# A family of projective two-weight linear codes

Ziling Heng<EMAIL_ADDRESS>Dexiang Li<EMAIL_ADDRESS>Jiao Du
<EMAIL_ADDRESS>Fuling Chen<EMAIL_ADDRESS>School of Science,
Chang’an University, Xi’an 710064, China  School of Mathematics and
Information Science, Henan Normal University, Xinxiang 453007, China  Henan
Engineering Laboratory for Big Data Statistical Analysis and Optimal Control,
Henan Normal University, Xinxiang 453007, China

###### Abstract

Projective two-weight linear codes are closely related to finite projective
spaces and strongly regular graphs. In this paper, a family of $q$-ary
projective two-weight linear codes is presented, where $q$ is a power of 2.
The parameters of both the codes and their duals are excellent. As
applications, the codes are used to derive strongly regular graphs with new
parameters and secret sharing schemes with interesting access structures.

###### keywords:

Linear code , strongly regular graph , secret sharing schemes

###### MSC:

94B05 , 05E30 , 15A03

fn1fn1footnotetext:  The research of this paper was supported in part by the
National Natural Science Foundation of China under Grant 11901049 and
11971004, in part by the the Natural Science Basic Research Program of Shaanxi
under Grant 2020JQ-343, and in part by the Young Talent Fund of University
Association for Science and Technology in Shaanxi, China, under Grant
20200505.

##  1 Introduction

An $[n,k,d]$ _linear code_ ${\mathcal{C}}$ of length $n$ over
${\mathbb{F}}_{q}$ is a $k$-dimensional linear subspace of
${\mathbb{F}}_{q}^{n}$ with minimal Hamming distance $d$, where
${\mathbb{F}}_{q}$ denotes the finite field with $q$ elements for a prime
power $q$. There exist some bounds on the parameters $n,k,d$ [13]. An
$[n,k,d]$ code is called (distance) optimal if no $[n,k,d+1]$ codes exists. An
$[n,k,d-1]$ code is called almost (distance) optimal if $[n,k,d]$ code is
optimal. The _dual_ code ${\mathcal{C}}^{\perp}$ of a linear code
${\mathcal{C}}$ is defined as
${\mathcal{C}}^{\perp}=\left\\{\textbf{c}^{\perp}\in{\mathbb{F}}_{q}^{n}:\langle\textbf{c}^{\perp},\textbf{c}\rangle=0\
\forall\ \textbf{c}\in{\mathcal{C}}\right\\}$. Clearly, the dimension of
${\mathcal{C}}^{\perp}$ is $n-k$ if ${\mathcal{C}}$ has dimension $k$. A
linear code ${\mathcal{C}}$ is said to be _projective_ if the minimal distance
of its dual is at least 3.

Let $A_{i}$ denote the number of codewords with weight $i$ in a code
${\mathcal{C}}$ of length $n$. The sequence $(1,A_{1},A_{2},\cdots,A_{n})$ is
called the _weight distribution_ of ${\mathcal{C}}$ and the polynomial
$1+A_{1}z+A_{2}z^{2}+\cdots+A_{n}z^{n}$ is called the _weight enumerator_ of
${\mathcal{C}}$. Denote by $t=\sharp\\{A_{i}:A_{i}>0,\ 1\leq i\leq n\\}$. Then
a code with weight distribution $(1,A_{1},A_{2},\cdots,A_{n})$ is called a
$t$-weight code. Weight distribution is an interesting research subject in
coding theory as it not only contains the information of the capabilities of
error detection and correction, but also allows the computation of the error
probability of error detection and correction of a given code. Many linear
codes with only a few weights were reported in the literature [1, 3, 4, 5, 6,
8, 14, 15, 16, 17, 18, 19, 20]. In particular, projective two-weight codes are
very precious as they are closely related to finite projective spaces,
strongly regular graphs and combinatorial designs [1, 2, 5]. However,
projective two-weight codes are rare and only a few families are known [1, 3,
9].

Let $D=\\{d_{1},d_{2},\cdots,d_{n}\\}\subseteq{\mathbb{F}}_{q^{m}}$ be a
nonempty set. Let ${\mathrm{Tr}}_{q^{m}/q}$ be the trace function from
${\mathbb{F}}_{q^{m}}$ to ${\mathbb{F}}_{q}$. Define a linear code of length
$n$ as

$\displaystyle{\mathcal{C}}_{D}=\\{({\mathrm{Tr}}_{q^{m}/q}(bd_{1}),{\mathrm{Tr}}_{q^{m}/q}(bd_{2}),\cdots,{\mathrm{Tr}}_{q^{m}/q}(bd_{n})):b\in{\mathbb{F}}_{q^{m}}\\}.$
(1)

The set $D$ is called the _defining set_ and the ordering of the elements in
$D$ doesn’t affect the parameters and weight distribution of
${\mathcal{C}}_{D}$. Recently, many good codes were obtained with different
proper defining sets [3, 4, 8, 9, 11, 15, 16, 17, 19, 20].

Let $s$ be a divisor of a positive integer $m$, ${\mathrm{N}}_{q^{m}/q^{s}}$
the norm function from ${\mathbb{F}}_{q^{m}}^{*}$ to
${\mathbb{F}}_{q^{s}}^{*}$. In this paper, we study the $q$-ary linear code
${\mathcal{C}}_{D}$ in (1) with the defining set

$D=\left\\{x\in{\mathbb{F}}_{q^{m}}^{*}:{\mathrm{Tr}}_{q^{s}/q}({\mathrm{N}}_{q^{m}/q^{s}}(x))+c=0\right\\},\
c\in{\mathbb{F}}_{q}.$

Known results on the code ${\mathcal{C}}_{D}$ are summarized as follows:

  1. 1.

If $q$ is an odd prime, $c=0$ and $m=2s$, the code ${\mathcal{C}}_{D}$ is a
two-weight code and its complete weight enumerator was given in [11].

  2. 2.

If $c=0$ and $s=2$, the code ${\mathcal{C}}_{D}$ is a two-weight code and its
weight enumerator was given in [9].

  3. 3.

If $c\in{\mathbb{F}}_{q}^{*},s=2$ and $\gcd(m/2,q-1)=1$, the code
${\mathcal{C}}_{D}$ is a two-weight code and its weight enumerator was given
in [9].

The objective of this paper is to investigate the weight distribution of
${\mathcal{C}}_{D}$ for $m=2s$, any prime power $q$ and any
$c\in{\mathbb{F}}_{q}$. The parameters of its dual are also determined. It is
interesting that ${\mathcal{C}}_{D}$ is a projective two-weight code if
$m=2s$, $q$ is even and $c\in{\mathbb{F}}_{q}^{*}$. As an application,
strongly regular graphs with new parameters can be derived with the projective
two-weight codes. As another application, all codes obtained in this paper can
be used to construct secret sharing schemes with interesting access schemes.

##  2 Preliminaries

Let $p$ be a prime and $q=p^{e}$. Denote by $\zeta_{p}$ the primitive $p$-th
root of complex unity. For any $a\in{\mathbb{F}}_{q}$, the function
$\varphi_{a}(x)=\zeta_{p}^{{\mathrm{Tr}}_{q/p}(ax)},\ x\in{\mathbb{F}}_{q}$,
is called an _additive character_ of ${\mathbb{F}}_{q}$. If $a=0$,
$\varphi_{0}(x)=1$ for all $x\in{\mathbb{F}}_{q}$ and $\varphi_{0}$ is called
the trivial additive character of ${\mathbb{F}}_{q}$. If $a=1$, $\varphi_{1}$
is said to be the canonical additive character of ${\mathbb{F}}_{q}$. The
orthogonality relation of additive characters (see [12]) is given by

$\sum_{x\in{\mathbb{F}}_{q}}\varphi_{1}(ax)=\left\\{\begin{array}[]{rl}q&\mbox{
for }a=0,\\\ 0&\mbox{ for }a\in{\mathbb{F}}_{q}^{*}.\end{array}\right.$

Let ${\mathbb{F}}_{q}^{*}={\mathbb{F}}_{q}\setminus\\{0\\}$ and $\alpha$ be a
primitive element of it. Define the _multiplicative characters_ of
${\mathbb{F}}_{q}$ by $\psi_{j}(\alpha^{k})=\zeta_{q-1}^{jk}\mbox{ for
}k=0,1,\cdots,q-1$, where $0\leq j\leq q-2$. In particular, $\psi_{0}$ is
called the trivial multiplicative character and $\eta:=\psi_{(q-1)/2}$ is
referred to as the quadratic multiplicative character of ${\mathbb{F}}_{q}$.
The orthogonality relation of multiplicative characters (see [12]) is given by

$\sum_{x\in{\mathbb{F}}_{q}^{*}}\psi_{j}(x)=\left\\{\begin{array}[]{rl}q-1&\mbox{
for }j=0,\\\ 0&\mbox{ for }j\neq 0.\end{array}\right.$

For an additive character $\varphi$ and a multiplicative character $\psi$ of
${\mathbb{F}}_{q}$, the _Gauss sum_ $G(\psi,\varphi)$ over ${\mathbb{F}}_{q}$
is defined by

$G(\psi,\varphi)=\sum_{x\in{\mathbb{F}}_{q}^{*}}\psi(x)\varphi(x).$

We call $G(\eta,\varphi)$ the quadratic Gauss sum over ${\mathbb{F}}_{q}$ for
nontrivial $\varphi$.

######  Lemma 1.

[12, Th. 5.15] Let $q=p^{e}$ with $p$ odd. Let $\chi$ be the canonical
additive character of ${\mathbb{F}}_{q}$. Then

$\displaystyle G(\eta,\varphi)$ $\displaystyle=$
$\displaystyle(-1)^{e-1}(\sqrt{-1})^{(\frac{p-1}{2})^{2}e}\sqrt{q}$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lll}(-1)^{e-1}\sqrt{q}&\mbox{ for
}p\equiv 1\pmod{4},\\\ (-1)^{e-1}(\sqrt{-1})^{e}\sqrt{q}&\mbox{ for }p\equiv
3\pmod{4}.\end{array}\right.$

######  Lemma 2.

[12, Th. 5.33] Let $\varphi$ be a nontrivial additive character of
${\mathbb{F}}_{q}$ with $q$ odd, and let
$f(x)=a_{2}x^{2}+a_{1}x+a_{0}\in{\mathbb{F}}_{q}[x]$ with $a_{2}\neq 0$. Then

$\sum_{c\in{\mathbb{F}}_{q}}\varphi(f(c))=\varphi(a_{0}-a_{1}^{2}(4a_{2})^{-1})\eta(a_{2})G(\eta,\varphi).$

######  Lemma 3.

[12, Cor. 5.35] Let $\varphi_{b}$ be a nontrivial additive character of
${\mathbb{F}}_{q}$ with $b\in{\mathbb{F}}_{q}^{*}$, and let
$f(x)=a_{2}x^{2}+a_{1}x+a_{0}\in{\mathbb{F}}_{q}[x]$ with $q$ even. Then

$\sum_{c\in{\mathbb{F}}(q)}\varphi_{b}(f(c))=\left\\{\begin{array}[]{ll}\chi_{b}(a_{0})q&\mbox{
if }a_{2}=ba_{1}^{2},\\\ 0&\mbox{ otherwise. }\end{array}\right.$

##  3 A class of $q$-ary two-weight linear codes

Let $s$ be a positive integer and $m=2s$. Let $q$ be a power of a prime $p$.
In this section, we study the code ${\mathcal{C}}_{D}$ in (1) with the
defining set

$D=\left\\{x\in{\mathbb{F}}_{q^{m}}^{*}:{\mathrm{Tr}}_{q^{s}/q}(x^{q^{s}+1})+c=0\right\\}\
\text{ for }c\in{\mathbb{F}}_{q}.$

Denote by $n=\sharp D$. The following lemma is useful for determining the
length of ${\mathcal{C}}_{D}$.

######  Lemma 4.

[7, Lemma 7] For
$a\in{\mathbb{F}}_{q^{s}}^{*},b\in{\mathbb{F}}_{q^{m}},c\in{\mathbb{F}}_{q}$
and $m=2s$, define an exponential sum by

$\Delta(a,b,c)=\sum_{y\in{\mathbb{F}}_{q}^{*}}\varphi(yc)\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\varphi\left(y\left({\mathrm{Tr}}_{q^{s}/q}(ax^{q^{s}+1})+{\mathrm{Tr}}_{q^{m}/q}(bx)\right)\right),$

where $\varphi$ is the canonical additive character of ${\mathbb{F}}_{q}$.
Then

$\displaystyle\Delta(a,b,c)=\begin{cases}(q^{s}+1)(1-q)&\text{ if
$c=0,{\mathrm{Tr}}_{q^{s}/q}\left(\frac{b^{q^{s}+1}}{a}\right)=0$},\\\
q^{s}+1-q&\text{ if
$c=0,{\mathrm{Tr}}_{q^{s}/q}\left(\frac{b^{q^{s}+1}}{a}\right)\neq 0$},\\\
q^{s}+1&\begin{subarray}{c}\mbox{ if }c\neq 0,\
{\mathrm{Tr}}_{q^{s}/q}\left(\frac{b^{q^{s}+1}}{a}\right)\neq 0,\
c\neq{\mathrm{Tr}}_{q^{s}/q}\left(\frac{b^{q^{s}+1}}{a}\right),\mbox{ or
}c\neq 0,\
{\mathrm{Tr}}_{q^{s}/q}\left(\frac{b^{q^{s}+1}}{a}\right)=0,\end{subarray}\\\
q^{s}+1-q^{s+1}&\text{ if
$c={\mathrm{Tr}}_{q^{s}/q}\left(\frac{b^{q^{s}+1}}{a}\right)\neq
0$}.\end{cases}$

######  Lemma 5.

The length $n$ of ${\mathcal{C}}_{D}$ is given by

$\displaystyle n=\left\\{\begin{array}[]{ll}(q^{s}+1)(q^{s-1}-1)&\text{ if
}c=0,\\\ q^{s-1}(q^{s}+1)&\text{ if
}c\in{\mathbb{F}}_{q}^{*}.\end{array}\right.$

###### Proof.

By the orthogonal relation of additive characters and Lemma 4,

$\displaystyle
n=\frac{1}{q}\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\sum_{y\in{\mathbb{F}}_{q}}\zeta_{p}^{{\mathrm{Tr}}_{q/p}\left(y{\mathrm{Tr}}_{q^{s}/q}(x^{q^{s}+1})+yc\right)}=\frac{q^{m}-1}{q}+\frac{1}{q}\sum_{y\in{\mathbb{F}}_{q}^{*}}\varphi(yc)\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\varphi\left(y{\mathrm{Tr}}_{q^{s}/q}(x^{q^{s}+1})\right)=\frac{q^{m}-1}{q}+\frac{1}{q}\Delta(1,0,c).$

Then the desired conclusion follows from Lemma 4. ∎

Now we determine the value of the following exponential sums used to obtain
the weight distribution of ${\mathcal{C}}_{D}$.

######  Lemma 6.

Let $b\in{\mathbb{F}}_{q^{m}}^{*}$ with $m=2s$. Denote by

$S_{c}(b)=\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)\varphi\left(y{\mathrm{Tr}}_{q^{m}/q}(bx)+z{\mathrm{Tr}}_{q^{s}/q}(x^{q^{s}+1})\right),\
c\in{\mathbb{F}}_{q}.$

If $c=0$ and $q$ is any prime power, then

$\displaystyle S_{0}(b)$ $\displaystyle=$
$\displaystyle\begin{cases}-(q-1)^{2}(q^{s}+1)&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$},\\\ (q-1)(q^{s}-q+1)&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}.\end{cases}$

If $c\in{\mathbb{F}}_{q}^{*}$ and $q$ is even, then

$\displaystyle S_{c}(b)$ $\displaystyle=$
$\displaystyle\begin{cases}(q^{s}+1)(q-1)&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$},\\\ q-1-q^{s}&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}.\end{cases}$

If $c\in{\mathbb{F}}_{q}^{*}$ and $q=p^{e}$ is odd with $\eta$ the quadratic
character of ${\mathbb{F}}_{q}^{*}$,

$S_{c}(b)=\begin{cases}(q^{s}+1)(q-1)&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$},\\\
(q-1)-q^{s}\left(1+(-1)^{(\frac{p-1}{2})^{2}e}q\eta\left(-c{\mathrm{Tr}}_{q^{s}/q}\left(b^{q^{s}+1}\right)\right)\right)&\text{
if ${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}.\\\ \end{cases}$

###### Proof.

Recall that $m=2s$. By the transitivity of trace functions, we have

$\displaystyle S_{c}(b)$ $\displaystyle=$
$\displaystyle\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\varphi\left(z\left({\mathrm{Tr}}_{q^{s}/q}(x^{q^{s}+1})+{\mathrm{Tr}}_{q^{m}/q}\left(\frac{y}{z}bx\right)\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\varphi\left(z\left({\mathrm{Tr}}_{q^{s}/q}(x^{q^{s}+1})+{\mathrm{Tr}}_{q^{s}/q}\left({\mathrm{Tr}}_{q^{m}/q^{s}}\left(\frac{y}{z}bx\right)\right)\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\varphi\left(z\left({\mathrm{Tr}}_{q^{s}/q}\left(x^{q^{s}+1}+\frac{ybx}{z}+\frac{yb^{q^{s}}x^{q^{s}}}{z}\right)\right)\right).$

Note that

$\displaystyle
x^{q^{s}+1}+\frac{ybx}{z}+\frac{yb^{q^{s}}x^{q^{s}}}{z}=\left(x+\frac{yb^{q^{s}}}{z}\right)^{q^{s}+1}-\frac{y^{2}b^{q^{s}+1}}{z^{2}}$

as $y,z\in{\mathbb{F}}_{q}^{*}$. Hence

$\displaystyle S_{c}(b)$ $\displaystyle=$
$\displaystyle\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\varphi\left(z\left({\mathrm{Tr}}_{q^{s}/q}\left(\left(x+\frac{yb^{q^{s}}}{z}\right)^{q^{s}+1}-\frac{y^{2}b^{q^{s}+1}}{z^{2}}\right)\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi\left(zc-{\mathrm{Tr}}_{q^{s}/q}\left(\frac{y^{2}b^{q^{s}+1}}{z}\right)\right)\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\varphi\left(z{\mathrm{Tr}}_{q^{s}/q}\left(x+\frac{yb^{q^{s}}}{z}\right)^{q^{s}+1}\right)$
$\displaystyle=$
$\displaystyle-\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi\left(zc-{\mathrm{Tr}}_{q^{s}/q}\left(\frac{y^{2}b^{q^{s}+1}}{z}\right)\right)\varphi\left({\mathrm{Tr}}_{q^{s}/q}\left(\frac{y^{2}b^{q^{s}+1}}{z}\right)\right)$
$\displaystyle+\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi\left(zc-{\mathrm{Tr}}_{q^{s}/q}\left(\frac{y^{2}b^{q^{s}+1}}{z}\right)\right)\sum_{x\in{\mathbb{F}}_{q^{m}}}\varphi\left(z{\mathrm{Tr}}_{q^{s}/q}\left(\left(x+\frac{yb^{q^{s}}}{z}\right)^{q^{s}+1}\right)\right)$
$\displaystyle=$
$\displaystyle-\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)+\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi\left(zc-{\mathrm{Tr}}_{q^{s}/q}\left(\frac{y^{2}b^{q^{s}+1}}{z}\right)\right)\sum_{x^{\prime}\in{\mathbb{F}}_{q^{m}}}\varphi\left({\mathrm{Tr}}_{q^{s}/q}(zx^{\prime
q^{s}+1})\right)$ $\displaystyle=$
$\displaystyle-\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)+\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi\left(zc-{\mathrm{Tr}}_{q^{s}/q}\left(\frac{y^{2}b^{q^{s}+1}}{z}\right)\right)\sum_{x^{\prime}\in{\mathbb{F}}_{q^{m}}}\phi\left(zx^{\prime
q^{s}+1})\right),$

where we used the substitution $x+\frac{yb^{q^{s}}}{z}\mapsto x^{\prime}$ in
the forth equality and $\phi$ denotes the canonical additive character of
${\mathbb{F}}_{q^{s}}$. Since

$\displaystyle\sum_{x^{\prime}\in{\mathbb{F}}_{q^{m}}}\phi\left(zx^{\prime
q^{s}+1})\right)=1+\sum_{x^{\prime}\in{\mathbb{F}}_{q^{m}}^{*}}\phi\left(zx^{\prime
q^{s}+1})\right)=1+(q^{s}+1)\sum_{x^{\prime}\in{\mathbb{F}}_{q^{s}}^{*}}\phi\left(zx^{\prime}\right)=-q^{s}$

for $z\in{\mathbb{F}}_{q}^{*}$, we have

$\displaystyle S_{c}(b)$ $\displaystyle=$
$\displaystyle-\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)-q^{s}\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi\left(zc-{\mathrm{Tr}}_{q^{s}/q}\left(\frac{y^{2}b^{q^{s}+1}}{z}\right)\right)$
(4) $\displaystyle=$
$\displaystyle-\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)-q^{s}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)\sum_{y\in{\mathbb{F}}_{q}^{*}}\varphi\left(-{\mathrm{Tr}}_{q^{s}/q}\left(\frac{b^{q^{s}+1}}{z}\right)y^{2}\right)$
$\displaystyle=$
$\displaystyle-\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)+q^{s}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)-q^{s}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)\sum_{y\in{\mathbb{F}}_{q}}\varphi\left(-{\mathrm{Tr}}_{q^{s}/q}\left(\frac{b^{q^{s}+1}}{z}\right)y^{2}\right).$

Now we determine the value of $S_{c}(b)$ in the following two cases:

  1. 1.

Let $q$ be even. By Lemma 3,

$\sum_{y\in{\mathbb{F}}_{q}}\varphi\left(-{\mathrm{Tr}}_{q^{s}/q}\left(\frac{b^{q^{s}+1}}{z}\right)y^{2}\right)=\begin{cases}q&\text{
if ${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$},\\\ 0&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}.\end{cases}$

By Equation (4), we then have

$\displaystyle S_{c}(b)$ $\displaystyle=$
$\displaystyle\begin{cases}-\sum\limits_{y\in{\mathbb{F}}_{q}^{*}}\sum\limits_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)+q^{s}\sum\limits_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)-q^{s+1}\sum\limits_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)&\text{
if ${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$},\\\
-\sum\limits_{y\in{\mathbb{F}}_{q}^{*}}\sum\limits_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)+q^{s}\sum\limits_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)&\text{
if ${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}.\end{cases}$

Since

$\sum\limits_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)=\begin{cases}q-1&\text{ if
$c=0$,}\\\ -1&\text{ if $c\neq 0$}.\end{cases}$

the value of $S_{c}(b)$ follows.

  2. 2.

Let $q$ be odd and $\eta$ the quadratic character of ${\mathbb{F}}_{q}^{*}$.
If $c\in{\mathbb{F}}_{q}^{*}$, by Lemmas 2 and 1,

$\displaystyle\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)\sum_{y\in{\mathbb{F}}_{q}}\varphi\left(-{\mathrm{Tr}}_{q^{s}/q}\left(\frac{b^{q^{s}+1}}{z}\right)y^{2}\right)$
$\displaystyle=$
$\displaystyle\begin{cases}q\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)&\text{
if ${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$}\\\
G(\eta,\varphi)\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)\eta\left(-{\mathrm{Tr}}_{q^{s}/q}\left(\frac{b^{q^{s}+1}}{z}\right)\right)&\text{
if ${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}\\\ \end{cases}$
$\displaystyle=$ $\displaystyle\begin{cases}-q&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$}\\\
G(\eta,\varphi)\eta\left(-c{\mathrm{Tr}}_{q^{s}/q}\left(b^{q^{s}+1}\right)\right)\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)\eta(cz)&\text{
if ${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}\\\ \end{cases}$
$\displaystyle=$ $\displaystyle\begin{cases}-q&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$}\\\
G(\eta,\varphi)^{2}\eta\left(-c{\mathrm{Tr}}_{q^{s}/q}\left(b^{q^{s}+1}\right)\right)&\text{
if ${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}\\\ \end{cases}$
$\displaystyle=$ $\displaystyle\begin{cases}-q&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$},\\\
(-1)^{(\frac{p-1}{2})^{2}e}q\eta\left(-c{\mathrm{Tr}}_{q^{s}/q}\left(b^{q^{s}+1}\right)\right)&\text{
if ${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}.\\\ \end{cases}$

By Equation (4), we then obtain

$S_{c}(b)=\begin{cases}(q^{s}+1)(q-1)&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$},\\\
(q-1)-q^{s}\left(1+(-1)^{(\frac{p-1}{2})^{2}e}q\eta\left(-c{\mathrm{Tr}}_{q^{s}/q}\left(b^{q^{s}+1}\right)\right)\right)&\text{
if ${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}.\\\ \end{cases}$

If $c=0$, by Lemmas 2 and 1,

$\displaystyle\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)\sum_{y\in{\mathbb{F}}_{q}}\varphi\left(-{\mathrm{Tr}}_{q^{s}/q}\left(\frac{b^{q^{s}+1}}{z}\right)y^{2}\right)$
$\displaystyle=$ $\displaystyle\begin{cases}q(q-1)&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$}\\\
G(\eta,\varphi)\sum_{z\in{\mathbb{F}}_{q}^{*}}\eta\left(-{\mathrm{Tr}}_{q^{s}/q}\left(\frac{b^{q^{s}+1}}{z}\right)\right)&\text{
if ${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}\\\ \end{cases}$
$\displaystyle=$ $\displaystyle\begin{cases}q(q-1)&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$}\\\
G(\eta,\varphi)\sum_{z\in{\mathbb{F}}_{q}^{*}}\eta\left(-{\mathrm{Tr}}_{q^{s}/q}\left(b^{q^{s}+1}\right)z\right)&\text{
if ${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}\\\ \end{cases}$
$\displaystyle=$ $\displaystyle\begin{cases}q(q-1)&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$},\\\ 0&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$},\\\ \end{cases}$

where the second equation holds due to $\eta(z)=\eta(1/z)$ and the third one
holds due to the orthogonal relation of multiplicative characters. By Equation
(4), we then obtain

$S_{0}(b)=\begin{cases}-(q^{s}+1)(q-1)^{2}&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$},\\\ (q-1)(q^{s}-q+1)&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}.\\\ \end{cases}$

The proof is completed. ∎

In the following, we study the weight enumerator of ${\mathcal{C}}_{D}$.

######  Theorem 7.

Let $s$ be a positive integer and $m=2s$. If $c=0,s>1$ and $q$ is any prime
power, then ${\mathcal{C}}_{D}$ is a $q$-ary
$[(q^{s}+1)(q^{s-1}-1),m,(q^{2s-2}-q^{s-1})(q-1)]$ linear code with weight
enumerator

$1+(q^{s}+1)(q^{s}-q^{s-1})z^{(q^{2s-2}-q^{s-1})(q-1)}+(q^{s}+1)(q^{s-1}-1)z^{q^{2s-2}(q-1)}.$

If $c\in{\mathbb{F}}_{q}^{*}$ and $q$ is even, then ${\mathcal{C}}_{D}$ is a
$q$-ary $[q^{s-1}(q^{s}+1),m,q^{2s-2}(q-1)]$ linear code with weight
enumerator

$1+(q^{s}+1)(q^{s-1}-1)z^{q^{2s-2}(q-1)}+(q^{s}+1)(q^{s}-q^{s-1})z^{q^{2s-2}(q-1)+q^{s-1}}.$

If $c\in{\mathbb{F}}_{q}^{*}$ and $q$ is odd, then ${\mathcal{C}}_{D}$ is a
$q$-ary $[q^{s-1}(q^{s}+1),m,q^{2s-2}(q-1)]$ linear code with weight
enumerator

$1+\frac{(q^{s}+1)(q^{s}+q^{s-1}-2)}{2}z^{q^{2s-2}(q-1)}+\frac{q^{s-1}(q-1)(q^{s}+1)}{2}z^{q^{2s-2}(q-1)+2q^{s-1}}.$

###### Proof.

For any codeword

${\mathbf{c}}(b)=\left({\mathrm{Tr}}_{q^{m}/q}(bd_{1}),{\mathrm{Tr}}_{q^{m}/q}(bd_{2}),\cdots,{\mathrm{Tr}}_{q^{m}/q}(bd_{n})\right)\in{\mathcal{C}}_{D}$

with $b\in{\mathbb{F}}_{q^{m}}^{*}$, by the orthogonal relation of additive
characters, its Hamming weight

$\displaystyle{\mathtt{wt}}({\mathbf{c}}(b))$ $\displaystyle=$ $\displaystyle
n-\sharp\left\\{x\in{\mathbb{F}}_{q^{m}}^{*}:{\mathrm{Tr}}_{q^{m}/q}(bx)=0\mbox{
and }{\mathrm{Tr}}_{q^{s}/q}(x^{q^{s}+1})+c=0\right\\}$ $\displaystyle=$
$\displaystyle
n-\frac{1}{q^{2}}\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\sum_{y\in{\mathbb{F}}_{q}}\zeta_{p}^{y{\mathrm{Tr}}_{q^{m}/q}(bx)}\sum_{z\in{\mathbb{F}}_{q}}\zeta_{p}^{z\left({\mathrm{Tr}}_{q^{s}/q}(x^{q^{s}+1})+c\right)}$
$\displaystyle=$ $\displaystyle
n-\frac{1}{q^{2}}\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\sum_{y\in{\mathbb{F}}_{q}}\sum_{z\in{\mathbb{F}}_{q}}\varphi(zc)\varphi\left(y{\mathrm{Tr}}_{q^{m}/q}(bx)+z{\mathrm{Tr}}_{q^{s}/q}(x^{q^{s}+1})\right)$
$\displaystyle=$ $\displaystyle
n-\frac{q^{m}-1}{q^{2}}-\frac{1}{q^{2}}\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\varphi\left(y{\mathrm{Tr}}_{q^{m}/q}(bx)\right)-\frac{1}{q^{2}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\varphi(zc)\varphi\left(z{\mathrm{Tr}}_{q^{s}/q}(x^{q^{s}+1})\right)$
$\displaystyle-\frac{1}{q^{2}}\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\varphi(zc)\varphi\left(y{\mathrm{Tr}}_{q^{m}/q}(bx)+z{\mathrm{Tr}}_{q^{s}/q}(x^{q^{s}+1})\right)$
$\displaystyle=$ $\displaystyle
n-\frac{q^{m}-1}{q^{2}}-\frac{1}{q^{2}}\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\varphi\left(y{\mathrm{Tr}}_{q^{m}/q}(bx)\right)-\frac{1}{q^{2}}\sum_{z\in{\mathbb{F}}_{q}^{*}}\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\varphi(zc)\varphi\left(z{\mathrm{Tr}}_{q^{s}/q}(x^{q^{s}+1})\right)-\frac{1}{q^{2}}S_{c}(b),$

where $S_{c}(b)$ was defined in Lemma 6. Let $\chi$ be the canonical additive
character of ${\mathbb{F}}_{q^{m}}$. By the transitivity of trace functions,

$\displaystyle\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\varphi\left(y{\mathrm{Tr}}_{q^{m}/q}(bx)\right)=\sum_{y\in{\mathbb{F}}_{q}^{*}}\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\chi(ybx)=-(q-1).$
(5)

By Lemma 4,

$\displaystyle\sum_{z\in{\mathbb{F}}_{q}^{*}}\sum_{x\in{\mathbb{F}}_{q^{m}}^{*}}\varphi(zc)\varphi\left(z{\mathrm{Tr}}_{q^{s}/q}(x^{q^{s}+1})\right)=\Delta(1,0,c)=\left\\{\begin{array}[]{ll}(q^{s}+1)(1-q)&\text{
if }c=0,\\\ q^{s}+1&\text{ if }c\in{\mathbb{F}}_{q}^{*}.\end{array}\right.$
(8)

Now we discuss the Hamming weight in several cases as follows.

  1. 1.

Let $c=0$. By Equations (3), (5), (8) and Lemmas 5 and 6, we directly have

$\displaystyle{\mathtt{wt}}({\mathbf{c}}(b))$ $\displaystyle=$
$\displaystyle\begin{cases}q^{2s-2}(q-1)&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$},\\\ (q^{2s-2}-q^{s-1})(q-1)&\text{
if ${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}.\end{cases}$

For $b\in{\mathbb{F}}_{q^{m}}^{*}$,
${\mathrm{Tr}}_{q^{s}/q}\left(b^{q^{s}+1}\right)=0\Leftrightarrow
b^{q^{s}+1}\in\ker({\mathrm{Tr}}_{q^{s}/q})\backslash\\{0\\}.$ For fixed
$b^{\prime}\in\ker({\mathrm{Tr}}_{q/p})\backslash\\{0\\}$, there exist
$q^{s}+1$ elements $b$ such that $b^{q^{s}+1}=b^{\prime}$. Hence
$A_{q^{2s-2}(q-1)}=(q^{s}+1)(q^{s-1}-1)$ and

$A_{(q^{2s-2}-q^{s-1})(q-1)}=q^{2s}-1-A_{q^{2s-2}(q-1)}=(q^{s}+1)(q^{s}-q^{s-1}).$

Then the weight distribution follows in this case.

  2. 2.

Let $c\in{\mathbb{F}}_{q}^{*}$ and $q$ be even.By Equations (3), (5), (8) and
Lemmas 5 and 6, we directly have

$\displaystyle{\mathtt{wt}}({\mathbf{c}}(b))$ $\displaystyle=$
$\displaystyle\begin{cases}q^{2s-2}(q-1)&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$},\\\ q^{2s-2}(q-1)+q^{s-1}&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq 0$}.\end{cases}$

Similarly to Case 1), we have $A_{q^{2s-2}(q-1)}=(q^{s}+1)(q^{s-1}-1)$ and
$A_{q^{2s-2}(q-1)+q^{s-1}}=(q^{s}+1)(q^{s}-q^{s-1})$. Then the weight
distribution follows in this case.

  3. 3.

Let $c\in{\mathbb{F}}_{q}^{*}$ and $q=p^{e}$ be odd. By Equations (3), (5),
(8) and Lemmas 5 and 6, we directly derive

${\mathtt{wt}}({\mathbf{c}}(b))=\begin{cases}q^{2s-2}(q-1)&\text{ if
${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})=0$},\\\ q^{2s-2}(q-1)\text{ or
}q^{2s-2}(q-1)+2q^{s-1}&\text{ if ${\mathrm{Tr}}_{q^{s}/q}(b^{q^{s}+1})\neq
0$}.\\\ \end{cases}$

The dimension of ${\mathcal{C}}_{D}$ is $m$ as $A_{0}=1$. Since $0\in D$, the
minimal distance $d^{\perp}$ of the dual of ${\mathcal{C}}_{D}$ can’t be 1.
Then $d^{\perp}\geq 2$. By the first two Pless power moments, we have

$\displaystyle\begin{cases}A_{q^{2s-2}(q-1)}+A_{q^{2s-2}(q-1)+2q^{s-1}}=q^{m}-1,\\\
q^{2s-2}(q-1)A_{q^{2s-2}(q-1)}+(q^{2s-2}(q-1)+2q^{s-1})A_{q^{2s-2}(q-1)+2q^{s-1}}=q^{m-1}(qn-n).\end{cases}$

Solving this equation system yields
$A_{q^{2s-2}(q-1)}=\frac{(q^{s}+1)(q^{s}+q^{s-1}-2)}{2}$ and
$A_{q^{2s-2}(q-1)+2q^{s-1}}=\frac{q^{s-1}(q-1)(q^{s}+1)}{2}$. Then the weight
distribution follows in this case.

The proof is completed. ∎

Examples of the linear code ${\mathcal{C}}_{D}$ are summarized in Table 1.
Some of them are optimal or almost optimal and some of them have the same
parameters as the best linear codes according to the Code Tables at
https://www.codetables.de/.

Table 1: Examples of ${\mathcal{C}}_{D}$ in Theorem 7 Length | Dimension | Minimal distance | $c$ | $m$ | $q$ | Optimality
---|---|---|---|---|---|---
5 | 4 | 2 | 0 | 4 | 2 | Optimal
27 | 6 | 12 | 0 | 6 | 2 | Optimal
51 | 4 | 36 | 0 | 4 | 4 | Best known
5 | 2 | 4 | 1 | 2 | 4 | Optimal
64 | 4 | 48 | 1 | 4 | 4 | Optimal
9 | 2 | 8 | 1 | 2 | 8 | Optimal
10 | 2 | 8 | 1 | 2 | 9 | Almost Optimal
30 | 4 | 18 | 1 | 4 | 3 | Almost optimal

Denote by $(1,A_{1}^{\perp},A_{2}^{\perp},\cdots,A_{n}^{\perp})$ the weight
distribution of ${\mathcal{C}}_{D}^{\perp}$. There exists a relationship
called Pless power moments between $(1,A_{1},A_{2},\cdots,A_{n})$ and
$(1,A_{1}^{\perp},A_{2}^{\perp},\cdots,A_{n}^{\perp})$ [10, Page 259]. Pless
power moments are useful for determining the minimal distance of
${\mathcal{C}}_{D}^{\perp}$.

######  Theorem 8.

Let $m=2s$. If $q=2,s\geq 3$ and $c=0$, then ${\mathcal{C}}_{D}^{\perp}$ is a
$[(q^{s}+1)(q^{s-1}-1),(q^{s}+1)(q^{s-1}-1)-m,3]$ linear code. If $q>2,s>1$
and $c=0$, then ${\mathcal{C}}_{D}^{\perp}$ is a
$[(q^{s}+1)(q^{s-1}-1),(q^{s}+1)(q^{s-1}-1)-m,2]$ linear code. If $q$ is even
and $c\in{\mathbb{F}}_{q}^{*}$, then ${\mathcal{C}}_{D}^{\perp}$ is a
$[q^{s-1}(q^{s}+1),q^{s-1}(q^{s}+1)-m,3]$ linear code. If $q$ is odd and
$c\in{\mathbb{F}}_{q}^{*}$, then ${\mathcal{C}}_{D}^{\perp}$ is a
$[q^{s-1}(q^{s}+1),q^{s-1}(q^{s}+1)-m,2]$ linear code.

###### Proof.

Since $0\not\in D$, the minimal distance $d^{\perp}$ of the dual satisfies
$d^{\perp}\geq 2$. By the second Pless power moment,

$\displaystyle\sum_{j=0}^{n}j^{2}A_{j}=q^{m-2}(q-1)n(qn-n+1)-2q^{m-2}A_{2}^{\perp}.\
$ (9)

  1. 1.

If $c=0$, by Theorem 7 and Equation (9),
$A_{2}^{\perp}=\frac{(q-1)(q-2)(q^{s}+1)(q^{s-1}-1)}{2}$. Hence
$A_{2}^{\perp}=0$ if and only if $q=2$ as $s>1$. Let $q=2$, by the third Pless
power moment,

$\displaystyle\sum_{j=0}^{n}j^{3}A_{j}=2^{2s-3}[n^{3}(n+3)-6A_{3}^{\perp}].$
(10)

By Theorem 7 and Equation (10),
$A_{3}^{\perp}=\frac{2^{3s-4}(2^{s}-3)-2^{s-2}(2^{s+1}-3)+1}{3}>0$ if $s\geq
3$ and $q=2$. Hence $d^{\perp}=3$ if $q=2,s\geq 3$ and $d^{\perp}=2$ if
$q>2,s>1$.

  2. 2.

If $c\in{\mathbb{F}}_{q}^{*}$ and $q$ is even, by Theorem 7 and Equation (9),
we have $A_{2}^{\perp}=0$. Then $d^{\perp}\geq 3$. By the third Pless power
moment,

$\displaystyle\sum_{j=0}^{n}j^{3}A_{j}=q^{2s-3}[(q-1)n(q^{2}n^{2}-2qn^{2}+3qn-q+n^{2}-3n+2)-6A_{3}^{\perp}].$
(11)

By Theorem 7 and Equation (11),
$A_{3}^{\perp}=\frac{q^{s-1}(q^{s}+1)(q-1)\left(q^{s-1}(2q-3)+q^{2s-2}(q-1)^{2}+2-q\right)}{6}>0$
for $c\in{\mathbb{F}}_{q}^{*}$ and even $q$. Then $d^{\perp}=3$.

  3. 3.

If $c\in{\mathbb{F}}_{q}^{*}$ and $q$ is odd, by Theorem 7 and Equation (9),
we have $A_{2}^{\perp}=\frac{q^{s-1}(q-1)(q^{s}+1)}{2}>0$. Then $d^{\perp}=2$.

The proof is completed. ∎

##  4 Strongly regular graphs with new parameters

A connected graph of $N$ vertices is called _strongly regular_ with parameters
$(N,K,\lambda,\mu)$ if it is regular with valency $K$ and if the number of
vertices joined to two given vertices is $\lambda$ or $\mu$ according as the
two given vertices are adjacent or non-adjacent.

Let $G=[\textbf{y}_{1},\textbf{y}_{2},\cdots,\textbf{y}_{n}]$ be the generator
matrix of a $[n,k]$ linear code ${\mathcal{C}}$ over ${\mathbb{F}}_{q}$, where
$\textbf{y}_{i}\in{\mathbb{F}}_{q}^{k}$. Let
$\textbf{V}={\mathbb{F}}_{q}^{k}$,
$\textbf{O}=\\{\langle\textbf{y}_{i}\rangle:i=1,2,\cdots,n\\}$ and
$\Omega=\\{\textbf{v}\in\textbf{V}:\langle\textbf{v}\rangle\in\textbf{O}\\}$.
Define a graph $G(\Omega)$ with vertices the vectors of V and where two
vertices are joint if and only if their difference is in $\Omega$. In [1, Th.
3.2], it was proved that $G(\Omega)$ is strongly regular if and only if
${\mathcal{C}}$ is a projective two-weight code. Assume that the nonzero
weights of ${\mathcal{C}}$ are $w_{1}$ and $w_{2}$. By [1, Coro. 3.7], the
parameters of $G(\Omega)$ are

$N=q^{k},\ K=n(q-1),\
\lambda=K^{2}+3K-q(w_{1}+w_{2})-Kq(w_{1}+w_{2})+q^{2}w_{1}w_{2},\
\mu=K^{2}+K-Kq(w_{1}+w_{2})+q^{2}w_{1}w_{2}.$

By Theorems 7 and 8, if $m=2s$, $q$ is even and $c\in{\mathbb{F}}_{q}^{*}$,
then ${\mathcal{C}}_{D}$ is a projective $[q^{s-1}(q^{s}+1),m,q^{2s-2}(q-1)]$
two-weight linear code with weight enumerator

$1+(q^{s}+1)(q^{s-1}-1)z^{q^{2s-2}(q-1)}+(q^{s}+1)(q^{s}-q^{s-1})z^{q^{2s-2}(q-1)+q^{s-1}}.$

Hence ${\mathcal{C}}_{D}$ yields a strongly regular graph $G(\Omega)$ with the
following parameters:

$\displaystyle N=q^{2s},\ K=q^{s-1}(q^{s}+1)(q-1),\
\lambda=q^{s-1}\left(2q-3+q^{s-1}(q-1)^{2}\right),\
\mu=(q-1)q^{s-1}(q^{s}-q^{s-1}+1).$

Compared with known strongly regular graphs [1], $G(\Omega)$ has new
parameters.

##  5 Summary and concluding remarks

The contributions of this paper are summarized as follows:

  1. $\bullet$

If $m=2s$, the weight distribution of ${\mathcal{C}}_{D}$ was determined in
Theorem 7 for any prime power $q$ and any $c\in{\mathbb{F}}_{q}$;

  2. $\bullet$

The parameters of the dual of ${\mathcal{C}}_{D}$ was also determined in
Theorem 8;

  3. $\bullet$

By Theorems 7 and 8, ${\mathcal{C}}_{D}$ is a projective two-weight code if
$q$ is even and $c\in{\mathbb{F}}_{q}^{*}$. This class of projective two-
weight codes yield strongly regular graphs with new parameters in Section 4.

It is easy to prove that ${\mathcal{C}}_{D}$ in Theorem 7 is minimal for
$s\geq 3$ as $w_{\min}/w_{\max}>\frac{q-1}{q}$ [18, Lemma 3], where
$w_{\min},w_{\max}$ denote the maximum and minimum weights of
${\mathcal{C}}_{D}$, respectively. Hence the code ${\mathcal{C}}_{D}$ in
Theorem 7 can also be used to construct secret sharing schemes with
interesting access structures [18]. As another application, the projective
two-weight code ${\mathcal{C}}_{D}$ for even $q$ and
$c\in{\mathbb{F}}_{q}^{*}$ holds $1$-design by the well-known Assmus-Mattson
theorem [2].

## References

  * [1] R. Calderbank, W. M. Kanter, The geometry of two-weight codes, Bull. Lond. Math. Soc. 18 (1986), pp. 97–122.
  * [2] C. Ding, Designs from Linear Codes, World Scientific, Singapore, 2018.
  * [3] C. Ding, Linear codes from some 2-designs. IEEE Trans. Inf. Theory 60 (2015), pp. 3265–3275.
  * [4] K. Ding, C. Ding, A class of two-weight and three-weight codes and their applications in secret sharing. IEEE Trans. Inf. Theory 61 (2015), pp. 5835–5842.
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  * [10] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes. Cambridge, U.K.: Cambridge Univ. Press (2003).
  * [11] C. Li, S. Bae, J. Ahn, S. Yang, Z. Yao, Complet eweight enumerators of some linear codes and their applications, Des. Codes Cryptogr. 81 (2016), pp. 153–168.
  * [12] R. Lidl, H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 1997.
  * [13] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, 1977.
  * [14] S. Mesnager, A. S${\i}$nak, Several classes of minimal linear codes with few weights from weakly regular plateaued functions, IEEE Trans. Inf. Theory 66(4) (2019), pp. 2296–2310.
  * [15] C. Tang, N. Li, Y. Qi, Z. Zhou, T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Trans. Inf. Theory 62 (2016), pp. 1166–1176.
  * [16] C. Tang, Y. Qi, M. Huang, Two-weight and three-weight linear codes from square functions, IEEE Commun. Lett. 20 (2016), pp. 29–32.
  * [17] S. Yang, C. Zhao, Z. Yao, A class of three-weight linear codes and their complete weight enumerators, Cryptogr. Commun. 9 (2017), pp. 133–149.
  * [18] J. Yuan, C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory 52 (2016), pp. 206–212.
  * [19] D. Zheng, J. Bao, Four classes of linear codes from cyclotomic cosets, Des. Codes Cryptogr. 86 (2018), pp. 1007–1022.
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<EMAIL_ADDRESS>
<EMAIL_ADDRESS>]

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
# Lower bound of the expressibility of ansatzes for Variational Quantum
Algorithms

Tamojit Ghosh  [  Department of Physics, Indian Institute of Technology
Madras, Chennai-600036, Tamil Nadu, India  Department of Physical Sciences,
Indian Institute of Science Education and Research Kolkata, Mohanpur- 741246,
West Bengal, India  Arijit Mandal  [  Department of Physics, Indian Institute
of Technology Madras, Chennai-600036, Tamil Nadu, India  Department of
Physical Sciences, Indian Institute of Science Education and Research Kolkata,
Mohanpur- 741246, West Bengal, India  Shreya Banerjee  [  Department of
Physical Sciences, Indian Institute of Science Education and Research Kolkata,
Mohanpur- 741246, West Bengal, India  Departement de physique and Institut
quantique, Université de Sherbrooke, Sherbrooke- J1K 2R1, Québec, Canada
Center for Excellence in Quantum Technology, School of Computer Science,
University of Petroleum and Energy Studies, Dehradun- 248007, Uttarakhand,
India  Prasanta K. Panigrahi  [  Department of Physical Sciences, Indian
Institute of Science Education and Research Kolkata, Mohanpur- 741246, West
Bengal, India

###### Abstract

The expressibility of an ansatz used in a variational quantum algorithm is
defined as the uniformity with which it can explore the space of unitary
matrices. The expressibility of a particular ansatz has a well-defined upper
bound [1]. In this work, we show that the expressibiliity also has a well-
defined lower bound in the hypothesis space. We provide an analytical
expression for the lower bound of the covering number, which is directly
related to expressibility. We also perform numerical simulations to to support
our claim. To numerically calculate the bond length of a diatomic molecule, we
take hydrogen ($H_{2}$) as a prototype system and calculate the error in the
energy for the equilibrium energy point for different ansatzes. We study the
variation of energy error with circuit depths and show that in each ansatz
template, a plateau exists for a range of circuit depths, which we call the
set of acceptable points, and the corresponding expressibility is known as the
best expressive region. We report that the width of this best expressive
region in the hypothesis space is inversely proportional to the average error.
Our analysis reveals that alongside trainability, the lower bound of
expressibility also plays a crucial role in selecting variational quantum
ansatzes.

††preprint: APS/123-QED

##  I Introduction

The promise of quantum computing is the ability to outperform classical
computers in various tasks [2, 3], thus effectively displaying ‘quantum
advantage’ [4]. But before we can reach the point of noiseless error free
quantum computing, it has become imperative to come up with algorithms that
can be performed efficiently on today’s Noisy Intermediate-Scale Quantum
(NISQ) devices [4, 5]. Despite some limitations [6, 7], Variational Quantum
Algorithms (VQA) have turned out to be one of the best applications for near
term quantum devices [8, 9, 10, 11]. They use parameterised quantum circuits
in order to construct a quantum state subject to certain optimization routines
which are performed on a classical computer. This act of deputising part of
the routine to a classical computer helps keep the quantum circuits shallow
and makes implementation viable on a NISQ device [12].

One of the very first VQAs developed was the Variational Quantum Eigensolver
(VQE) [9] which is designed to find the groundstate energy and the
corresponding wavefunction of a given system Hamiltonian.The variational
principle in quantum mechanics says that given a Hamiltonian $\hat{H}$ of a
system and a trial wavefunction $\ket{\psi}$, the ground state energy $E_{0}$
of the system is upper bounded by the following relation:

$E_{0}\leq\frac{\bra{\psi}H_{0}\ket{\psi}}{\braket{\psi}{\psi}},$

where equality is achieved if and only if the wavefunction $\ket{\psi}$ is
equal to the ground state wavefunction. VQE uses the variational principle in
Quantum Mechanics and starts with an initial parameterized trial wavefunction
(the ansatz) [13]. It then finds the expectation value with respect to the
Hamiltonian and optimizes the parameters in the wavefunction in order to
minimize the energy at each step. The entire process is an optimization
problem which (ideally) helps us obtain a good approximation to the ground
state energy and the corresponding wavefunction. It has wide varying
applications in Quantum Chemistry [14], condensed matter physics and material
science [15], medical drug discovery [16], and nuclear physics [17].

One of the most important steps in designing any Variational Quantum Algorithm
is the selection of its initial ansatz. There is a wide variety of ansatzes
that can be prepared like the Hardware-Efficient Ansatzes [18], Symmetry
preserving ansatzes [19], problem inspired ansatz like the Unitary Coupled
Cluster (UCC) [20], quantum alternating operator ansatz [21], variational
hamiltonian ansatz [22], variable structure ansatz [23]. The Hardware-
Efficient Ansatzes (HEAs) are very versatile and can be easily tailored to the
quantum device they are being used in. This feature and the ease of their
construction led to their wide usage for various tasks [24, 25, 26]. There is
a need to determine the quality of each ansatz. One of such measures is the
expressibility. It is defined as the ability of an ansatz to uniformly explore
the space of all possible unitaries ($\mathcal{U}(d)$). Formally, the
expressibility of an ansatz is measured by the distance between the uniform
distribution of unitaries sampled by changing the parameters in the ansatz and
the distribution of unitaries uniformly sampled from the maximally expressive
Haar distribution [27, 28, 29]. However, their expression for quantifying
expressibility by use of unitary t-design [30] is expensive to calculate and
has several constraints. Other measures to quantify expressibility have been
developed which deal with particular ansatzes [31, 32]. Recently, the first
step towards a general expression for the expressibility of an ansatz in terms
of the local qudit dimension, number of trainable gates, and the types of
gates used was taken [1]. It uses the covering number[33] of the hypothesis
space of the ansatz as a quantifier to get an upper bound for the
expressibility of the said ansatz. The expressibility of an ansatz is not a
complete measure of its performance. It has been shown that ansatzes like HEAs
with a large depth suffer from barren plateaus [34] due to their high
expressibility [27]. It must be noted that vanishing gradients have been seen
for shallow circuits too [35]. This leads to trainability issues for these
ansatzes [34, 35, 27, 36, 37]

In this paper, we extend the results from [1] to get an expression for the
lower bound for the expressibility of an ansatz. This is an important result
that, as far the best of our knowledge, has not been found yet. This helps us
tighten the bound given by [1] and brings us closer to a generic expression
for expressibility. We take VQEs with HEAs as a case study and perform
numerical simulations of the groundstate energy of a Hydrogen molecule using
QISKIT. Different ansatz templates are compared according to their
performance. The error of the VQE algorithm with respect to the classical
result (obtained by diagonalizing the Hamiltonian) is used as a quantifier of
the performance of repeating layers of the ansatz templates. We find that each
ansatz templates performs well for certain intermediate number of layers. We
draw a relationship between the ’best expressive region’ of an ansatz template
and the ’average error’ across this region.

The presence of a lower bound to the expressibility of an ansatz used in VQE
tells us that if our solution corresponds to a wavefunction in a low
expressive region then an initial choice of a highly expressive trial ansatz
will never be able to reach the solution leading to an erroneous VQE result.
This is in contrast to earlier literature [27] which blame the
underperformance of highly expressive ansatz solely on trainability issues. We
find that it is a result of both expressibility and trainability problems.

##  II Preliminaries

###  II.1 Variational Quantum Eigensolver

Figure 1: General workflow of Variational Quantum Algorithms (VQAs).

We consider a wavefunction $\ket{\psi(\theta)}$ containing a set of tunable
parameter $\theta$ is obtained by applying a unitary $U(\theta)$ on $N$ qubit
register initialized to $\ket{0}$, i.e., $\ket{\psi_{0}}=\ket{0}^{\otimes N}$.
Now the expectation value of the Hamiltonian $\hat{H}$ with respect to
$\ket{\psi(\theta)}$ is calculated and minimized by optimizing the parameters.
We can quantify this better by saying that let $\ket{\psi(\theta)}$ be the
result of the action of an unitary $U(\theta)$ (where $\theta$ is a set of
tunable parameters) applied on an initial quantum register of $N$ qubits (let
the initial state be $\ket{0}^{\otimes N}\;\equiv\;\ket{\psi_{0}}$). Thus, the
VQE cost function can be written down as,

$\displaystyle E_{0}^{VQE}$ $\displaystyle=$
$\displaystyle\min_{\theta}\bra{\psi_{0}}U^{\dagger}(\theta)\hat{H}U(\theta)\ket{\psi_{0}}$
(1) $\displaystyle=$
$\displaystyle\min_{\theta}\bra{\psi(\theta)}\hat{H}\ket{\psi(\theta)}.$

The hybrid algorithm tries to optimize the expectation value, computed on a
quantum computer, by classically varying the parameters $\theta$. This is done
by writing the Hamiltonian $\hat{H}$ as a weighted sum of spin operators.

$\hat{H}=\sum_{i=1}^{k}c_{i}\hat{P}_{i}$ (2)

where $c_{i}$ are the weights, $\hat{P}_{i}$ is a Pauli string on $N$ qubits
i.e. $\hat{P}_{i}\in\left\\{I,X,Y,Z\right\\}^{\otimes N}$ and $k$ denotes the
number of such Pauli strings needed to represent the Hamiltonian.

###  II.2 Covering number and the hypothesis space

$Q$ is ‘$\epsilon$-covering’ for a metric space $\mathcal{M}$ with distance
$d$ if for every $y\in\mathcal{M}$ and $x\in Q$, $d(x,y)\leq\epsilon$. Then
the cardinality $\mathcal{N}(\mathcal{M},d,\epsilon)$ of the smallest $Q$ is
called “$\epsilon$ covering number” of set $\mathcal{M}$ [38].

$|Q|_{smallest}=\mathcal{N}(\mathcal{M},\epsilon,d)$ (3)

We now define the hypothesis space $\mathcal{H}$ as:

$\mathcal{H}=\left\\{Tr(\hat{U}(\theta)^{\dagger}\hat{O}\hat{U}(\theta)\rho)|\theta\in\Theta\right\\}$
(4)

where $\rho\in\mathbf{C}^{d^{N}\times d^{N}}$ is the $N$-qudit input quantum
state, $\hat{O}\in\mathbf{C}^{d^{N}\times d^{N}}$ is the problem Hamiltonian
and
$\hat{U}(\theta)=\prod_{i=1}^{N_{g}}\hat{u}_{i}(\theta_{i})\in\mathcal{U}(d^{N})$
is the ansatz that we are using. $\Theta$ is the parameter space that contains
the parameters $\theta_{i}$. $\mathcal{U}(d^{k})$ denote the set of unitary
matrices of dimension $d^{k}$. This is also the set for all possible gates
that we can use in the ansatz of our variational algorithm.
$\hat{u}_{p}(\theta_{p})$ is the $p^{th}$ quantum gate with
$\hat{u}_{p}(\theta_{p})\in\mathcal{U}\left(d^{k}\right)$ i.e. it can act on a
maximum of $k$ qudits such that $k\leq N$. We choose the ansatz
$\hat{U}(\theta)$ such that it has $N_{g}$ gates among which $N_{gt}$ are
trainable (or parameterized) gates.

Next, we define the operator group $\mathcal{H}_{circ}$ which consists of the
different parameterised forms of a particular ansatz that we use in our
variational algorithm.

$\mathcal{H}_{circ}=\\{\hat{U}(\theta)^{\dagger}\hat{O}\hat{U}(\theta)|\theta\in\Theta\\}$
(5)

For $0<\epsilon\leq 1/10$, the $\epsilon$-covering number for the unitary
group $\mathcal{U}(d^{k})$ with respect to the operator norm distance
$||\cdot||$, is bounded by the relation [38] :

$\left(\frac{3}{4\epsilon}\right)^{d^{2k}}\leq\mathcal{N}(\mathcal{U}(d^{k}),\epsilon,||\cdot||)\leq\left(\frac{7}{\epsilon}\right)^{d^{2k}}$
(6)

where the operator norm of an $n\times n$ matrix $A$ is defined as [1]:

$||A||=\sqrt{\mu_{1}\left(AA^{\dagger}\right)}$ (7)

where $\mu_{1}\left(AA^{\dagger}\right)$ is the largest eigenvalue of the
matrix $AA^{\dagger}$.

##  III Results

###  III.1 Presence of a lower bound

[1] presents an upper bound of the expressibility of an ansatz. It relates the
expressibility of an ansatz to the complexity of the hypothesis space of the
same. They use the covering number of the hypothesis space as a measure of the
complexity. For $0<\epsilon\leq 1/10$, the covering number of the hypothesis
space $\mathcal{H}$ is upper bounded by:

$\mathcal{N}(\mathcal{H},\epsilon,|\cdot|)\leq\left(\frac{7N_{gt}||\hat{O}||}{\epsilon}\right)^{d^{2k}N_{g}t}$
(8)

where $|\cdot|$ is the norm for the hypothesis space. It can be seen that the
complexity of the hypothesis space (thus the expressibility of the ansatz
used) depends on the type of gates (as directed by the parameter $k$) that we
used to build said ansatz.

We follow a similar process as in [1] to find a lower bound to the covering
number of the hypothesis space.Let $S$ be the $\epsilon$-covering set for the
set $\mathcal{U}(d^{k})$.

$|S|=\mathcal{N}(\mathcal{U}(d^{k}),\epsilon,||\cdot||)$ (9)

We want to find the covering set for the set $\mathcal{H}_{circ}$. For this,
we define another set $\tilde{S}$ such that:

$\tilde{S}:=\\{\prod_{i\in\\{N_{gt}\\}}\hat{u}_{i}(\theta_{i})\prod_{j\in(N_{g}-N_{gt})}\hat{u}_{j}|\hat{u}_{i}(\theta_{i})\in
S\\}$ (10)

The ordered $\hat{u}_{i}$ in $\tilde{S}$ are the particular ones used in the
operator group $\mathcal{H}_{circ}$. Now, we show that $\tilde{S}$ is an
‘$\epsilon^{\prime}$-covering set’ of $\mathcal{H}_{circ}$ and find
$\epsilon^{\prime}$. First, we replace the $N_{gt}$ trainable gates
$\hat{u}_{i}(\theta_{i})$ in $\tilde{S}$ with the nearest element in $S$ (all
$\epsilon$ distance away as $S$ is $\epsilon$-covering set of
$\mathcal{U}(d^{k})$) . Now, we have :

$\hat{U}_{\epsilon}(\theta)\in\tilde{S}$

We will now prove that for any element $a\in\mathcal{H}_{circ}$ and element
$b\in\tilde{S}$, $d(a,b)\leq\epsilon^{\prime}$, where $d$ is the distance
measure for the operator group i.e. $||\cdot||$. It can be shown that the
distance measure satisfies:

$\begin{split}&||\hat{U}(\theta)^{\dagger}\hat{O}\hat{U}(\theta)-\hat{U}_{\epsilon}(\theta)^{\dagger}\hat{O}\hat{U}_{\epsilon}(\theta)||\\\
&\leq||\hat{U}-\hat{U}_{\epsilon}||\;||\hat{O}||\\\ &\leq
N_{gt}||\hat{O}||\epsilon=\epsilon^{\prime}\end{split}$ (11)

where we use the triangle inequality for the first inequality and
$||\hat{U}-\hat{U}_{\epsilon}||\leq N_{gt}\epsilon$ for the second inequality.
From our earlier definition, we can say that the set $\tilde{S}$ is an
‘$N_{gt}||\hat{O}||\epsilon$ -covering set’ for $\mathcal{H}_{circ}$. Thus:

$|\tilde{S}|=\mathcal{N}(\mathcal{H}_{circ},N_{gt}||\hat{O}||\epsilon,||\cdot||)$
(12)

From Equations 6 and 9, we have:

$\left(\frac{3}{4\epsilon}\right)^{d^{2k}}\leq|S|$ (13)

We must remember that we replaced $N_{gt}$ trainable gates in $\tilde{S}$
using gates from $S$ and we have $|S|^{N_{gt}}$ combinations for the gates in
$\tilde{S}$. This gives us:

$\left(\frac{3}{4\epsilon}\right)^{d^{2k}N_{gt}}\leq|\tilde{S}|$ (14)

Finally, we use Equations 12 and 14 to get a lower bound to the covering
number of the operator group $\mathcal{H}_{circ}$:

$\left(\frac{3}{4\epsilon}\right)^{d^{2k}N_{gt}}\leq\mathcal{N}(\mathcal{H}_{circ},N_{gt}||\hat{O}||\epsilon,||\cdot||)$
(15)

We rescale $\epsilon$ by using $\epsilon=\frac{2\epsilon}{N_{gt}||\hat{O}||}$

$\left(\frac{3N_{gt}||\hat{O}||}{8\epsilon}\right)^{d^{2k}N_{gt}}\leq\mathcal{N}(\mathcal{H}_{circ},2\epsilon,||\cdot||)$
(16)

Next, we draw a relation between the covering number of the operator space
$\mathcal{H}_{circ}$ and the hypothesis space $\mathcal{H}$. In order to do
that, it can be shown from [1] that a Bi-Lipschitz mapping exists between the
two spaces.

Let ($\mathcal{H}_{1},d_{1}$) and ($\mathcal{H}_{2},.d_{2}$) be two metric
spaces and the map F: $\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}$ be Bi-
Lipschitz with:

$\begin{split}d_{2}(f(x),f(y))&\leq Kd_{1}(x,y);\forall
x,y\in\mathcal{H}_{1}\\\ d_{2}(f(x),f(y))&\geq kd_{1}(x,y);\forall
x,y\in\mathcal{H}_{1}\;with\;d_{1}(x,y)<r\end{split}$ (17)

Then their respective covering numbers follow the relations:

$\mathcal{N}(\mathcal{H}_{1},\frac{2\epsilon}{k},d_{1})\leq\mathcal{N}(\mathcal{H}_{2},\epsilon,d_{2})\leq\mathcal{N}(\mathcal{H}_{1},\frac{\epsilon}{K},d_{1})$
(18)

where the left inequality requires

$\epsilon\leq\frac{kr}{2}$ (19)

In our case,

$\mathcal{H}_{1}=\mathcal{H}_{circ}$ (20)

where
$\mathcal{H}_{circ}=\\{\hat{U}(\theta)^{\dagger}\hat{O}\hat{U}(\theta)|\theta\in\Theta\\}$
is the operator group with the corresponding distance metric
$d_{1}=||\cdot||$.

The other space:

$\mathcal{H}_{2}=\mathcal{H}$ (21)

where
$\mathcal{H}=\left\\{Tr(\hat{U}(\theta)^{\dagger}\hat{O}\hat{U}(\theta)\rho)|\theta\in\Theta\right\\}$
is the hypothesis space with the distance metric $d_{2}=|\cdot|$.

In order to find $K$, we follow the similar process of replacing the gates
used in the ansatz $\hat{U}(\theta)$ with the nearest gates in the covering
set $S$ to get a new ansatz $\hat{U}_{\epsilon}(\theta)$. This is used in the
first inequality in Equation 17 to give [1]:

$\begin{split}d_{2}(Tr(\hat{U}_{\epsilon}(\theta)^{\dagger}\hat{O}\hat{U}_{\epsilon}(\theta)\rho)-Tr(\hat{U}(\theta)^{\dagger}\hat{O}\hat{U}(\theta)\rho))&=\\\
d_{1}(\hat{U}_{\epsilon}(\theta)^{\dagger}\hat{O}\hat{U}_{\epsilon}(\theta)-\hat{U}(\theta)^{\dagger}\hat{O}\hat{U}(\theta))&\end{split}$
(22)

hence

$K=1$ (23)

For a Bi-Lipshitz map, it can be shown [39] that through isometry of the map:

$k=\frac{1}{K}$ (24)

Thus, since $K=1$, hence it follows that $k=1$.

Using this and the left inequality in Equation 18, we get:

$\mathcal{N}(\mathcal{H}_{circ},2\epsilon,||\cdot||)\leq\mathcal{N}(\mathcal{H},\epsilon,|\cdot|)$
(25)

Finally,using Equation 16, we get the lower bound to the covering number of
the hypothesis space $\mathcal{H}$.

$\left(\frac{3N_{gt}||\hat{O}||}{8\epsilon}\right)^{d^{2k}N_{gt}}\leq{N}(\mathcal{H},\epsilon,|\cdot|)$
(26)

It must be noted that the left inequality in Equation 18 requires
$\epsilon\leq\frac{kr}{2}$ where $d_{1}(x,y)\leq r$. Hence, we can use the
result from the Equation 11 to get:

$r=N_{gt}||\hat{O}||\epsilon$ (27)

Plugging all of this into the condition, we get:

$N_{gt}\geq\frac{2}{||\hat{O}||}$ (28)

This gives us a lower bound on the number of trainable gates that we need in
our ansatz for our result to be valid.

Figure 2:  Different ansatz templates used for numerical simulations. Ansatz
templates 1 (a) and 2 (b) are taken from Ref. [40]. We develop ansatz
templates 3 (c) and 4 (d) to make some variation in the number of trainable
gates in the template.The highlighted block in each template gets repeated for
each layer.

###  III.2 Numerical simulation

All of the following simulations have been done on Python on a system with 16
GB of RAM and a 2.40GHz processor with 4 Cores and 8 Logical Processors. We
take help of the QISKIT library provided by IBM Quantum in order to construct
the variational algorithm.

We use the PySCF Driver in an ’sto3g’ basis to set up our Hamiltonian. Also,
we use the Parity Mapper to encode into qubits. The NumPyMinimumEigensolver
function is used to classically get the solution and the inbuilt VQE function
of QISKIT simulated on a ’statevector simulator’ is used to run the
variational code. The GradientDescent function of QISKIT is used as a
classical optimizer with learning rate $0.4$ and tolerance $10^{-6}$ (in
Hartree). We perform the simulations for the hydrogen molecule ($H_{2}$) and
for four different ansatzes. The methodology that we follow for the ansatzes
can be summarised in the following steps:

  * •

We take an ansatz and run our VQE algorithm for $H_{2}$ molecule at every
atomic distance from $0.3$ to $2.1$ $\mathrm{\text{Å}}$ in steps of $0.2$. We
obtain the minimum distance for each atomic distance. There is a distance at
which the minimum energy is the lowest amongst all others. That would be the
atomic distance according to the variational method.

  * •

We perform ten such trials. Now, it must be noted that this was done only for
a single layer ( we call this the ansatz template [28]). We increase the
number of layers and repeat the same steps. We get an atomic distance and the
atomic energy from each such endeavour.

  * •

$H_{2}$ being a simple molecule, the corresponding Hamiltonian can be
diagonalized classically. We get the actual atomic energy for $H_{2}$ at
atomic distance by selecting the minimum eigenvalue of the Hamiltonian.

  * •

We use this value as reference and calculate the error for each layer used.
The error is calculated for a single layer by taking the difference between
the minimum VQE energy for each layer and the real energy. The average across
ten trials gives us the error. We do this for each layer.

  * •

Finally we plot an error vs layer number plot.

We do this for four different ansatz templates (Fig. 2).

Figure 3: Variation of energy error with circuit depth (i.e., the number of
repetitions of the circuit building block) for ansatz template (a) 1, (b) 2,
(c) 3, and (d) 4 as depicted in Fig. 2. The error in energy ($\Delta E$) is
the deviation of the minimum energy obtained by our VQE algorithm from the
minimum energy obtained by diagonalizing our $H_{2}$ Hamiltonian. The ‘Depth’
is the number of times we have repeated our ansatz template. The blue vertical
line at each datapoint indicates the error bar which give the standard
deviation of each error point. For the point whose lower error bar takes it to
negative values, we take a prior and set the lower bar to the mean itself. The
curves dip for an intermediate depth and then rise up again. The plateau
region, i.e., the set of acceptable points in each ansatz template is
highlighted in grey color.

The results of our simulations are shown in Fig. 3. The first similarity that
we notice across the four of them is that the error (i.e. the difference
between the VQE result and the actual result from diagonalization) for all of
them seems to have a dip or a minimum point and then rises back up again with
increase in depth of the ansatz template used. Each of the templates has a few
layers for which it performs the best. Further increase or decrease of depth
leads to a worse result. Ansatz template $1$ was repeated only till depth $10$
as it is the most ‘complicated’ out of the four and is not expected to show a
further dip. Furthermore, it is noted during the simulations that the time
taken to execute the code for each ansatz template increases exponentially
with increase in depth, as is expected.

##  IV Discussions

In order for an ansatz to perform well, there is a definite need to balance
its expressibility and trainability [27]. For our case, the hypothesis space
of the ansatz used for this particular problem ($O$) is given by Eq. 4. As we
have noted during our numerical simulations, our problem ($O$) already has a
solution (a minimum energy that we want to find), regardless of the ansatz
used. How well our ansatz performs will be decided by how close it can get to
the actual solution (the actual groundstate wavefunction in this case). This
means that the hypothesis space of the ansatz must contain the solution for it
to perform well.

Figure 4: Schematic for the expressivity of an ansatz in hypothesis space
$\mathcal{H}$ and its distance from the target in the same space. The
expressivity of an ansatz is denoted by the shaded elliptic region, and the
target is denoted by the red star. The upper panel describes (a) low, (b)
high, and (c) moderate expressivity of an ansatz when the expressivity is
defined by an upper bound only. The lower panel describes the (d) low, (e)
high, and (c) moderate expressivity of an ansatz when a lower bound of
expressivity also exists. (b) and (c) denotes that beyond a critical point the
expressivity of an ansatz does not create any barrier in reaching the target.
But (e) and (f) indicates that a very high expressive ansatz also never reach
to the target because of the existence of its lower bound.

It was already known that the expressibility of an ansatz has an upper bound
[1]. The direct consequence of this was that the hypothesis space of the
ansatz was thought to be a filled out object (like an ellipse) in some
abstract space. When the expressibility of the ansatz used is low (Fig: 4(a)),
the corresponding hypothesis space does not include the target and thus the
ansatz would be unable to reach the correct solution. On the other hand, when
the expressibility of the ansatz is very high (Fig. 4(b)), the hypothesis
space is very big and the algorithm has to cover a lot of space in order to
reach the target. The optimization algorithm becomes difficult to train along
with increasing chances of encountering barren plateaus [27]. This again leads
to a ‘bad’ result. Thus, we can see that, in order to get a good solution
(close to our actual target), we need an ansatz with moderate expressibility
with a hypothesis space which covers the target but is not spread out too much
from it (Fig. 4(c)) [27].

But now, by including the lower bound too, we understand that the hypothesis
space of the ansatz is not a continuous space. Instead, it can be thought of
as an annular space. It has a lower bound below which the algorithm cannot
search for the target solution. This is illustrated in Fig. 4(d)-4(f). The
presence of a lower bound to the expressibility of an ansatz tells us that if
our solution corresponds to a wavefunction in a low expressive region then an
initial choice of a highly expressive trial ansatz will never be able to reach
the solution leading to an erroneous VQE result. The most important
consequence of this endeavour is that as we can see from Fig. 4(e), the bad
performance of an ansatz with a high expressibility is not just because of the
difficulty of training but also due to its lack of expressibility. Hence,
trainability and expressibility are intrinsically connected.

Using the upper bound provided by [1] and the lower bound in this paper, we
get a relation for the covering number of the hypothesis space (and thus the
expressibility of the corresponding ansatz).

$\left(\frac{3N_{gt}||\hat{O}||}{8\epsilon}\right)^{d^{2k}N_{gt}}\leq\mathcal{N}(\mathcal{H},\epsilon,|\cdot|)\leq\left(\frac{7N_{gt}||\hat{O}||}{\epsilon}\right)^{d^{2k}N_{gt}}$
(29)

Since the logarithms of the bounds in Equation 29 are easier to calculate
numerically than the exponentials present, we prefer to work with the former.

$\begin{split}&d^{2k}N_{gt}\log\left(\frac{3N_{gt}||\hat{O}||}{8\epsilon}\right)\\\
&\leq\log\mathcal{N}(\mathcal{H},\epsilon,|\cdot|)\\\ &\leq
d^{2k}N_{gt}\log\left(\frac{7N_{gt}||\hat{O}||}{\epsilon}\right)\end{split}$
(30)

For our case, $d=2$ (as we have qubits) , $k=2$ (as only single and two qubit
gates are used in our ansatz templates). We know that the parameter $\epsilon$
can take values between $0$ and $\frac{1}{10}$. We select
$\epsilon=\frac{1}{100}$. The operator norm for the Hamiltonian is different
for different atomic distances. We work with its value when $d=0.8$(Angstrom)
as the energy is always minimum between $d=0.7$ and $d=0.9$ for $H_{2}$. Thus
we have $||\hat{O}||=||H||=1.16863955$. Additionally, plugging $||H||$ into
Equation 28, we get the trivial lower bound on the number of trainable gates
we need as $N_{gt}\geq 1$.

The only parameter in Equation 30 that changes from ansatz to ansatz is the
number of trainable gates $N_{gt}$. Plugging in the rest of the parameters, we
get the bounds as a function of $N_{gt}$.

$\begin{split}&16N_{gt}\log\left(115.037956N_{gt}\right)\\\
&\leq\log\mathcal{N}(\mathcal{H},\epsilon,|\cdot|)\\\ &\leq
16N_{gt}\log\left(43.8239831N_{gt}\right)\end{split}$ (31)

The number of trainable gates changes with the layer number differently for
different ansatz templates, as we tabulate in Table 1.

Table 1:  Ansatz template Number |  Single Layer $N_{gt}$ |  $N_{gt}$ for layer $n$
---|---|---
1 |  $3\times 4+3\times 4=12$ |  $12n+12$
2 |  $2\times 4+2\times 4=16$ |  $8n+8$
3 |  $4\times 4+4\times 2=24$ |  $16n+8$
4 |  $2\times 4=8$ |  $8n$

We can use this to get a numerical value for the upper bound and the lower
bound of the expressibility of each layer for every ansatz. The first thing
that it allows us to do is to define an average expressibility value for each
layer of the ansatz. This completes the search for a general expression for
expressibility of an ansatz and is also a much more easily obtainable value
than what was put forward in previous literature [28, 29]. We then change the
x-axes in Fig. 3 in order to get an error vs average expressibility graph for
all four of our ansatz templates. As we can see in Fig. 5, the error of our
result obtained from the variational method is more for high and low average
expressibility, but the error is less for an intermediate values of average
ansatz expressibility. This agrees with previous results [27].

Figure 5:  Variation of error in energy ($\Delta E$) with average
expressibility (in scales of $10^{4}$) for ansatz template (a) 1, (b) 2, (c)
3, and (d) 4. The x-axis from Fig:3 is replaced with our expression for
average expressibility. The error seems to increase for low and high values of
average expressibility. There are several points making up the minima of each
curve. Ansatz depths with these values of average expressibility seem to
perform the best (the least error). The set of acceptable points in each
ansatz template is again highlighted in grey color.

We notice that for each ansatz template, there are certain layers for which
the error is the least among all others. We call these set of points the set
of acceptable points. Next, we define the expressibility range from the upper
bound of the rightmost acceptable point and the lower bound of the leftmost
acceptable point in Fig: 3 as the “best expressive range” for each of the
ansatz templates. For these set of layers, the ansatz template performs the
best over the entire range of repetitions. Even if we keep on adding more
layers, we will not get a better performance. Secondly, we also calculate the
average error over these sets of acceptable points for the four ansatz
templates that we have. The result is that, as we have noted in Table 2, we
notice a direct connection between the span of the “best expressive range” and
the average error. For low average error across the set of acceptable points,
the span of the best expressive range is the widest. There is a direct
correlation between the two.

Table 2:  We tabulate the best expressive range and the average error over this range for the four different ansatz templates. The best expressive range is selected by choosing the points with the lowest error points from Fig. 5. We can see a direct correlation between the span of the best expressive range and the average error over the range. They are inversely proportional. The ansatz template that performs the best does so over a wider range of layers than the ones that perform poorly. Ansatz template number |  Best expressive range |  Average error (over best expressive range)
---|---|---
$1$ |  $3399.50$ |  $0.004145$
$2$ |  $6342.89$ |  $0.002861$
$3$ |  $6740.02$ |  $0.002046$
$4$ |  $5176.76$ |  $0.003473$

##  V Conclusions

In this work, we have shown that there exists a lower bound to the covering
number of the hypothesis space of an ansatz. We have found a correlation
between the aforementioned lower bound to the complexity of the hypothesis
space and correspondingly to the expressibility of the ansatz used. This
changes how the hypothesis space exists in an abstract space. By our findings,
it is more close to an annular region than a continuous space. This makes the
expressibility and trainability of an ansatz more inter-wined with the ansatz
performances [27, 28]. We have also performed numerical simulations to plot
the error of a variational algorithm vs the layer number of the ansatz
template. We noticed that the intermediate layer numbers which have the
moderate expressibility, the error is the lowest, We have also found that the
average error across the set of acceptable points and the ‘best expressive
region’ for an ansatz template are inversely proportional.

Our analytic result and numerical simulations have been done in a noise-free
situation. The next step should be to extend it to include noise. Also, in
this paper we have just looked at expressibility as a measure of ansatz
performance and have linked our results to trainability. But it would be
interesting to look at expressibility and trainability together as a single
evaluation parameter in order to better judge ansatzes. Our results show that
the performance of an ansatz is an intrinsic property of the “ansatz and
problem” system and not just an issue of trainability. We hope that, in the
future, this result can be used to select better ansatzes for practical
application of VQAs.

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# Comment on ”Horizon-scale tests of gravity theories and fundamental physics
from the Event Horizon Telescope image of Sagittarius A*”

Naoki Tsukamoto<EMAIL_ADDRESS>Department of Physics, Faculty of
Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo
162-8601, Japan

###### Abstract

Vagnozzi et al. constrained the additional parameter of spacetimes with a
photon sphere from the observation of the shadow of Sagittarius A* under the
assumption that a distance to the Sagittarius A* and its mass parameter was
estimated from other observations. They claimed that a Damour-Solodukhin
wormhole with an additional parameter $\lambda$ is not an asymptotically-flat
spacetime and that they gave the first robust observational constraint on the
parameter $\lambda$ of the Damour-Solodukhin wormhole. However, they
overlooked the fact that: (A) the Damour-Solodukhin wormhole spacetime is
asymptotically flat, (B) the throat of the Damour-Solodukhin wormhole works as
an effective photon sphere for $\lambda>\sqrt{2}/2$, and (C) not only a usual
mass parameter but also the parameter $\lambda$ contributes the mass of the
Damour-Solodukhin wormhole. Because of the overlook (C), we realize that their
constraint on the parameter $\lambda$ is invalid. This is because their method
corresponds to the following way: They estimated the mass parameter from the
other observations under the assumption $\lambda=0$ even though the value of
the parameter $\lambda$ strongly affects the determination of the mass
parameter, and then, they used the value of the mass parameter to constrain
$\lambda$ from the observation of the shadow. We conclude that we should
constrain the mass parameter and $\lambda$ from the shadow observation and
other observations without the assumption $\lambda=0$.

##  I Introduction

In Ref. Vagnozzi:2022moj , Vagnozzi et al. gave the constraints of an
additional parameter of spacetimes with a photon sphere which is formed by
unstable circular photon orbits from the observation of a black hole candidate
which is called Sagittarius A* at the center of our galaxy observed by Event
Horizon Telescope Collaboration EventHorizonTelescope:2022wkp  under the
asusumption that the mass of the Sagittarius A* and its distance are given by
other observations GRAVITY:2020gka ; EventHorizonTelescope:2019jan .

Vagnozzi et al. considered the shadow of a Damour-Solodukhin wormhole
Damour:2007ap . Its metric in coordinates ($\tilde{t}$, $r$, $\theta$, $\phi$)
is given by

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2\tilde{M}}{r}+\lambda^{2}\right)\mathrm{d}\tilde{t}^{2}+\frac{\mathrm{d}r^{2}}{1-\frac{2\tilde{M}}{r}}$
(1)
$\displaystyle+r^{2}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}\right),$

where $\tilde{M}$ is a mass parameter and $\lambda$ is the additional
parameter. It has a wormhole throat at $r=2\tilde{M}$. We assume the $Z_{2}$
symmetry of the wormhole spacetime against the throat. We treat $\lambda$ as a
free positive parameter on this paper in the same way as Ref. Vagnozzi:2022moj
.

At a spatial infinity $r\rightarrow\infty$, the metric (1) in the coordinates
($\tilde{t}$, $r$, $\theta$, $\phi$) becomes

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=$
$\displaystyle-\left(1+\lambda^{2}\right)\mathrm{d}\tilde{t}^{2}+\mathrm{d}r^{2}+r^{2}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}\right).$

The coordinates ($\tilde{t}$, $r$, $\theta$, $\phi$) are different from ones
which are often used in a Minkowski spacetime. Therefore, the physical
interpretation of $\lambda$ and the constants of motion under the coordinates
($\tilde{t}$, $r$, $\theta$, $\phi$) can be uncertain for observers at far
away from the wormhole or it would not be simple for them.

In Sec. II, we comment on overlooks in Ref. Vagnozzi:2022moj :(A) its
asymptotically flatness, (B) an effective photon sphere on its throat for
$\lambda>\sqrt{2}/2$, and (C) the contribution of the parameter $\lambda$ to
its mass. We give a conclusion in Sec. III. On this paper, we use the units in
which a light speed and Newton’s constant are unity.

##  II Comments on Vagnozzi et al. Vagnozzi:2022moj

###  II.1 Asymptotically flatness

Vagnozzi et al. Vagnozzi:2022moj  claimed that the wormhole is not
asymptotically flat. This statement contradicts Ref. Damour:2007ap  by Damour
and Solodukhin. Here, we clearly show that it is asymptotically flat by
transforming of the time coordinate $\tilde{t}$ into a new one $t$ defined by
111The new time coordinate used in Refs. Bueno:2017hyj ; Ovgun:2018fnk ;
Tsukamoto:2020uay . Note that definition of the coordinate $t$ given by Eq.
(2.6) in Ref. Tsukamoto:2020uay  has a typo and it should be read as Eq. (3)
on this paper.

$\displaystyle t=\sqrt{1+\lambda^{2}}\tilde{t}.$ (3)

In the coordinates ($t$, $r$, $\theta$, $\phi$), the metric (1) is rewritten
in

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=$
$\displaystyle-\left[1-\frac{2\tilde{M}}{(1+\lambda^{2})r}\right]\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{1-\frac{2\tilde{M}}{r}}$
(4)
$\displaystyle+r^{2}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}\right)$

and it is asymptotically flat. 222 We keep to use the mass parameter
$\tilde{M}$. However, $\tilde{M}$ is often redefined as a new mass parameter
$M$ by a relation Bueno:2017hyj ; Ovgun:2018fnk ; Tsukamoto:2020uay
$\displaystyle M=\frac{\tilde{M}}{1+\lambda^{2}}$ (5) and the metric (4) is
rewriten in  $\displaystyle\mathrm{d}s^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2M}{r}\right)\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{1-\frac{2M\left(1+\lambda^{2}\right)}{r}}$
(6)
$\displaystyle+r^{2}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}\right).$
The mass parameter $\tilde{M}$ is an Arnowitt-Deser-Misner (ADM) mass in the
line element (4) as shown in Ref. Tsukamoto:2020uay .333The ADM mass
$M(1+\lambda^{2})$ was obtained as shown Eq. (A6) in Ref. Tsukamoto:2020uay .
It is equal to $\tilde{M}$ from Eq. (5) on this paper.

###  II.2 Effective photon sphere on the throat for $\lambda>\sqrt{2}/2$

Vagnozzi et al. obtained the shadow size, which is Eq. (38) in Ref.
Vagnozzi:2022moj , as

$\displaystyle r_{\mathrm{sh}}\simeq\frac{3\sqrt{3}}{1+\lambda^{2}}\tilde{M},$
(7)

where we have recovered $\tilde{M}$, which is denoted by $M$ on Ref.
Vagnozzi:2022moj , while they used units that the mass parameter is unity.
They used $\simeq$ because they misunderstood that the wormhole spacetime is
not asymptotically flat. We should read Eq. (7) as

$\displaystyle r_{\mathrm{sh}}=\frac{3\sqrt{3}}{1+\lambda^{2}}\tilde{M}.$ (8)

Equation (8) is equal to the critical impact parameter $b_{\mathrm{m}}$ of a
ray under the coordinates ($t$, $r$, $\theta$, $\phi$) obtained by Övgün
Ovgun:2018fnk  who is one of the authors of Ref. Vagnozzi:2022moj . See Eq.
(2.26) in Ref. Ovgun:2018fnk . 444 In Ref. Ovgun:2018fnk , $b_{m}$ is denoted
by $u_{m}$. Note that $M$ on Ref. Ovgun:2018fnk  and $\tilde{M}$ on this paper
has the relation of Eq. (5) on this paper. We notice that an effective photon
sphere on the throat was overlooked in Ref. Ovgun:2018fnk .

However, Eq. (8) is correct only for $\lambda\leq\sqrt{2}/2$. We explain the
reason why it is vaild only for $\lambda\leq\sqrt{2}/2$ by following Ref.
Tsukamoto:2020uay . The Damour-Solodukhin wormhole spacetime has a photon
sphere, which is formed unstable circular photon orbits, at
$r=r_{\mathrm{m}}$. For $\lambda<\sqrt{2}/2$, $r_{\mathrm{m}}$ is defined by

$\displaystyle r_{\mathrm{m}}\equiv\frac{3\tilde{M}}{1+\lambda^{2}}.$ (9)

For $\lambda=\sqrt{2}/2$, the radius $r_{\mathrm{m}}$ of the photon sphrere
given by Eq. (9) is equal to the radius of the throat, i.e.,

$\displaystyle r_{\mathrm{m}}=2\tilde{M}.$ (10)

In this case, the photon sphere on the throat is a kind of a marginally
unstable photon sphere. For $\lambda>\sqrt{2}/2$, the throat works as an
effective photon sphere. Thus, we define $r_{\mathrm{m}}$ as, for
$\lambda>\sqrt{2}/2$,

$\displaystyle r_{\mathrm{m}}\equiv 2\tilde{M}.$ (11)

A ray coming from spatical infinity falls into the throat for
$b<b_{\mathrm{m}}$, where $b=b(r_{0})$ is the positive impact parameter of the
ray, where $r_{0}$ is the closest distance of the ray from the wormhole, and

$\displaystyle b_{\mathrm{m}}\equiv b(r_{0}=r_{\mathrm{m}})$ (12)

is the critical impact parameter of the ray. On the other hand, for
$b>b_{\mathrm{m}}$, the ray coming from the spatial infinity is deflected by
the wormhole and it goes back to the same spatial infinity. In the critical
case with $b=b_{\mathrm{m}}$, the ray rotates around the (effective) photon
sphere infinity times.

The image angle of the (effective) photon sphere, which would be the observed
angle of a ring image around the shadow, is given by $b_{\mathrm{m}}/D$, where
$D$ is a distance between the wormhole and the observer. If the distance is
obtained from other observations, the size of the (effective) photon sphere is
determined by the critical impact parameter $b_{\mathrm{m}}$. For
$\lambda\leq\sqrt{2}/2$, the critical impact parameter $b_{\mathrm{m}}$ is
obtained as

$\displaystyle b_{\mathrm{m}}=\frac{3\sqrt{3}}{1+\lambda^{2}}\tilde{M}.$ (13)

It is almost equal to the shadow size (7) obtained by Vagnozzi et al.
Vagnozzi:2022moj  and it is equivalent with the critical impact parameter
obtained by Övgün Ovgun:2018fnk . On the other hand, for $\lambda>\sqrt{2}/2$,
it is given by Tsukamoto:2020uay

$\displaystyle b_{\mathrm{m}}=\frac{2\sqrt{1+\lambda^{2}}}{\lambda}\tilde{M}.$
(14)

Thus, the shadow size (7) obtained by Vagnozzi et al. Vagnozzi:2022moj  should
be read as Eq. (14) or

$\displaystyle r_{\mathrm{sh}}=\frac{2\sqrt{1+\lambda^{2}}}{\lambda}\tilde{M}$
(15)

for $\lambda>\sqrt{2}/2$.

The throat of a rotating Damour-Solodukhin wormhole, which is often called
Kerr-like wormhole, can affect its shadow in shape and size. In Ref.
Amir:2018pcu , the effect of the throat of the rotating wormhole on the shadow
was overlooked. Kasuya and Kobayashi modified the shadow of the rotating
wormhole by taking an account for the effect of the throat Kasuya:2021cpk .
See also Ref. Shaikh:2018kfv  for the effect of a throat on a wormhole shadow.

###  II.3 Contribution of $\lambda$ to its mass

The coodinates ($t$, $r$, $\theta$, $\phi$) make the meaning of $\lambda$
clear. From Eq. (4), we realize that not only the mass parameter $\tilde{M}$
but also the parameter $\lambda$ contributes the mass of the Damour-Solodukhin
wormhole.

However, Vagnozzi et al. Vagnozzi:2022moj  did not mention the effect of
$\lambda$ to the mass and they assumed that the mass parameter $\tilde{M}$
could be estimated by other observations GRAVITY:2020gka ;
EventHorizonTelescope:2019jan . The assumption is invalid since the value of
$\lambda$ strongly affects the determination of $\tilde{M}$. Thus, the
constraint on $\lambda$ from the shadow is invalid.

##  III Conclusion

In Ref. Vagnozzi:2022moj , Vagnozzi et al. gave the constraint of an
additional parameter of spacetimes with a photon sphere from the shadow
observation under the assumption that the mass parameter $\tilde{M}$ is given
by other observations. We should keep in mind that their constraint on the
additional parameter is valid only if it does not contribute the mass of the
spacetime. Unfortunately, they overlooked the contribution of the parameter
$\lambda$ of the Damour-Solodukhin wormhole to its mass. We realize that they
should not have estimated the mass parameter $\tilde{M}$ of the Damour-
Solodukhin wormhole from the other observations under the assumption
$\lambda=0$ since $\lambda$ strongly affects the mass of the wormhole. Thus,
we conclude that their constraint on the parameter $\lambda$ is invalid. We
suggest that they should give constraints on $\lambda$ and $\tilde{M}$ from
the shadow observation and other observations without the assumption
$\lambda=0$.

On this paper, we have concentraced on the Damour-Solodukhin wormhole
spacetime but we will give short comments on other spacetimes discussed in
Ref. Vagnozzi:2022moj . Their constraints on parameters in almost trivial
spacetimes would be valid but we should be careful to treat other non-trivial
spacetimes discussed in Ref. Vagnozzi:2022moj . We should reconsider several
non-trivial cases such as Eqs. (72), (92), (111), (129), and (132) because not
only the usual mass parameter but also another parameter can contribute the
mass of the spacetime.

## Acknowledgements

The author is grateful to Hideki Maeda for his comments on the asymptotically
(locally) flatness and the ADM mass.

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