliams
identities that are preserved under the Gray map, which help us find binary
FSD codes as the images of FSD codes over $S$. Using $n$-block $\lambda $%
-circulant matrices for suitable integers $n$ and units $\lambda $, we were
able to obtain many binary FSD codes with a higher minimum distance than
best known self-dual codes of lengths 34, 40, 44, 54, 58, 70, 72 and 74. Ten
new even $\left[ 72,36,14\right]$ codes were also constructed. In addition
to these, eight new near-extremal FSD even codes of length 44 and
twenty-five new near-extremal FSD even codes of length 36 were obtained.

The rest of the paper is organized as follows: In Section 2, we give the
necessary preliminaries about FSD codes as well as the rings over which the
constructions are described. In particular, the MacWilliams identities are
given for the Lee weight enumerators of codes over $S$, to demonstrate that
the image of FSD codes are binary FSD codes. In Section 3, the main
construction methods are described with the proofs. Section 4 includes
numerical results, where many good FSD codes of different lengths
constructed through our constructions are tabulated.

\section{Preliminaries}

In this paper, we will work mainly on the binary field $\mathbb{F}_{2}$ and
two of its ring extensions, namely $\mathbb{F}_{2}+u\mathbb{F}_{2}$,  where $%
u^{2}=0$ and $S=\mathbb{F}_{2}[u]/(u^{3}-1)$. In order to understand
the constructions better, we will recall some of the properties of
these rings, leaving aside the binary field as an obvious case. We
refer the reader to \cite{huffman} for the definitions and notations
on codes.

Let $R$ be one of the rings; $\mathbb{F}_{2}$, $R_{1}$ or $S$ and $\mathcal{C%
}$  a code of length $m$ over $R$. The dual of $\mathcal{C}$ is defined as
$\mathcal{C}^{\perp }=\{\overline{y}\in R^{m}\ |\ \overline{y}\cdot
\overline{x}=0$ for all $\overline{x}\in \mathcal{C}\}$, where $\cdot$
denotes the Euclidean inner product $\overline{a}\cdot \overline{b}=\sum
a_{i}b_{i}$. A code $\mathcal{C}$ is said to be self-orthogonal if $\mathcal{%
C}\subseteq \mathcal{C}^{\perp}$, self-dual if $\mathcal{C}=\mathcal{C}%
^{\perp }$ and iso-dual if $\mathcal{C}$ is permutation equivalent to $%
\mathcal{C}^{\perp }$. The code is said to be even if all the codewords have
even Lee weight.

\begin{definition}
A code $\mathcal{C}$ is called FSD if the code $\mathcal{C}$
and its dual $\mathcal{C}^{\perp }$ have the same weight enumerator. A
binary FSD even code of length $m$ is called {\it extremal} if $d=2\left\lfloor
m/4\right\rfloor +2$ and {\it near-extremal} if $d=2\left\lfloor m/4\right\rfloor $%
.
\end{definition}

The ring $\mathbb{F}_{2}+u\mathbb{F}_{2}\simeq \mathbb{F}_{2}[u]/(u^{2})$ is
a commutative ring of characteristic 2 and it contains the binary field as a
subring. It has just one non-trivial ideal, given by $(u)$. Type II, Type
IV, self-dual codes and cyclic codes over $\mathbb{F}_{2}+u\mathbb{F}_{2}$
have been studied extensively in \cite{Dougherty1}. Through its different
extensions that appeared later in the literature, we now refer to this ring
as $R_{1}.$

We recall that a linear code $\mathcal{C}$ of length $m$ over the ring $%
R_{1} $ is an $R_{1}$-submodule of $R_{1}^{m}$.

The elements of $R_1$ are $0,1,u,1+u$ and their Lee weights are defined as $%
0,1,2,1$, respectively.

A Gray map $\phi $ is defined as $\phi :R_{1}^{m}\longrightarrow \mathbb{F}%
_{2}^{2m}$%
\begin{equation}
\phi \left( \overline{a}+\overline{b}u\right) =\left( \overline{b},\overline{%
a}+\overline{b}\right),  \label{Gray}
\end{equation}%
where $\overline{a},\overline{b}$ in $\mathbb{F}_{2}^{m}$. $\phi $ is a
distance preserving, duality preserving isometry from $\left(
R_{1}^{m},d_{L}\right) $ to $\left( \mathbb{F}_{2}^{2m},d_{H}\right) $,
where $d_{L}$ and $d_{H}$ denote the Lee and Hamming distance in $R_{1}^{m}$
and $\mathbb{F}_{2}^{2m}$, respectively. This means if $\mathcal{C}$ is a
linear code over $R_{1}$ with parameters $\left[ m,2^{k},d\right] $, (here $%
2^{k}$ means the number of the codewords), then $\phi \left( \mathcal{C}%
\right) $ is a binary linear code of parameters $\left[ 2m,k,d\right] $.
Moreover, $\phi (\mathcal{C}^{\perp })=\phi (\mathcal{C})^{\perp }$.

Since the Gray map is a duality-preserving map, self-dual codes over $R_1$
have self-dual binary images, but more importantly for our work we have the
following theorem, as a consequence of the duality-preserving property of $%
\phi$:

\begin{theorem}
\label{R1image} If $\mathcal{C}$ is an iso-dual code over $R_{1}$ of
length $m$, then $\phi (\mathcal{C})$ is an iso-dual binary code of length $%
2m$. If $\mathcal{C}$ is an FSD code over $R_{1}$ of length $m$, then $%
\phi (\mathcal{C})$ is an FSD binary code of length $2m$.
\end{theorem}

\subsection{The ring $S = \mathbb{F}_2[u]/(u^3-1)$ and the MacWilliams
Identity}

The ring $\mathbb{F}_2[u]/(u^3-1)$, which will be denoted by $S$ here, was
used quite effectively in \cite{u3=1} to construct many new binary self-dual
codes of length $68$. Thus we recall some properties of the ring $S$ from
\cite{u3=1}, leaving some of the details for the work in question.

We start by observing that the ring is  structurally  quite familiar, namely $%
S\cong \mathbb{F}_{2}\times \mathbb{F}_{4}$. $S$ is not local and its ideal
structure is given by $I_{0}\subset I_{u+u^{2}},I_{1+u+u^{2}}\subset S$,
where;%
\begin{eqnarray*}
I_{1+u+u^{2}} &=&\left( 1+u+u^{2}\right) =\left\{ 0,1+u+u^{2}\right\} , \\
I_{u+u^{2}} &=&\left( 1+u\right) =\left\{ 0,u+u^{2},1+u,1+u^{2}\right\}
\text{.}
\end{eqnarray*}%
The units in $S$ are given by $\left\{ 1,u,u^{2}\right\} $ and the non-units
are
\[
\left\{ u+u^{2},1+u,1+u^{2},1+u+u^{2}\right\} \text{.}
\]

A linear code $\mathcal{C}$ of length $m$ over $S$ is an $S$-submodule of $%
S^{m}$ and has a generating matrix that is permutation equivalent to
\[
G=\left(
\begin{array}{cccc}
I_{k_{1}} & A & B & C \\
0 & \left( u+u^{2}\right) I_{k_{2}} & 0 & \left( u+u^{2}\right) D \\
0 & 0 & \left( 1+u+u^{2}\right) I_{k_{3}} & \left( 1+u+u^{2}\right) E%
\end{array}%
\right)
\]

We define a Gray map as follows:%
\begin{eqnarray*}
\varphi &:&S^{m}\rightarrow \mathbb{F}_{2}^{3m} \\
\overline{a}+\overline{b}u+\overline{c}u^{2} &\mapsto &\left( \overline{a},%
\overline{b},\overline{c}\right) .
\end{eqnarray*}

\begin{definition}
The Lee weight of an element $x=a+bu+cu^{2}\in S$ is the Hamming weight of
its Gray image, i.e. $w_{L}\left( a+bu+cu^{2}\right) =w_{H}\left( a\right)
+w_{H}\left( b\right) +w_{H}\left( c\right) $.
\end{definition}

Since the Gray map $\varphi$ introduced above does not preserve
orthogonality, we cannot immediately have a theorem such as Theorem \ref%
{R1image}. In order to have an analogous result, we need to show that the
MacWilliams identities for the Lee weight enumerators of codes over $S$
exist. We will quickly establish this by using Wood's results in \cite{Wood}.

Let us label the elements in $S= \{ g_1,g_2, \dots, g_{8} \}$ as $%
S=\{0,u^2,u,u+u^2,1,1+u^2,1+u,1+u+u^2\}$

\begin{definition}
The complete weight enumerator of a linear code $\mathcal{C}$ over $S$ is
defined as
\begin{equation*}
cwe_{\mathcal{C}}(X_{1},X_{2},\dots ,X_{8})=\sum_{\overline{c}\in \mathcal{C}%
}{X_{1}^{n_{g_{1}}(\overline{c})}X_{2}^{n_{g_{2}}(\overline{c})}\cdots
X_{8}^{n_{g_{8}}(\overline{c})}},
\end{equation*}
where $n_{g_i}(\overline{c})$ is the number of appearances of $g_i$ in the
vector $\overline{c}$.
\end{definition}

The complete weight enumerator gives us a lot of information about the code,
such as the size of the code, the minimum weight of the code and the weight
enumerator of the code for any weight function.

Now, since $S$ is a Frobenius ring, the MacWilliams identities for the
complete weight enumerator hold. To find the exact identities we define the
following character on $S:$

\begin{definition}
Define $\chi: S \rightarrow \mathbb{C}^{\times}$ by
\begin{equation*}
\chi(a+bu+cu^2) = (-1)^{c}.
\end{equation*}
\end{definition}

It is easy to verify that $\chi$ is a non-trivial character when restricted
to each non-zero ideal, hence it is a generating character for $S$.

Then we make up an $8 \times 8$ matrix $T$, by letting $T(i,j) =
\chi(g_ig_j).$ The matrix $T$ is given as follows:
\[
T=\left [
\begin{array}{rrrrrrrr}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\
1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\
1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 \\
1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1%
\end{array}
\right ].
\]

The following theorem then follows from \cite{Wood} (Theorem 8.1) quite
easily:

\begin{theorem}
\label{cweMacWilliams} Let $\mathcal{C}$ be a linear code over $S$ of length
$m$ and suppose $\mathcal{C}^{\perp }$ is its dual. Then we have
\begin{equation*}
cwe_{\mathcal{C}^{\perp }}(X_{1},X_{2},\dots ,X_{8})=\frac{1}{|\mathcal{C}|}%
cwe_{\mathcal{C}}(T\cdot (X_{1},X_{2},\dots ,X_{8})^{t}),
\end{equation*}%
where $()^{t}$ denotes the transpose.
\end{theorem}

Now, in $S$, we have $w_{L}(0)=0$; $w_{L}(1)=w_{L}(u)=w_{L}(u^{2})=1$; $%
w_{L}(1+u)=w_{L}(1+u^{2})=w_{L}(u+u^{2})=2$ and $w_{L}(1+u+u^{2})=3$. Thus,
identifying the variables corresponding to the elements with the same weight
we get the symmetrized weight enumerator, namely,
\begin{equation}
swe_{\mathcal{C}}(X,Y,Z,W)=cwe_{\mathcal{C}}(X,Y,Y,Z,Y,Z,Z,W).
\end{equation}

Now, combining Theorem \ref{cweMacWilliams} and the definition of the
symmetrized weight enumerator, we obtain the following theorem:

\begin{theorem}
\label{sweMacWilliams} Let $\mathcal{C}$ be a linear code over $S$ of length
$m$ and let $\mathcal{C}^{\perp }$ be its dual. Then we have
\begin{equation*}
swe_{\mathcal{C}^{\perp }}(X,Y,Z,W)=
\end{equation*}%
\begin{equation*}
\frac{1}{|\mathcal{C}|}swe_{\mathcal{C}}(X+3Y+3Z+W,X+Y-Z-W,X-Y-Z+W,X-3Y+3Z-W).
\end{equation*}
\end{theorem}

We are now ready to obtain the MacWilliams identity for the Lee weight
enumerator. Define the Lee weight enumerator for codes over $S$ as the
homogeneous polynomial
\begin{equation}
Lee_{\mathcal{C}}(W,X)=\sum_{\overline{c}\in \mathcal{C}}W^{3n-w_{L}(%
\overline{c})}X^{w_{L}(\overline{c})}.  \label{Leeenum}
\end{equation}

Considering the weights that the variables $X,Y,Z,W$ of the symmetrized
weight enumerator represent, we easily get the following theorem:

\begin{theorem}
\label{swelee} Let $\mathcal{C}$ be a linear code over $S$ of length $m$.
Then
\begin{equation*}
Lee_{\mathcal{C}}(W,X)=swe_{\mathcal{C}}(W^{3},W^{2}X,WX^{2},X^{3}).
\end{equation*}
\end{theorem}

Now combining Theorem \ref{sweMacWilliams} and Theorem \ref{swelee} we
obtain the following theorem:

\begin{theorem}
\label{leeMacWilliams} Let $\mathcal{C}$ be a linear code over $S$ of length
$m$ and let $\mathcal{C}^{\perp }$ be its dual. With $Lee_{\mathcal{C}}(W,X)$
denoting its Lee weight enumerator as given in (\ref{Leeenum}), we have
\begin{equation*}
Lee_{\mathcal{C}^{\perp }}(W,X)=\frac{1}{|\mathcal{C}|}Lee_{\mathcal{C}%
}(W+X,W-X).
\end{equation*}
\end{theorem}

Note that this is exactly the same identity that is satisfied by the Hamming
weight enumerators of binary codes. The existence of the identity gives us
the following corollary that is of central importance for our work:

\begin{corollary}
If $\mathcal{C}$ is an FSD code over $S$ of length $m$, then $\varphi (%
\mathcal{C})$ is a binary FSD code of length $3m$. Moreover, the Lee weight
enumerator of $\mathcal{C}$ is the same as the Hamming weight enumerator of $%
\varphi (\mathcal{C})$.
\end{corollary}

\section{Constructions\label{constructions}}

In this section, we introduce the construction methods for FSD codes over
commutative Frobenius rings by block $\lambda $-circulant matrices.

Let $R$ be a commutative Frobenius ring and $v=\left( v_{1},v_{2},\ldots
,v_{m}\right) $  an element of the $R$-module $R^{m}$. The $\lambda $%
-cyclic shift of $v$ is defined as $\sigma _{\lambda }\left( v\right)
=\left( \lambda v_{m},v_{1},v_{2},\ldots ,v_{m-1}\right) $, where $\lambda
\in R$. An $m\times m$ square matrix $A$ is called $\lambda $-circulant if
every row is a $\lambda $-cyclic shift of the previous one, in other words, $%
A $ is in the following form;
\[
\left(
\begin{array}{ccccc}
a_{1} & a_{2} & a_{3} & \cdots & a_{m} \\
\lambda a_{m} & a_{1} & a_{2} & \cdots & a_{n-1} \\
\lambda a_{m-1} & \lambda a_{m} & a_{1} & \cdots & a_{m-2} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\lambda a_{2} & \lambda a_{3} & \lambda a_{4} & \cdots & a_{1}%
\end{array}%
\right) .
\]%
If $\lambda =1$ the matrix is simply called circulant. Recently,  in \cite%
{karadeniz},  Karadeniz
et al. applied the double and bordered double circulant constructions for a
family of commutative Frobenius rings of characteristic $2$. The constructions were generalized to double $\lambda$%
-circulant codes in \cite{batoul}. $\lambda$-circulant matrices share
properties of circulant matrices such as they commute with each other and
they are permutation equivalent to their transposes.

\begin{theorem}
\label{pure}Let $A_{i}$ be a $\lambda $-circulant matrix of size $m\times m$
for $1\leq i\leq n$ over $R$. Suppose that there is a weight function on $R$ such that $wt(a) = wt(-a)$ for all $a \in R$. Then the code $\mathcal{C}$ generated by
\begin{equation}
G=\left[ I_{mn}\
\begin{array}{|c}
M%
\end{array}%
\right] \text{ where }M=\left[
\begin{array}{cccc}
A_{1} & A_{2} & \cdots & A_{n} \\
A_{n} & A_{1} & \cdots & A_{n-1} \\
\vdots & \vdots & \ddots & \vdots \\
A_{2} & \cdots & A_{n} & A_{1}%
\end{array}%
\right]  \label{matrix}
\end{equation}%
is an FSD code over $R$. Moreover, $\mathcal{C}$ is iso-dual if $char\left(
R\right) =2$. The code is called an $n$-block $\lambda $-circulant~code.
\end{theorem}

\begin{proof}
Let $\mathcal{C}$ be the code generated by $G$. Then the dual $\mathcal{C}%
^{\perp }$ of $\mathcal{C}$ is generated by
\begin{equation*}
G^{\prime }=\left[
\begin{array}{c|c}
M^{t} &  -I_{mn}%
\end{array}%
\right] \text{ .}
\end{equation*}%
Let $\Pi _{n}$ be the permutation
\begin{equation*}
\Pi _{n}=\left( 1,n\right) \left( 2,n-1\right) \cdots \left(
k-1,n-k+2\right) \left( k,n-k+1\right),
\end{equation*}%
where $k=\left\lfloor n/2\right\rfloor $. It is easily observed that $A_{i}$
and $A_{i}^{t}$ are permutation equivalent since when $\Pi _{m}$ is applied
to rows and then to columns of $A_{i}$, we obtain $A_{i}^{t}$. So the matrix
\begin{equation*}
M^{\prime }=\left[
\begin{array}{cccc}
A_{1}^{t} & A_{2}^{t} & \cdots & A_{n}^{t} \\
A_{n}^{t} & A_{1}^{t} & \cdots & A_{n-1}^{t} \\
\vdots & \vdots & \ddots & \vdots \\
A_{2}^{t} & \cdots & A_{n}^{t} & A_{1}^{t}%
\end{array}%
\right]
\end{equation*}%
is obtained by applying $\Pi _{m}$ to each block of $M$. Then $M^{t}$ is
obtained by applying $\Pi _{n}$ to $M^{\prime }$ block-wise. Let $\mathcal{C}%
^{\prime \prime }$ be the code generated by $G^{\prime \prime }=\left[
\begin{array}{c|c}
M^{T} & I_{mn}%
\end{array}%
\right] $. The matrices $G$ and $G^{\prime \prime }$ are permutation
equivalent and so are the codes they generate. On the other hand, $\mathcal{C}%
^{\perp }$ and $\mathcal{C}^{\prime \prime }$ have the same weight
enumerator since $wt(a) = wt(-a)$ for each $a \in R$. Hence
$\mathcal{C}$ is formally self-dual. Moreover, $\mathcal{C}
$ is isodual if $R$ is a ring of characteristic $2$ since $\mathcal{C}%
^{\perp }=\mathcal{C}^{\prime \prime }$.
\end{proof}

\begin{theorem}
\label{bordered}Let $M$ and $R$ be as in (\ref{matrix}); $a,b\in R$ and let $\boldsymbol{b%
}$ denote the all-$b$ row of length $mn$. Then the code generated by
\begin{equation*}
\left[ I_{mn+1}\
\begin{array}{|cc}
a & \boldsymbol{b} \\
\boldsymbol{b}^{t} & M%
\end{array}%
\right]
\end{equation*}%
is an FSD code over $R$. The code is iso-dual when $char\left( R\right) =2$
and it is called a bordered $n$-block $\lambda $-circulant code.
\end{theorem}

\begin{proof}
Let $\mathcal{C}$ be the code generated by $G$. Then the dual $\mathcal{C}%
^{\perp }$ of $\mathcal{C}$ is generated by
\begin{equation*}
G' = \left[ \begin{array}{cc|}
a & \boldsymbol{b} \\
\boldsymbol{b}^{t} & M^t%
\end{array}  -I_{mn+1}\
%
\right]
\end{equation*}%
Applying the same permutations as in the proof of Theorem \ref{pure}, we see that
$C$ is equivalent to $C''$, which is generated by
\[G'' = \left[ \begin{array}{cc|}
a & \boldsymbol{b} \\
\boldsymbol{b}^{t} & M^t%
\end{array}  I_{mn+1}\
%
\right]. \]
On the other hand, $\mathcal{C}%
^{\perp }$ and $\mathcal{C}^{\prime \prime }$ have the same weight
enumerator since $wt(a) = wt(-a)$ for each $a \in R$. Hence
$\mathcal{C}$ is formally self-dual. Moreover, $\mathcal{C}
$ is isodual if $R$ is a ring of characteristic $2$ since $\mathcal{C}%
^{\perp }=\mathcal{C}^{\prime \prime }$.
\end{proof}
The constructions given in theorems \ref{pure} and \ref{bordered} can be
generalized further by considering $\lambda _{0}$-circulating blocks of
matrices. Such a code is called $\lambda _{0}$-$n$-block $\lambda $%
-circulant code. We illustrate this construction in the following example;

\begin{example}
\label{lambda}Let $R=R_1$ and $r_{1}=\left( 1+u,u\right) $, $r_{2}=\left( 1,1+u\right) $%
, $r_{3}=\left( 1+u,1+u\right) $ and $r_{4}=\left( 0,u\right) $ $\
$be the first rows of circulant matrices $A_{1}$, $A_{2}$, $A_{3}$
and $A_{4}$, respectively. Let $\mathcal{C}$ be the $\lambda _{0}$
block circulant code over $R_{1}$ generated by
\begin{equation*}
G=\left[ I_{8}\
\begin{array}{|c}
\begin{array}{cccc}
A_{1} & A_{2} & A_{3} & A_{4} \\
\lambda _{0}A_{4} & A_{1} & A_{2} & A_{3} \\
\lambda _{0}A_{3} & \lambda _{0}A_{4} & A_{1} & A_{2} \\
\lambda _{0}A_{2} & \lambda _{0}A_{3} & \lambda _{0}A_{4} & A_{1}%
\end{array}%
\end{array}%
\right],
\end{equation*}%
where $\lambda _{0}=1+u$. Then $\mathcal{C}$ is an iso-dual code.
The binary code $\varphi \left( \mathcal{C}\right) $ is a binary
even FSD code of parameters $\left[ 32,16,8\right]$, which is
optimal. A partial weight distribution of the code is given by
$1+364z^{8}+2048z^{10}+6720z^{12}+\cdots $. Even FSD codes of these
parameters have been studied extensively in \cite{gulliver2}.
\end{example}
More examples for the generalization of Theorem \ref{bordered} are
given in Tables \ref{tab:44},\ref{tab:40}  and \ref{tab:54}.
\section{Computational results}

In this section, the methods from Section \ref{constructions} are
applied to the rings $\mathbb{F}_{2}$, $R_{1}$ and $S$. FSD codes
with parameters better than self-dual codes are tabulated. New
near-extremal FSD even codes are constructed for lengths $36$, $38$
and $44$. For the rest of the paper $(.)^{\ast }$ is used to denote
that the code is optimal as a linear code and $(.)^{b}$ is used to
denote that the minimum distance of the code is the distance of the
best known linear code with respect to the online database
\cite{grassl}.
\begin{remark}
In the tables that follow, $n$ denotes the number of blocks,
$m~$denotes the length of each block, $a$ and $b$ denote the borders
of the block circulant matrix (In the case of bordered double
circulant constructions given in Theorem \ref{bordered}), $A_{i} $
is a $\lambda $-circulant matrix for each $1\leq i$ $\leq n$ and
$\lambda _{0}$ is as in Example \ref{lambda}.  The order of the
automorphism group of $\mathcal{C}$ is denoted by $\left\vert
Aut(\mathcal{C})\right\vert $. In order to
fit the upcoming tables we denote the element $1+u$ of $R_{1}$ as $3$ and $%
1+u+u^{2}$ of $S$ as $7$. All computations were run on an average
PC, using the magma algebra system (\cite{Magma}).
\end{remark}
\begin{example}
Let $A_{1}$ and $A_{2}$ be $2\times 2$ $\left( 1+u\right) $-circulant
matrices over $R_{1}$ with first rows $r_{1}=\left( 1,1+u\right) $ and $%
r_{2}=\left( 1+u,u\right) $. Then the bordered $2$-block $\left( 1+u\right) $%
-circulant code generated by
\begin{equation*}
G=\left[ I_{5}\
\begin{array}{|c}
\begin{array}{ccccc}
u & 1 & 1 & 1 & 1 \\
1 & 1 & 1+u & 1+u & u \\
1 & 1 & 1 & u & 1+u \\
1 & 1+u & u & 1 & 1+u \\
1 & u & 1+u & 1 & 1%
\end{array}%
\end{array}%
\right]
\end{equation*}%
is an iso-dual code over $R_{1}$. The binary image of the code is an odd binary FSD
$\left[ 20,10,6\right]^{\ast }$ code that is optimal as a linear code.
A partial weight distribution of the code is $%
1+40z^{6}+160z^{7}+130z^{8}+\cdots $ and its automorphism group has order $%
2^{8}\times 3\times 5$.
\end{example}

\begin{example}
Let $\mathcal{C}$ be the pure double circulant code with the first row
\begin{equation*}
r=\left( 11000100101111110100111\right).
\end{equation*}%
Then $\mathcal{C}$ is an iso-dual code by Theorem \ref{pure}. The
code $\mathcal{C}$ is an optimal linear $\left[
46,23,11\right]^{\ast }$ code and a partial weight distribution of
$\mathcal{C}$ is given by
\begin{equation*}
1+3312z^{11}+9660z^{12}+121440z^{15}+235290z^{16}+\cdots .
\end{equation*}%
With respect to the online database of linear codes, \cite{grassl},
a binary linear $\left[ 46,23,11\right]^{\ast }$ code is constructed
as a shortened puncturing of the unique Type II $\left[
48,24,12\right]^{\ast }$. The code $\mathcal{C}$ is equivalent to
the one kept in \cite{grassl}.
\end{example}

\subsection{Binary even FSD $\left[ 72,36,14\right]$ codes}

The existence of a Type II binary self-dual $\left[ 72,36,16\right]$
code is an open problem that has generated a lot of interest in the
last four decades. Equivalent to the same problem is the existence
of a binary Type I self-dual $\left[ 72,36,14\right]$ code. The best
known binary self-dual codes of length 72 have minimum distance 12,
see \cite{kovic,gurel}. The first examples of binary $\left[
72,36,14\right]$ FSD even codes were constructed in
\cite{karadeniz}. In this section, we construct $10$ binary FSD even
$\left[ 72,36,14\right]$ codes, which are different than the codes
given in \cite{karadeniz}. The results are given in Table
\ref{tab:72}.

\begin{example}
Let $\lambda =1+u$, $n=1$ and let $\mathcal{C}$ be the double $\lambda $%
-circulant code with the first row $%
r_{1}=(u,0,u,u,0,u,0,0,0,u+1,0,1,u+1,u+1,1,0,1,1,)$ over $R_{1}$. Then $%
\mathcal{C}$ is an iso-dual code. The Gray image $\varphi \left( \mathcal{C}%
\right) $ of $\mathcal{C}$ is a binary  FSD even code of parameters $\left[ 72,36,14%
\right]$. A partial weight enumerator of the code is given by
$1+9036z^{14}+121959z^{16}\cdots $.
\end{example}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|l|l|l|}
\hline $n$ & $r_{1},\ldots ,r_{n}$ & Partial weight distribution \\
\hline
4 & $000110011,111101100,101111100,100011011$ & $1+9144z^{14}+120897z^{16}+%
\cdots $ \\ \hline
4 & $000100101,011101011,101000000,110110000$ & $1+8748z^{14}+123525z^{16}+%
\cdots $ \\ \hline
4 & $101110001,000100010,001010011,101011101$ & $1+8748z^{14}+123660z^{16}+%
\cdots $ \\ \hline
4 & $000100101,011101011,000101111,000100110$ & $1+9072z^{14}+121383z^{16}+%
\cdots $ \\ \hline
4 & $000100110,001010101,101000000,110110000$ & $1+8640z^{14}+124479z^{16}+%
\cdots $ \\ \hline
3 & $111100111010,110101101100,111011001011$ & $1+8640z^{14}+125433z^{16}+%
\cdots $ \\ \hline
3 & $000111110111,001101011000,100110001101$ & $1+9288z^{14}+120537z^{16}+%
\cdots $ \\ \hline
1 & $100100010000010111111111000111010000$ & $1+8820z^{14}+122841z^{16}+%
\cdots $ \\ \hline 1 & $100 1 0 0 0 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 1
1 0 0 0 1 0 0 1 1 1 0 1 1$ & $1+8748z^{14}+123831z^{16}+\cdots $ \\
\hline
\end{tabular}%
\vspace*{0.2cm}\caption{\rm\em Binary FSD $\left[ 72,36,14\right]$ codes as $n$-block
circulant codes over $\mathbb{F}_{2}$} \label{tab:72}
\end{center}
\end{table}

\subsection{New near extremal binary FSD even codes}

Extremal binary FSD even codes do not exist for many lengths and the
non-existence of near-extremal binary FSD even codes have been studied in
\cite{han}. In \cite{gulliver}, possible weight enumerators of near-extremal
binary FSD even codes of lengths $16$ to $46$ has been determined. In this
section, we construct new binary near-extremal FSD even codes for lengths $36
$, $38$ and $44$, where possible weight enumerators are;%
\begin{eqnarray*}
W_{36} &=&1+\left( 225+\alpha \right) y^{8}+\left( 2016-6\alpha \right)
y^{10}+\cdots , \\
W_{38} &=&1+\left( 171+\alpha \right) y^{8}+\left( 1862-5\alpha \right)
y^{10}+\cdots , \\
W_{44} &=&1+\left( 1320+\alpha \right) z^{10}+\left( 10461-8\alpha \right)
z^{12}+\cdots .
\end{eqnarray*}

The existence of near-extremal binary FSD even codes is known for weight
enumerators; $\alpha =$0, 9, 13, 18, 27, 36, 45, 47, 63, 64, 72, $-$4, $-$9,
$-$21, $-$27, $-$36, $-$38, $-$45, $-$54, $-$55 and $-$72 for $W_{36}$; $%
\alpha =$0, 22, 31, 44, 66, 176, $-19$, $-22$, $-44$, $-53$, $-66$,
$-176$ for $W_{44}$.

In this work, we were able to construct binary near-extremal FSD even codes
with weight enumerators; $\alpha =$3, 6, 12, 15, 21, 24, 30, 33, 39, 42, 48,
60, 96, 144, $-3$, $-$6, $-$12, $-$15, $-$18, $-$24, $-$30, $-$33, $-$42, $-$%
144 in $W_{36}$; $\alpha =$10, 30, 50, 80, $-$10, $-$11, $-$30 and $-$70 in $%
W_{44}$.
\begin{example}
For $n=1$ and $\lambda =1$, let $\mathcal{C}$ be the double circulant
code over $S$ of length $12$ with the first row of $A_{1}$ as \
$r_{1}=\left( 1,u^{2},1+u,u+u^{2},1,1+u^{2}\right) $. Then
$\mathcal{C}$ is iso-dual by Theorem \ref{pure}. The Gray image of
$\mathcal{C}$ is a binary near-extremal FSD even code with weight
enumerator $\alpha =144$ in $W_{36}$. The weight enumerator of the
code is different than the ones in \cite{gulliver}.
\end{example}
More new near extremal FSD even codes are constructed in Table
\ref{tab:36}.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c||c|}
\hline
$\lambda $ & $r_1$ & $a,b$ & $\left\vert Aut(\mathcal{C})\right\vert $ & $\alpha $ \\
\hline
$1$ & $\left( u,1+u^{2},0,7,1+u^{2}\right) $ & $1+u^{2},u$ & 30 & $3$ \\
\hline
$1$ & $\left( 1+u^{2},1+u^{2},u,0,7\right) $ & $1+u^{2},1$ & 15 & $6$ \\
\hline
$u$ & $\left( 0,1+u^{2},1,u+u^{2},u\right) $ & $u+u^{2},u$ & 3 & $12$ \\
\hline $u$ & $\left( 1+u,7,0,1,0\right) $ & $1+u,1$ & 3 & $15$ \\
\hline
$u$ & $\left( u,0,u+u^{2},u^{2},1+u\right) $ & $1+u^{2},1$ & 3 & $21$ \\
\hline
$u$ & $\left( 1+u^{2},u^{2},u,u^{2},1\right) $ & $u+u^{2},1$ & 3 & $24$ \\
\hline $u$ & $\left( 1,1,0,u^{2},7\right) $ & $1+u^{2},u^{2}$ & 3 &
$30$ \\ \hline
$1$ & $\left( u^{2},0,1+u^{2},1,1+u\right) $ & $1+u^{2},1$ & 15 & $33$ \\
\hline $u$ & $\left( 1+u,7,7,0,1+u^{2}\right) $ & $1+u,1$ & 3 & $39$
\\ \hline $u$ & $\left( 1+u^{2},7,1,u^{2},u^{2}\right) $ & $1+u,u$ &
3 & $42$ \\ \hline
$1$ & $\left( u,u^{2},u,1+u^{2},u\right) $ & $u+u^{2},u$ & 15 & $48$ \\
\hline $u$ & $\left( 7,u^{2},1+u^{2},7,1\right) $ & $1+u,u$ & 3 &
$60$ \\ \hline $1$ & $\left( u+u^{2},u+u^{2},1,1,1+u^{2}\right) $ &
$1+u,u^{2}$ & 60 & $96$
\\ \hline
$u$ & $\left( 7,u,u+u^{2},0,0\right) $ & $1+u^{2},u$ & 3 & $-3$ \\
\hline $u$ & $\left( 0,u^{2},1,u,7\right) $ & $1+u^{2},1$ & 3 & $-6$
\\ \hline $1$ & $\left( 0,7,1+u^{2},0,1\right) $ & $u+u^{2},u$ & 15
& $-12$ \\ \hline $u$ & $\left( 1+u,u,1+u,0,u^{2}\right) $ &
$1+u^{2},u$ & 3 & $-15$ \\ \hline $u$ & $\left( 7,u,1+u,1,7\right) $
& $1+u,u^{2}$ & 3 & $-18$ \\ \hline $1$ & $\left(
1+u^{2},u,1,1+u^{2},0\right) $ & $1+u,u$ & 15 & $-24$ \\ \hline $u$
& $\left( 7,0,u,u,1\right) $ & $1+u^{2},u$ & 3 & $-30$ \\ \hline $u$
& $\left( 1+u^{2},u+u^{2},7,u,0\right) $ & $1+u,u$ & 3 & $-33$ \\
\hline
$1$ & $\left( 1+u,1+u,u,u^{2},0\right) $ & $1+u^{2},u^{2}$ & 15 & $-42$ \\
\hline
$1$ & $\left( 1+u^{2},1+u^{2},u,1+u,u\right) $ & $1+u^{2},u^{2}$ & 1440 & $%
-144$ \\ \hline
\end{tabular}%
\vspace*{0.2cm}
\caption{\rm\em 24 new near extremal FSD even codes of length $36$ over $S$ by
Theorem \ref{bordered} }\label{tab:36}
\end{center}
\end{table}

For a list of known near-extremal FSD even codes of length $38$ we refer to
\cite{gulliver}. In the following example we construct a code with a new
weight enumerator by applying bordered 2-block circulant construction on $%
\mathbb{F}_{2}$.

\begin{example}
Let $\mathcal{C}$ be the bordered 2-block circulant code with $r_{1}=\left(
111100111\right) ,$ $r_{2}=\left( 101011100\right) $ and $a=1=b$ over $%
\mathbb{F}_{2}$. The code $\mathcal{C}$ is an even code and it is FSD by
Theorem \ref{bordered}. $\left\vert Aut\left( \mathcal{C}\right) \right\vert
=2\times 3^{2}$ and $\mathcal{C}$ has a weight enumerator with $\alpha =-9$
in $W_{38}$.
\end{example}

\begin{example}
Let $\mathcal{C}$ be the binary bordered 3-block circulant code with $%
r_{1}=\left( 1110101\right) ,$ $r_{2}=\left( 1100100\right) ,$ $r_{3}=\left(
0010100\right) $, $a=0$ and $b=1$. Then $\mathcal{C}$ is a near-extremal FSD
even code with weight enumerator $\alpha =-11$ in $W_{44}$. The existence of
an FSD code with this weight enumerator was unknown. The automorphism group
of the code has order $3\times 7$.
\end{example}

We were able to construct seven near extremal binary FSD even codes of
length $44$ with new weight enumerators as Gray images of FSD codes over $%
R_{1}$, which are listed in Table \ref{tab:44}.

\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c||c|}
\hline
$n$ & $\lambda _{0}$ & $\lambda $ & $r_{1},\ldots ,r_{n}$ & $a,b$ & $%
\left\vert Aut(\mathcal{C})\right\vert $ & $\alpha $ \\ \hline
$2$ & $3$ & $1$ & $\left( u01u3\right) ,\left( 03u01\right) $ & $3,3$ & $%
2\times 5$ & $30$ \\ \hline
$2$ & $3$ & $1$ & $\left( 0330u\right) ,\left( 33101\right) $ & $3,3$ & $%
2\times 5$ & $50$ \\ \hline
$5$ & $1$ & $1$ & $\left( 13\right) ,\left( 10\right) ,\left( uu\right)
,\left( 3u\right) \left( 33\right) $ & $1,1$ & $2^{2}5$ & $10$ \\ \hline
$5$ & $1$ & $1$ & $\left( 30\right) ,\left( 11\right) ,\left( 1u\right)
,\left( u3\right) \left( u1\right) $ & $1,1$ & $2^{2}5$ & $80$ \\ \hline
$5$ & $1$ & $1$ & $\left( u0\right) ,\left( u3\right) ,\left( 03\right)
,\left( 11\right) \left( 13\right) $ & $1,1$ & $2^{2}5$ & $-10$ \\ \hline
$5$ & $1$ & $1$ & $\left( 3u\right) ,\left( 0u\right) ,\left( 1u\right)
,\left( uu\right) \left( 13\right) $ & $1,1$ & $2^{2}5$ & $-30$ \\ \hline
$5$ & $1$ & $1$ & $\left( 3u\right) ,\left( 03\right) ,\left( 33\right)
,\left( 13\right) \left( u0\right) $ & $1,1$ & $2^{2}5$ & $-70$ \\ \hline
\end{tabular}%
 \vspace*{0.2cm}\caption{\rm\em Optimal $\left[ 44,22,10\right]$ FSD even codes as images
of bordered $n$-block circulant codes over $R_{1}$}\label{tab:44}
\end{center}
\end{table}



\begin{remark}
Throughout the paper the binary FSD codes that have been listed have
better minimum distances than the best known self-dual codes, except
for the ones listed in Tables \ref{tab:36} and \ref{tab:76}, which
have the same distance as the corresponding extremal self-dual code.
\end{remark}

Theorem \ref{pure} is applied over $\mathbb{F}_{2}$ in Table
\ref{tab:70}. Its generalization is applied over $R_1$ and $S$
 in Table \ref{tab:40} and in Table \ref{tab:54}, respectively.
Theorem \ref{bordered} is applied in Tables \ref{tab:34}, \ref{tab:58},
\ref{tab:74} and \ref{tab:76}.

\begin{table}[H]
\begin{center}
\begin{tabular}{||c|c|c|c|c||}
\hline $\lambda _{0}$ & $\lambda$ & $r_{1}$ & $r_{2}$ & Partial
weight distribution \\ \hline
$1$ & $1$ & $\left( 01113\right) $ & $\left( 013u0\right) $ & $%
1+280z^{9}+1052z^{10}+\cdots $ \\ \hline
$1$ & $1$ & $\left( 1u0uu\right) $ & $\left( 1330u\right) $ & $%
1+320z^{9}+992z^{10}+\cdots $ \\ \hline
$1$ & $1$ & $\left( u1u03\right) $ & $\left( u303u\right) $ & $%
1+340z^{9}+972z^{10}+\cdots $ \\ \hline
$3$ & $1$ & $\left( 1013u\right) $ & $\left( 3u303\right) $ & $%
1+260z^{9}+1030z^{10}+\cdots $ \\ \hline
$3$ & $1$ & $\left( 11u01\right) $ & $\left( 33100\right) $ & $%
1+280z^{9}+1010z^{10}+\cdots $ \\ \hline
$3$ & $1$ & $\left( 10u33\right) $ & $\left( u1033\right) $ & $%
1+300z^{9}+1020z^{10}+\cdots $ \\ \hline
$3$ & $1$ & $\left( u3u00\right) $ & $\left( 110u1\right) $ & $%
1+320z^{9}+1000z^{10}+\cdots $ \\ \hline
$3$ & $1$ & $\left( u3311\right) $ & $\left( 00013\right) $ & $%
1+320z^{9}+982z^{10}+\cdots $ \\ \hline
$3$ & $1$ & $\left( 0u31u\right) $ & $\left( 1111u\right) $ & $%
1+360z^{9}+922z^{10}+\cdots $ \\ \hline
$3$ & $3$ & $\left( 01133\right) $ & $\left( u31uu\right) $ & $%
1+340z^{9}+982z^{10}+\cdots $ \\ \hline
$3$ & $3$ & $\left( 03u30\right) $ & $\left( u1331\right) $ & $%
1+240z^{9}+1166z^{10}+\cdots $ \\ \hline
$3$ & $3$ & $\left( 00u33\right) $ & $\left( 3u03u\right) $ & $%
1+300z^{9}+1022z^{10}+\cdots $ \\ \hline
$3$ & $3$ & $\left( 1030u\right) $ & $\left( 01311\right) $ & $%
1+320z^{9}+1012z^{10}+\cdots $ \\ \hline
\end{tabular}%
\vspace*{0.2cm}
\caption{\rm\em Binary $\left[ 40,20,9\right]^{b}$ FSD codes as Gray images
of $2$-block circulant codes over $R_{1}$} \label{tab:40}
\end{center}
\end{table}

\begin{table}[htbp]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline $n$ & $m$ & $r_{1},\ldots ,r_{n}$ & $a,b$ & $\left\vert
Aut(\mathcal{C})\right\vert $ & Partial weight distribution \\
\hline
$2$ & $8$ & $11000001,11010100$ & $0,1$ & $2^{6}$ & $1+172z^{8}+352z^{9}+%
\cdots $ \\ \hline
$2$ & $8$ & $10001010,10110001$ & $0,1$ & $2^{7}$ & $1+204z^{8}+384z^{9}+%
\cdots $ \\ \hline
$2$ & $8$ & $10001101,00011001$ & $1,1$ & $2^{6}$ & $1+220z^{8}+384z^{9}+%
\cdots $ \\ \hline
$2$ & $8$ & $10001011,11101111$ & $0,1$ & $2^{6}$ & $1+236z^{8}+416z^{9}+%
\cdots $ \\ \hline
$2$ & $8$ & $10111110,10000101$ & $0,1$ & $2^{7}$ & $1+268z^{8}+448z^{9}+%
\cdots $ \\ \hline
$4$ & $4$ & $1110,1000,0110,0111$ & $0,1$ & $2^{9}3$ & $1+140z^{8}+320z^{9}+%
\cdots $ \\ \hline
$4$ & $4$ & $1001,1000,1000,0111$ & $0,1$ & $2^{9}3$ & $1+204z^{8}+512z^{9}+%
\cdots $ \\ \hline
$4$ & $4$ & $1010,0110,0001,1100$ & $0,1$ & $2^{9}5^{2}3$ & $%
1+332z^{8}+640z^{9}+\cdots $ \\ \hline
$1$ & $16$ & $1000101110011100$ & $0,1$ & $2^{4}$ & $1+192z^{8}+298z^{9}+%
\cdots $ \\ \hline
$1$ & $16$ & $0111001100101101$ & $0,1$ & $2^{4}$ & $1+200z^{8}+380z^{9}+%
\cdots $ \\ \hline
$1$ & $16$ & $1100101001101001$ & $0,1$ & $2^{4}$ & $1+208z^{8}+282z^{9}+%
\cdots $ \\ \hline
$1$ & $16$ & $0001111011101111$ & $0,1$ & $2^{5}$ & $1+208z^{8}+388z^{9}+%
\cdots $ \\ \hline
$1$ & $16$ & $1110011000111110$ & $0,1$ & $2^{4}$ & $1+210z^{8}+284z^{9}+%
\cdots $ \\ \hline
$1$ & $16$ & $1000101110110000$ & $0,1$ & $2^{4}$ & $1+212z^{8}+392z^{9}+%
\cdots $ \\ \hline
$1$ & $16$ & $1100001111101001$ & $1,1$ & $2^{7}$ & $1+220z^{8}+384z^{9}+%
\cdots $ \\ \hline
$1$ & $16$ & $0110111010000110$ & $0,1$ & $2^{4}$ & $1+224z^{8}+266z^{9}+%
\cdots $ \\ \hline
$1$ & $16$ & $1010111011110010$ & $0,1$ & $2^{4}$ & $1+224z^{8}+282z^{9}+%
\cdots $ \\ \hline
$1$ & $16$ & $0101010000011001$ & $0,1$ & $2^{4}$ & $1+240z^{8}+266z^{9}+%
\cdots $ \\ \hline $1$ & $17$ & $10101110111110110$ & pure & $17$ &
$1+153z^{8}+527z^{9}+\cdots $ \\ \hline $1$ & $17$ &
$01111110100011111$ & pure & $17$ & $1+170z^{8}+527z^{9}+\cdots $ \\
\hline $1$ & $17$ & $11101000001001011$ & pure & $17$ &
$1+187z^{8}+459z^{9}+\cdots $ \\ \hline
$1$ & $17$ & $11001111111100110$ & pure & $2\times 17$ & $%
1+187z^{8}+493z^{9}+\cdots $ \\ \hline $1$ & $17$ &
$11011101000100100$ & pure & $17$ & $1+204z^{8}+442z^{9}+\cdots $ \\
\hline $1$ & $17$ & $11111010111011001$ & pure & $17$ &
$1+221z^{8}+408z^{9}+\cdots $ \\ \hline $1$ & $17$ &
$10011011011000010$ & pure & $17$ & $1+255z^{8}+816z^{9}+\cdots $ \\
\hline
\end{tabular}%
\vspace*{0.2cm}
\caption{\rm\em Optimal binary linear $\left[ 34,17,8\right]$ codes as odd
FSD codes over $\mathbb{F}_{2}$} \label{tab:34}
\end{center}
\end{table}

\begin{table}[htbp]
\begin{center}
\resizebox{1\textwidth}{!}{\begin{tabular}{|l|l|l|l|l|} \hline $n$
& $\lambda _{0}$ & $\lambda$ & $r_{1},\ldots ,r_{n}$ & Partial
weight distribution \\ \hline 3 & $u^{2}$ & $1$ & $\left(
1+u,u,u\right) ,\left( u^{2},u+u^{2},0\right) ,\left(
1+u,1,u+u^{2}\right) $ & $1+729z^{11}+3096z^{12}+\cdots $ \\ \hline
3 & $u^{2}$ & $1$ & $\left( 1+u,1+u^{2},0\right) ,\left(
1+u,0,1\right) ,\left( u^{2},1,7\right) $ &
$1+864z^{11}+3123z^{12}+\cdots $ \\ \hline 3 & 1 & $1$ & $\left(
1,1,0\right) ,\left( 1,0,1+u^{2}\right) ,\left(
u^{2},u+u^{2},1+u^{2}\right) $ & $1+1080z^{11}+5706z^{12}+\cdots $
\\ \hline 3 & $1$ & $1$ & $\left( 0,7,1+u\right) ,\left(
1+u,1+u^{2},1+u^{2}\right) ,\left( u,1,u\right) $ &
$1+1404z^{11}+5031z^{12}+\cdots $ \\ \hline 1 & - & $u$ & $\left(
1+u,1+u^{2},1+u^{2},1,u+u^{2},7,u+u^{2},u,1\right) $ &
$1+621z^{11}+2970z^{12}+\cdots $ \\ \hline 1 & - & $u$ & $\left(
u+u^{2},u^{2},1,u+u^{2},u,u+u^{2},1,u+u^{2},u+u^{2}\right) $ & $%
1+648z^{11}+3051z^{12}+\cdots $ \\ \hline
1 & - & $u$ & $\left( 1,1,u,u,u,1+u^{2},1+u+u^{2},1+u^{2},0\right) $ & $%
1+729z^{11}+3051z^{12}+\cdots $ \\ \hline 1 & - & $u$ & $\left(
u^{2},u+u^{2},u+u^{2},7,1+u,u+u^{2},1+u,7,u\right) $ &
$1+864z^{11}+2916z^{12}+\cdots $ \\ \hline
1 & - & $u$ & $\left( 1+u,1+u^{2},1+u^{2},1,u+u^{2},1+u,u,1,1\right) $ & $%
1+918z^{11}+3132z^{12}+\cdots $ \\ \hline
1 & - & $u^{2}$ & $\left( 1,1+u^{2},1,7,u+u^{2},u,1,7,1+u\right) $ & $%
1+891z^{11}+2889z^{12}+\cdots $ \\ \hline
\end{tabular}}
\vspace*{0.2cm}
\caption{\rm\em Binary FSD $\left[ 54,27,11\right]^{b}$ codes as Gray
images of FSD codes over $S$} \label{tab:54}
\end{center}
\end{table}

\begin{table}[htbp]
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline $r_{1}$ & $r_{2}$ & $r_{3}$ & $r_{4}$ & Partial weight
distribution \\ \hline
$1111001$ & $1011001$ & $0111111$ & $1000101$ & $1+3290z^{12}+40565z^{14}+%
\cdots $ \\ \hline
$1011110$ & $0000110$ & $0011011$ & $1111111$ & $1+3388z^{12}+39879z^{14}+%
\cdots $ \\ \hline
$1001111$ & $0111011$ & $0011000$ & $0101000$ & $1+3395z^{12}+39830z^{14}+%
\cdots $ \\ \hline
$0000100$ & $0100100$ & $1000101$ & $1001110$ & $1+3402z^{12}+39783z^{14}+%
\cdots $ \\ \hline
$0001001$ & $1101111$ & $1000101$ & $1110011$ & $1+3437z^{12}+39536z^{14}+%
\cdots $ \\ \hline
$0111111$ & $0100000$ & $0111001$ & $1001111$ & $1+3437z^{12}+39482z^{14}+%
\cdots $ \\ \hline
$0011100$ & $1010000$ & $0011111$ & $0011110$ & $1+3500z^{12}+39097z^{14}+%
\cdots $ \\ \hline
$1001100$ & $0110000$ & $0000101$ & $1001100$ & $1+3500z^{12}+39095z^{14}+%
\cdots $ \\ \hline
$1000110$ & $0011110$ & $1011111$ & $1001100$ & $1+3507z^{12}+38990z^{14}+%
\cdots $ \\ \hline
$1111001$ & $1010000$ & $1010100$ & $0011110$ & $1+3542z^{12}+38803z^{14}+%
\cdots $ \\ \hline
$1001011$ & $1010010$ & $1000111$ & $0111011$ & $1+3563z^{12}+38600z^{14}+%
\cdots $ \\ \hline
$1110011$ & $1001110$ & $0000110$ & $0000001$ & $1+3591z^{12}+38402z^{14}+%
\cdots $ \\ \hline
$0101100$ & $0101000$ & $1111010$ & $1001011$ & $1+3598z^{12}+38411z^{14}+%
\cdots $ \\ \hline
$0011000$ & $0100110$ & $0110110$ & $0111000$ & $1+3619z^{12}+38208z^{14}+%
\cdots $ \\ \hline
$1100001$ & $0101000$ & $1010001$ & $1000001$ & $1+3640z^{12}+38061z^{14}+%
\cdots $ \\ \hline
$1101110$ & $1000100$ & $0101100$ & $0101101$ & $1+3668z^{12}+37921z^{14}+%
\cdots $ \\ \hline
$0101100$ & $0010011$ & $0111001$ & $0011110$ & $1+3703z^{12}+37674z^{14}+%
\cdots $ \\ \hline
$1110101$ & $1100101$ & $0110111$ & $0111010$ & $1+3745z^{12}+37382z^{14}+%
\cdots $ \\ \hline
$0001010$ & $1000000$ & $1000011$ & $1101010$ & $1+3766z^{12}+37177z^{14}+%
\cdots $ \\ \hline
$1010111$ & $1010001$ & $1010011$ & $0110110$ & $1+3850z^{12}+36589z^{14}+%
\cdots $ \\ \hline
\end{tabular}%
\vspace*{0.2cm}
\caption{\rm\em Binary $\left[ 58,29,12\right]^{b}$ FSD codes as bordered
$4$-block circulant codes over $\mathbb{F}_{2}$ with $a=1=b$}
\label{tab:58}
\end{center}
\end{table}

\begin{table}[htbp]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline $r_{1}$ & $r_{2}$ & $r_{3}$ & $r_{4}$ & $r_{5}$ & Partial
weight distribution
\\ \hline
$1000111$ & $0101010$ & $0100110$ & $0100001$ & $0111001$ & $%
1\!+\!910z^{13}\!+\!5880z^{14}\!+\!\cdots $ \\ \hline
$1110011$ & $0001000$ & $1000110$ & $0100110$ & $0000011$ & $%
1\!+\!1120z^{13}\!+\!5740z^{14}\!+\!\cdots $ \\ \hline
$1101101$ & $0101000$ & $0011010$ & $0100010$ & $1100110$ & $%
1\!+\!1155z^{13}\!+\!6405z^{14}\!+\!\cdots $ \\ \hline
$0110111$ & $1001110$ & $0001000$ & $0001000$ & $0110100$ & $%
1\!+\!1295z^{13}\!+\!5810z^{14}\!+\!\cdots $ \\ \hline
$0010101$ & $1111000$ & $1001100$ & $1111001$ & $1001001$ & $%
1\!+\!1295z^{13}\!+\!5635z^{14}\!+\!\cdots $ \\ \hline
$1100100$ & $1110010$ & $0001110$ & $0110001$ & $1010111$ & $%
1\!+\!1400z^{13}\!+\!6020z^{14}\!+\!\cdots $ \\ \hline
$0111100$ & $0111101$ & $1110100$ & $1010101$ & $1001100$ & $%
1\!+\!1435z^{13}\!+\!5565z^{14}\!+\!\cdots $ \\ \hline
$1010011$ & $0010010$ & $1001100$ & $1100010$ & $0010111$ & $%
1\!+\!1505z^{13}\!+\!5670z^{14}\!+\!\cdots $ \\ \hline
$1001111$ & $0010110$ & $1000000$ & $0100110$ & $0101110$ & $%
1\!+\!1505z^{13}\!+\!6125z^{14}\!+\!\cdots $ \\ \hline
$0010111$ & $1001111$ & $0011011$ & $0100110$ & $1111000$ & $%
1\!+\!1540z^{13}\!+\!5530z^{14}\!+\!\cdots $ \\ \hline
\end{tabular}%
\vspace*{0.2cm}
\caption{\rm\em  Binary $\left[ 70,35,13\right]$ FSD codes as 5-block
circulant codes over $\mathbb{F}_{2}$} \label{tab:70}
\end{center}
\end{table}

\begin{table}[h!]
\begin{center}
\resizebox{1\textwidth}{!}{\begin{tabular}{|l|l|l|l|} \hline $n$
& $m$ & $r_{1},\ldots ,r_{n}$ & Partial weight distribution \\
\hline
$4$ & $9$ & $101000100,101111100,010001001,011100100$ & $%
1+6681z^{14}+98703z^{16}+\cdots $ \\ \hline
$4$ & $9$ & $100001110,110100110,000011001,001000010$ & $%
1+6714z^{14}+98538z^{16}+\cdots $ \\ \hline
$4$ & $9$ & $100111101,111010111,101000000,101100110$ & $%
1+6723z^{14}+98673z^{16}+\cdots $ \\ \hline
$4$ & $9$ & $111011010,010010001,111001000,110100000$ & $%
1+6876z^{14}+97980z^{16}+\cdots $ \\ \hline
$4$ & $9$ & $000111101,001011001,011100111,111011000$ & $%
1+7272z^{14}+95568z^{16}+\cdots $ \\ \hline
$3$ & $12$ & $001011000110,101011001001,000100000000$ & $%
1+6078z^{14}+102177z^{16}+\cdots $ \\ \hline
$3$ & $12$ & $000100001111,100101010110,111000101011$ & $%
1+6222z^{14}+102177z^{16}+\cdots $ \\ \hline
$3$ & $12$ & $111101000001,110110100101,010100100011$ & $%
1+6270z^{14}+100497z^{16}+\cdots $ \\ \hline
$3$ & $12$ & $001001100101,000000100110,010111110101$ & $%
1+6618z^{14}+99189z^{16}+\cdots $ \\ \hline
$3$ & $12$ & $000100011101,010010001100,101101010110$ & $%
1+7374z^{14}+95553z^{16}+\cdots $ \\ \hline
$2$ & $18$ & $101001111011011100,010110010001101110$ & $%
1+6390z^{14}+100446z^{16}+\cdots $ \\ \hline
$2$ & $18$ & $101000100111110001,010110011100111011$ & $%
1+6552z^{14}+99906z^{16}+\cdots $ \\ \hline
$2$ & $18$ & $100011101000011000,011111010001010010$ & $%
1+6732z^{14}+98538z^{16}+\cdots $ \\ \hline
$2$ & $18$ & $010111001111100011,011010100101111000$ & $%
1+6930z^{14}+98088z^{16}+\cdots $ \\ \hline
$2$ & $18$ & $001101101010011100,110111011100101100$ & $%
1+6957z^{14}+97035z^{16}+\cdots $ \\ \hline
\end{tabular}}
\vspace*{0.2cm}\caption{\rm\em Bordered $n$-block circulant $\left[ 74,37,14\right]^{b}$
FSD codes over $\mathbb{F}_{2}$ with $a=1=b$} \label{tab:74}
\end{center}
\end{table}

\begin{table}[h!]
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
$n$ & $m$ & $r_{1},\ldots ,r_{n}$ & $a,b$ & Partial weight distribution \\
\hline
$2$ & $9$ & $\left( u03033u00\right) ,\left( 01100u131\right) $ & $3,1$ & $%
1+4518z^{14}+80364z^{16}+\cdots $ \\ \hline
$2$ & $9$ & $\left( u10013303\right) ,\left( u00111uu0\right) $ & $1,1$ & $%
1+4554z^{14}+81072z^{16}+\cdots $ \\ \hline
$2$ & $9$ & $\left( 3u1011111\right) ,\left( u3u331010\right) $ & $3,1$ & $%
1+4740z^{14}+79836z^{16}+\cdots $ \\ \hline
$2$ & $9$ & $\left( 0u03013u3\right) ,\left( 1uu33311u\right) $ & $1,3$ & $%
1+5202z^{14}+77610z^{16}+\cdots $ \\ \hline
$2$ & $9$ & $\left( 133uu0310\right) ,\left( 3u310u101\right) $ & $3,1$ & $%
1+5208z^{14}+77916z^{16}+\cdots $ \\ \hline
$2$ & $9$ & $\left( uuu003131\right) ,\left( 0010uuuu1\right) $ & $1,1$ & $%
1+5262z^{14}+77232z^{16}+\cdots $ \\ \hline
$2$ & $9$ & $\left( 03u110100\right) ,\left( 111113033\right) $ & $3,1$ & $%
1+5340z^{14}+77526z^{16}+\cdots $ \\ \hline $3$ & $6$ & $\left(
11303u\right) ,\left( 30u3u3\right) ,\left( 330131\right) $ & $3,1$
& $1+4206z^{14}+82047z^{16}+\cdots $ \\ \hline $3$ & $6$ & $\left(
uu0133\right) ,\left( u31u10\right) ,\left( 00u0u0\right) $ & $3,3$
& $1+4716z^{14}+80295z^{16}+\cdots $ \\ \hline $3$ & $6$ & $\left(
103uu3\right) ,\left( 31u1u0\right) ,\left( 1101u3\right) $ & $1,1$
& $1+4842z^{14}+78663z^{16}+\cdots $ \\ \hline $3$ & $6$ & $\left(
331033\right) ,\left( 31u010\right) ,\left( 03u113\right) $ & $3,3$
& $1+4866z^{14}+79239z^{16}+\cdots $ \\ \hline $3$ & $6$ & $\left(
111313\right) ,\left( u033u1\right) ,\left( 0u311u\right) $ & $1,1$
& $1+4866z^{14}+79455z^{16}+\cdots $ \\ \hline $3$ & $6$ & $\left(
111300\right) ,\left( 1001u3\right) ,\left( u1303u\right) $ & $1,3$
& $1+5778z^{14}+75711z^{16}+\cdots $ \\ \hline
\end{tabular}%
\vspace*{0.2cm}
\caption{\rm\em Binary $\left[ 76,38,14\right]^{b}$ FSD codes as Gray
images of bordered $n$-block circulant codes over $R_{1}$}
\label{tab:76}
\end{center}
\end{table}
\clearpage

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\end{document}
