11institutetext: Departamento de Matemáticas, Estadística e I.O., Universidad
de La Laguna, Tenerife, Spain, 11email<EMAIL_ADDRESS>22institutetext: Institute of Mathematics, University of Valladolid, Castilla,
Spain, 22email<EMAIL_ADDRESS>Universidad Autónoma
Metropolitana, México, 33email<EMAIL_ADDRESS>
# Free Resolutions and Generalized Hamming Weights of binary linear
codes††thanks: This work was partially supported by the Spanish MICINN
PID2019-105896GB-I00 and MASCA and MACACO (ULL Research Projects). Third
author was supported in part by Grant PGC2018-096446-B-C21 funded by
MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. Fourth
author was supported by SEGIB-Fundación Carolina Grant and SNI México.

I. García-Marco 11 I. Márquez-Corbella 11 E. Martínez-Moro 22 Y. Pitones 33

###### Abstract

In this work, we explore the relationship between free resolution of some
monomial ideals and Generalized Hamming Weights (GHWs) of binary codes. More
precisely, we look for a structure smaller than the set of codewords of
minimal support that provides us some information about the GHWs. We prove
that the first and second generalized Hamming weight of a binary linear code
can be computed (by means of a graded free resolution) from a set of monomials
associated to a binomial ideal related with the code. Moreover, the remaining
weights are bounded by the Betti numbers for that set.

###### Keywords:

Generalized Hamming Weight Graded free resolution Second distance Binary code.

##  1 Introduction

The study of the Generalized Hamming Weights (GHWs) has been motivated by
several applications in cryptography [32] and they characterize the
performance of a linear code when it is used for a given channel. There are
few families of codes for which it is known the complete generalized weight
hierarchy as for example: first-order Reed-Muller codes, binary Reed-Muller
codes, Hamming code and its dual, Extended Hamming codes, Golay code, see
[32]. On the other side, there has been an extensive research on GHWs and the
second distance in particular for some classes of codes, see for example [8,
10, 12, 14, 15, 16, 25, 28, 30, 33] and the references therein. However, for
the general case of a linear code, few properties are known.

In their seminal paper [17], Johnsen and Verdure showed how the GHWs of a
linear code could be computed from a free resolution of a monomial ideal
associated to the set of codewords of minimal support of the code provided we
know this last set. That paper has produced a great avenue of research, see
for example [9, 11, 18, 19, 20, 21, 22, 23, 24].

In the present work, we explore if one can find a set (smaller than the set of
all codewords of minimal support) that provides us some information on the
GHWs in the case of binary codes. The selected set of codewords is the so
called Gröbner test set related to the binomial ideal associated to a code
defined in [5]. In that paper it was proven that one can decode using that set
and that the minimal distance of the code (thus the first GHW) was reflected
on it. Thus, somehow some of the relevant information of the code lies in it.
Moreover, in [5] it was also shown how to compute the Gröbner test set
avoiding some of the most common disadvantages when one uses Gröbner basis. In
this paper we will show how one can compute also the second GHW of a binary
linear code from that binomial ideal associated with the code without the need
of computing the complete set of codewords of minimal support of the code as
in [17]. Moreover, in Theorem 4.2 we bound the remaining GHWs with the
resolution of a monomial ideal associated to this new set.

The structure of the contribution is simple, in Section 2 we revise some
results on GHW, free resolutions and the ideal associated with a code. Section
3 shows briefly the false conjecture and experiments that drive us to this
study. In Section 4 we show our main result and finally in Section 5 we show
some future lines of research and some conjectures related to the topic that
we hope will be helpful for future research.

##  2 GHW and minimal supports

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements. Given two vectors
$\mathbf{x}=(x_{1},\ldots,x_{n})$ and
$\mathbf{y}=(y_{1},\ldots,y_{n})\in\mathbb{F}_{q}^{n}$, the _Hamming distance_
between $\mathbf{x}$ and $\mathbf{y}$ is defined as

$d_{H}(\mathbf{x},\mathbf{y})=|\\{i\mid x_{i}\neq y_{i}\\}|,$

where $|\cdot|$ denotes the cardinality of the set. The _Hamming weight_ of
$\mathbf{x}$ is given by
$\mathrm{w}_{H}(\mathbf{x})=d_{H}(\mathbf{x},\mathbf{0})$, where $\mathbf{0}$
denotes the zero vector in $\mathbb{F}_{q}^{n}$. The _support_ of $\mathbf{x}$
is the set $\mathrm{supp}(\mathbf{x})=\\{i\mid x_{i}\neq 0\\}$. A linear
subspace $\mathcal{C}$ in $\mathbb{F}_{q}^{n}$ is called a _linear code_. The
elements of $\mathcal{C}$ are called _codewords_. The _basic parameters_ of
$\mathcal{C}$ are its length, its dimension and its minimum distance, which
are denoted by $n(\mathcal{C})$, $k(\mathcal{C})$ and $\delta(\mathcal{C})$,
respectively. In this case, we call $\mathcal{C}$ an
$[n(\mathcal{C}),k(\mathcal{C}),\delta(\mathcal{C})]_{q}$ linear code. We
define a _generator matrix_ of $\mathcal{C}$ to be a matrix $G$ over
$\mathbb{F}_{q}$ of size $k(\mathcal{C})\times n(\mathcal{C})$ whose row
vectors span $\mathcal{C}$, while a _parity check matrix_ of $\mathcal{C}$ is
a matrix $H$ over $\mathbb{F}_{q}$ of size
$(n(\mathcal{C})-k(\mathcal{C}))\times n(\mathcal{C})$ whose null space is
$\mathcal{C}$.

###### Definition 1

Let $\mathcal{C}$ be a linear code, we will say that a codeword $\mathbf{m}$
has minimal support if it is non-zero and $\mathrm{supp}(\mathbf{m})$ is not
contained in the support of any other codewords. We will denote by
$\mathcal{M}_{\mathcal{C}}$ the set of codewords of minimal support of
$\mathcal{C}$.

Note that computing a set of codewords of minimal support is a hard problem
for a general linear code, in fact it implies finding the minimum distance of
the code and, thus, it is harder that the problem of maximum-likelihood
complete decoding, see [3, 2].

###### Definition 2

Let $\mathcal{C}$ be a linear code and $D$ be a subcode of $\mathcal{C}$, we
define _the support of $D$_, denoted $\mathrm{supp}(D)$ as the set of not-
always-zero bit positions of $D$, i.e.,

$\mathrm{supp}(D)=\\{i\mid\exists\mathbf{c}\in D\hbox{ with }c_{i}\neq 0\\}.$

It is clear that if $D$ is a one-dimensional subcode then, the support of $D$
is equal to the Hamming weight of any of its nonzero codewords, i.e.
$d_{1}(\mathcal{C})=\delta(\mathcal{C})$. Based on this idea, the _$h$ -th
generalized Hamming weight_ of $\mathcal{C}$, denoted $d_{h}(\mathcal{C})$, is
the size of the smallest support of an $h$-dimensional subcode of
$\mathcal{C}$ with $h=1,2,\ldots,k(\mathcal{C})$. That is, if $D_{h}$ is the
set of all linear subspaces of the linear code $\mathcal{C}$ of dimension $h$,
then

$d_{h}(\mathcal{C})=\min\\{|{\rm supp}(E)|\mid E\in D_{h}\\}.$

Some basic facts on the Generalized Hamming Weights (GHW) are provided in the
following proposition.

######  Proposition 1 ([32])

Let $\mathcal{C}$ be a linear code. Then:

  1. 1.

$1\leq
d_{1}(\mathcal{C})<d_{2}(\mathcal{C})<\ldots<d_{k(\mathcal{C})}(\mathcal{C})\leq
n(\mathcal{C})$

  2. 2.

(Generalized Singleton Bound) $d_{h}(\mathcal{C})\leq
n(\mathcal{C})-k(\mathcal{C})+h$.

From now on we assume that $\mathcal{C}$ is a nondegenerate code, that is,
$d_{k(\mathcal{C})}(\mathcal{C})~{}=~{}n(\mathcal{C})$.

The GHWs $d_{1}(\mathcal{C})$, …, $d_{k(\mathcal{C})}(\mathcal{C})$ are
completely determined by the underlying linear matroid structure of the code
in a non-trivial manner, and the way to obtain them is not efficient as it was
already pointed in [17]. Of course, as pointed before, calculating
$d_{1}(\mathcal{C})$ is equivalent to the problem of complete decoding linear
codes [2], hence one can not expect a computational efficient approach. From
now on, given a positive integer $\ell,$ we define
$\left[\ell\right]=\left\\{1,\ldots,\ell\right\\}$ and
$\left[\ell\right]_{0}=\left\\{0,\ldots,\ell\right\\}$.

###### Definition 3

Let $\mathcal{C}$ be a $[n,k]_{q}$ code and let $H$ be a parity check matrix
of $\mathcal{C}$. Let $H_{i}$ denote the $i$-th column of $H$ and define the
simplicial complex

$\Delta=\left\\{\sigma\in 2^{\left[n\right]}\mid\\{H_{i}\mid
i\in\sigma\\}\hbox{ is linearly independent over }\mathbb{F}_{q}\right\\}.$

Then, the pair $\mathcal{M}=(\left[n\right],\Delta)$ is the linear _matroid
associated with the code_ $\mathcal{C}$. The collection $\Delta$ of subsets of
$\left[n\right]$ are called _independent sets_ of this matroid. A subset of
$\left[n\right]$ that does not belong to $\Delta$ is called _dependent set_.
Minimal dependent subsets of $\left[n\right]$ are known as _circuits_ of
$\mathcal{M}$. A set is said to be a _cycle_ if it is a disjoint union of
circuits.

We refer to [27] for a brief introduction on the theory of simplicial
complexes, and to [29] for a thorough study of matroids.

###### Definition 4

Let $\mathbb{K}$ be any field. We denote by $I_{\Delta}$ the ideal in the
polynomial ring $R=\mathbb{K}[X_{1},\ldots,X_{n}]$ over a field $\mathbb{K}$
generated by all square-free monomials supported on elements that are not in
$\Delta$, i.e.

$\prod_{i\in\tau}X_{i}\hbox{ with }\tau\in 2^{\left[n\right]}\setminus\Delta$

That is, $I_{\Delta}$ is the ideal generated by monomials supported on the
circuits of $\mathcal{M}$, or equivalently in a matroid associated with a
code, supported on codewords of minimal support of $\mathcal{C}$.

$I_{\Delta}=\left\langle\prod_{i\in\mathrm{supp}(\mathbf{c})}X_{i}\mid\mathbf{c}\in\mathcal{M}_{\mathcal{C}}\right\rangle.$
(1)

The quotient $R_{\Delta}=R/I_{\Delta}$ is called the _Stanley-Reisner_ ring
associated with $\Delta$. $R_{\Delta}$ is a finitely generated standard graded
$\mathbb{K}$-algebra of dimension $n(\mathcal{C})-k(\mathcal{C})$. Thus, it
has a minimal graded free resolution. Moreover, since the generators of
$I_{\Delta}$ are supported in the set of circuits of a matroid, by [4], one
has that $\Delta$ is shellable and this implies that $R_{\Delta}$ is Cohen-
Macaulay. So, by the Auslander-Buchsbaum formula, the projective dimension of
$R_{\Delta}$ (i.e., the length of any minimal graded free resolution of
$R_{\Delta}$) is $k(\mathcal{C})$ and it looks like

$\begin{array}[]{ccccccccccccccc}0&\longrightarrow&F_{k(\mathcal{C})}&\longrightarrow&F_{k(\mathcal{C})-1}&\longrightarrow&\cdots&\longrightarrow&F_{1}&\longrightarrow&F_{0}&\longrightarrow&R_{\Delta}&\longrightarrow
0\end{array}$ (2)

where $F_{0}=R$ and each $F_{i}$ is a graded free $R$-module of the form

$F_{i}=\bigoplus_{j\in\mathbb{N}}R(-j)^{\beta_{i,j}}\hbox{ for
}i\in\left[k(\mathcal{C})\right]_{0}.$

We will refer to Equation (2) as a graded minimal free resolution of
$\mathcal{C}$. The nonnegative integers $\beta_{i,j}$ are called _Betti
numbers_ of $\mathcal{C}$ and they depend only on $\mathcal{C}$ and not on the
choice of the parity check matrix $H$, or the minimal free resolution of
$R_{\Delta}$ or even the chosen field $\mathbb{K}$ (this is because
$I_{\Delta}$ comes from the set of circuits of a matroid, see [4]).

In [17], Johnsen and Verdure described the GHWs of a linear code $\mathcal{C}$
in terms of the shifts of the minimal graded free resolution of a Stanley-
Reisner ideal $I_{\Delta}$ associated to $\mathcal{C}$. One of the main
disadvantages of this method is that the generators of $I_{\Delta}$ correspond
to the supports of all minimal support of $\mathcal{C}$. In general, the whole
set of codewords of minimal support can be huge and computationally expensive
to obtain and, in many cases, computing a minimal graded free resolution of an
ideal with that many generators is unaffordable. More precisely, their result
was:

###### Theorem 2.1

[17, Theorem 2] Let $\mathcal{C}$ be a $q$-ary linear code. Then,

$d_{i}(\mathcal{C})=\min\\{j\mid\beta_{i,j}\neq 0\\}\hbox{ for
}j\in\left[k(\mathcal{C})\right]$

######  Example 1 (Toy example)

Let $\mathcal{C}$ be the binary non-degenerate $[6,3]$ code with generator
matrix

$G=\left(\begin{array}[]{cccccc}1&0&0&0&0&1\\\ 0&1&1&0&1&0\\\
0&0&0&1&1&1\end{array}\right)\in\mathbb{F}_{2}^{3\times 6}$

One can check that the set of codewords of minimal support is

$\mathcal{M}_{\mathcal{C}}=\left\\{\begin{array}[]{ccc}w_{1}=100001,&w_{2}=100110,&w_{3}=011010,\\\
w_{4}=000111,&w_{5}=111100,&w_{6}=011101\end{array}\right\\}$

Its Stanley-Reisner ring is $R_{\Delta}=R/I_{\Delta}$ where
$R=\mathbb{F}_{2}[x_{1},\ldots,x_{6}]$ and the ideal $I_{\Delta}$ is generated
by the monomials associated to $\mathcal{M}_{\mathcal{C}}$. If we compute a
graded minimal free resolution of $R_{\Delta}$ we get the following Betti
diagram:

$\begin{array}[]{c|cccc}&0&1&2&3\\\ \hline\cr 0&1&0&0&0\\\ 1&0&1&0&0\\\
2&0&3&2&0\\\ 3&0&2&7&4\\\ \end{array}$

where the entry of the row indexed by $i$ and column indexed by $j$ indicates
the value $\beta_{j,i+j}$ (for example $\beta_{3,5}=7$). Hence, by Theorem
2.1, we have $d_{1}(\mathcal{C})=2$, $d_{2}(\mathcal{C})=4$ and
$d_{3}(\mathcal{C})=6$.

This result shows that the determination of the Betti numbers of the monomial
ideal $I_{\Delta}$ related to a code completely determine the weight
hierarchy. However, as mentioned before, this is usually a hard problem [2]
except in some special cases. For example, Johnsen and Verdure in [18]
explicitly determine the Betti Numbers for MDS codes, since the minimal free
resolution of these codes is linear. Moreover, in [18] the authors prove that
the resolution of the first order Reed-Muller code is pure. And a similar
result can be deduced for constant weight codes [11]. Thus, simplex codes or
dual Hamming codes, which are constant weight codes also have pure resolution,
although not necessarily linear.

###### Remark 1

The resolution (2) is said to be _pure_ of type
$(d_{0},\ldots,d_{k(\mathcal{C})})$ if for each $i\in\left[k\right]_{0}$, the
Betti number $\beta_{i,j}$ is nonzero if and only if $j=d_{i}$. If, in
addition $d_{1},\ldots,d_{k}$ are consecutive, then the resolution is said to
be _linear_.

The rest of this work is devoted to compute a simpler and smaller structure
than the whole set of codewords of minimal support that allows, at least
partially, know something about the GHW of the code.

###  2.1 Test-Sets of a binary code

From now on, we will restrict our study to binary codes. Let $\mathcal{C}$ be
a binary linear code. Let $X$ be a vector with $n=n(\mathcal{C})$ variables
$x_{1},\ldots,x_{n}$. A monomial in $X$ is a product of the form
$X^{\mathbf{a}}:=\prod_{i=1}^{n}x_{i}^{\,a_{i}},$ where
$\mathbf{a}=(a_{1},\ldots,a_{n})\in\mathbb{N}^{n}$. The total degree of
$X^{\mathbf{a}}$ is $\deg(X^{\mathbf{a}})=\sum_{i=1}^{n}a_{i}$. When
$\mathbf{a}=\mathbf{0}$, note that $X^{\mathbf{a}}=\mathbf{1}$. The polynomial
ring $\mathbb{K}[X]$ is the set of all polynomials in $X$ with coefficients in
$\mathbb{K}$, where $\mathbb{K}$ denotes an arbitrary finite field.

###### Remark 2

For abuse of notation we will write $X^{\mathbf{a}}$ with
$\mathbf{a}\in{\mathbb{F}_{2}^{n}}$. In this case, we understand that the
classes of $0,1$ are replaced by the same symbols regarded as integers.
Moreover, we will use the notation $X^{I}$ for the square free monomial with
support $I\subset[n]$, that is,

$X^{I}=\prod_{i\in I}x_{i}\hbox{ with }I\subseteq\left[n\right].$

Let $g=X^{A}-X^{B}$ be a binomial with $A,B\subseteq\left[n\right]$, we define
$\mathrm{supp}(g)=A\cup B$. We say that $g$ is in _standard form_ if $A\cap
B=\emptyset$ or, equivalently, if $\mathrm{supp}(g)=A\sqcup B,$ where $\sqcup$
denotes the disjoint union of $A$ and $B$.

###### Definition 5

Let $\mathbb{K}$ be any field, we define the _ideal associated with
$\mathcal{C}$ over $\mathbb{K}$_ as the binomial ideal:

$I(\mathcal{C})=\langle
X^{\mathbf{a}}-X^{\mathbf{b}}\mid\mathbf{a},\mathbf{b}\in\mathbb{F}_{2}^{n},\,\mathbf{a}+\mathbf{b}\in\mathcal{C}\rangle+\langle
x_{i}^{2}-1\mid i\in[n]\rangle\subseteq\mathbb{K}[X].$ (3)

Note that $I(\mathcal{C})$ is a zero-dimensional ideal since the quotient ring
$R=\mathbb{K}[X]/I(\mathcal{C})$ is a finite-dimensional vector space (i.e.
$\dim_{\mathbb{K}}(R)<\infty$). Moreover, its dimension is equal to the number
of _cosets_ in $\mathbb{F}_{2}^{n}/\mathcal{C}$. For
$\mathbf{a},\mathbf{b}\in\mathbb{F}_{2}^{n}$ one has that
$X^{\mathbf{a}}-X^{\mathbf{b}}\in I(\mathcal{C})$ if and only if
$\mathbf{a}-\mathbf{b}\in\mathcal{C}$. The following result shows how to
obtain a set of generators of the ideal $I(\mathcal{C})$ from a generator
matrix of $\mathcal{C}$.

###### Proposition 2

[5, Theorem 1] Let $\\{\mathbf{w}_{1},\ldots,\mathbf{w}_{k}\\}$ be the row
vectors of a generator matrix for $\mathcal{C}$. Then

$I(\mathcal{C})=\left\langle\left\\{X^{\mathbf{w}_{i}}-1\right\\}_{i\in\left[k\right]}\cup\left\\{x_{i}^{2}-1\right\\}_{i\in\left[n\right]}\right\rangle$

If we fix a term order $\prec$, then the _leading term_ of a polynomial $f$
with respect to $\prec$, denoted by $\mathrm{LT}_{\prec}(f)$, is the largest
monomial among all monomials which occur with non-zero coefficient in the
expansion of $f$. Let $I$ be an ideal in $\mathbb{K}[X]$, then the _initial
ideal_ $\mathrm{in}_{\prec}(I)$ is the monomial ideal generated by the leading
term of all the polynomials in $I$, i.e.,
$\mathrm{in}_{\prec}(I)=\langle\left\\{\mathrm{LT}_{\prec}(f)\mid f\in
I\right\\}\rangle$. By definition, $\mathrm{in}_{\prec}(I)$ is a monomial
ideal and, thus, it has a unique minimal generating set formed by monomials.
These monomials will be called minimal generators of $\mathrm{in}_{\prec}(I)$.

###### Definition 6

A finite set of nonzero polynomials $\mathcal{G}=\\{g_{1},\ldots,g_{m}\\}$ of
the ideal $I$ is a Gröbner basis of $I$ with respect to the term order $\prec$
if the leading terms of the elements of $\mathcal{G}$ generate the initial
ideal $\mathrm{in}_{\prec}(I)$. Moreover $\mathcal{G}$ is reduced if

  1. 1.

$g_{i}$ is monic (i.e., its leading coefficient is $1$) for all
$i\in\left[m\right]$, and

  2. 2.

none of the monomials appearing in the expansion of $g_{j}$ is divisible by
$\mathrm{LT}_{\prec}(g_{i})$ for all $i\neq j$.

For a given monomial order $\prec$, every ideal has a unique reduced Gröbner
basis (see, e.g., [1]). Since $I(\mathcal{C})$ is generated by binomials
(differences of monomials), then all its reduced Gröbner bases consist of
binomials (see [7]). In [5] it is shown that, if $\mathcal{C}$ is a binary
code and we fix a degree compatible term order $\prec$ on $\mathbb{K}[X]$,
then the reduced Gröbner basis $\mathcal{G}_{\prec}$ for the code ideal
$I(\mathcal{C})$ can be computed by a linear algebra (an FGLM-like) algorithm.
Moreover, the reduction provided by $\mathcal{G}_{\prec}$ gives a decoding
procedure. Along the way, they also prove that the support of every binomial
in $\mathcal{G}_{\prec}$ different from $x_{i}^{2}-1$ for $i=1,\ldots,n$
provides a codeword of minimal support of $\mathcal{C}$, and that there is a
word of Hamming weight $d_{1}(\mathcal{C})$ that can be obtained in this way.
More precisely:

###### Proposition 3

[5] Let $\mathcal{G}_{\prec}$ be the reduced Gröbner basis of $I(\mathcal{C})$
with respect to a degree compatible term order $\prec$. For every binomial
$X^{\mathbf{a}}-X^{\mathbf{b}}\in\mathcal{G}_{\prec}-\\{x_{i}^{2}-1\,|\,i\in[n]\\}$,
then $\mathbf{a}+\mathbf{b}\in\mathbb{F}_{2}^{n}$ is a codeword of minimal
support of $\mathcal{C}$. Moreover, there exists
$X^{\mathbf{a}}-X^{\mathbf{b}}\in\mathcal{G}_{\prec}$ such that
$\mathrm{w}_{H}(\mathbf{a}+\mathbf{b})=d_{1}(\mathcal{C})$.

This result motivates the definition of a $\mathcal{G}_{\prec}$-test, which by
the above proposition is a subset of the set of codewords of minimal support
and contains a word of minimum weight.

###### Definition 7

Given a binary code $\mathcal{C}$ and a degree compatible term order $\prec$,
we will call $\mathcal{G}_{\prec}$-test set of $\mathcal{C}$ to the subset of
codewords of minimal support of $\mathcal{C}$ whose supports are given by
those binomials in the reduced Gröbner basis of $I(\mathcal{C})$ different
from $x_{i}^{2}-1$ for all $i\in[n]$.

###### Example 2

We continue the toy Example 1. Then, the ideal associated with $\mathcal{C}$
over $R$ is defined as

$I(\mathcal{C})=\left\langle\left\\{\begin{array}[]{ccc}x_{1}x_{6}-1,&x_{2}x_{3}x_{5}-1,&x_{4}x_{5}x_{6}-1\end{array}\right\\}\cup\left\\{x_{i}^{2}-1\right\\}_{i\in\left[6\right]}\right\rangle\subseteq
R$

Now we consider the degree reverse lexicographic order $\prec$ with
$x_{6}\prec\ldots\prec x_{1}$. Then, the reduced Gröbner basis
$\mathcal{G}_{\prec}$ of $I(\mathcal{C})$ with respect to $\prec$ has $14$
elements. An a $\mathcal{G}_{\prec}$-test-set of $\mathcal{C}$ is given by
codewords of $\mathcal{M}_{\mathcal{C}}$ whose supports are given by those
binomials of $\mathcal{G}_{\prec}$ different from $x_{i}^{2}-1$, i.e. the set

$\left\\{\begin{array}[]{cccc}w_{1}=100001,&w_{3}=011010,&w_{4}=000111,&w_{6}=011101\end{array}\right\\}\subseteq\mathcal{M}_{\mathcal{C}}.$

##  3 On a false conjecture

Note that the second and the third author of this work have conjectured for a
long time the following:

Conjecture (false) If one considers the monomial ideal $M$ associated with the
supports of the binomials in the $\mathcal{G}_{\prec}$-test set, then
$d_{i}={\rm min}\\{j\,|\,\beta_{i,j}(R/M)\neq 0\\}$ for
$i\in\\{1,\ldots,k(\mathcal{C})\\}$, that is, the $\mathcal{G}_{\prec}$-test
set determines the GHWs of the codes.

This conjecture was supported by computational evidence (see Examples 3 and 4
for some example satisfying this conjecture), but unfortunately this is not
true as shown in the counterexample 5. Theorem 4.2 in Section 4 in this paper
will prove this fact for $i=2$ (and it was known for $i=1$ [5]). Note also
that in [26], the authors show that from the Graver basis associated to
$I(\mathcal{C})$ one can purge the set of codewords of minimal support of
$\mathcal{C}$, i.e. $\mathcal{M}_{\mathcal{C}}$.

###### Example 3

We continue the Example 1 where we have computed the Betti diagram associated
to $R_{\Delta}=R/I_{\Delta}$ where $I_{\Delta}$ is generated by the monomials
associated to $\mathcal{M}_{\mathcal{C}}$. Then, in Example 2 we compute a
$\mathcal{G}_{\prec}$-test-set $\mathcal{T}_{\mathcal{C}}$ of $\mathcal{C}$
with respect to the degree reverse lexicographic order $\prec$. Note that
$\mathcal{T}_{\mathcal{C}}\subset\mathcal{M}_{\mathcal{C}}$ with just $4$
elements. If we consider $M$ the corresponding monomial ideal related to
$\mathcal{T}_{\mathcal{C}}$ and compute a graded minimal free resolution of
$R/M$ we get the following Betti diagram:

$\begin{array}[]{c|cccc}&0&1&2&3\\\ \hline\cr 0&1&0&0&0\\\ 1&0&1&0&0\\\
2&0&2&1&0\\\ 3&0&4&4&2\\\ \end{array}$

The Betti numbers of $R/M$ are smaller than those of $R_{\Delta}$ and in this
example the sequence

$({\rm min}\\{j\,|\,\beta_{i,j}\neq 0\\};1\leq i\leq 3)=(2,4,6),$

coincides with the GHW of $\mathcal{C}$

The following example is a less trivial case where the difference between our
structure $\mathcal{T}_{\mathcal{C}}$ and the set of codewords of minimal
support of $\mathcal{C}$, $\mathcal{M}_{\mathcal{C}}$, is larger.

###### Example 4

Let $\mathcal{C}$ be the binary non-degenerate $[14,9]$-code with generator
matrix

$G=\left(\begin{array}[]{cccccccccccccccccccccc}1&1&0&1&0&0&0&0&0&1&0&0&0&0\\\
1&1&1&1&0&1&0&0&1&1&0&0&0&1\\\ 0&0&1&0&0&1&1&1&1&1&0&1&1&0\\\
1&1&0&1&1&1&0&0&1&1&0&0&0&1\\\ 0&1&1&1&1&0&0&1&0&0&1&0&1&1\\\
0&0&1&1&0&1&1&0&1&0&0&1&0&1\\\ 0&0&1&0&1&1&1&0&1&1&1&0&1&0\\\
0&1&0&1&1&0&1&0&1&1&0&1&1&0\\\ 1&1&1&1&1&1&0&1&0&0&0&0&1&0\\\
\end{array}\right)\in\mathbb{F}_{2}^{\,9\times 14}$

Its Stanley-Reisner ring is $R_{\Delta}=R/I_{\Delta}$ where
$R=\mathbb{F}_{2}[x_{1},\ldots,x_{14}]$ and $I_{\Delta}$ is minimally
generated by $147$ monomials. If we compute a graded minimal free resolution
of $R_{\Delta}$ we get the following Betti diagram:

$\begin{array}[]{c|ccccccccccc}&0&1&2&3&4&5&6&7&8&9\\\ \hline\cr
0&1&0&0&0&0&0&0&0&0&0\\\ 1&0&2&0&0&0&0&0&0&0&0\\\ 2&0&8&5&0&0&0&0&0&0&0\\\
3&0&34&82&48&8&0&0&0&0&0\\\ 4&0&52&441&897&753&289&42&0&0&0\\\
5&0&51&1345&7410&18309&25248&21008&10579&2990&366\par\end{array}$

Hence, by Theorem 2.1, we have that the GHW are

$(d_{1}(\mathcal{C}),\ldots,d_{9}(\mathcal{C}))=(2,4,6,7,9,10,12,13,14).$

Moreover, if we consider $\prec$ the graded degree lexicographic order with
$x_{1}\succ\cdots\succ x_{14}$, we get that the $\mathcal{G}_{\prec}$-test set
has 24 elements. If we consider $M$ the corresponding monomial ideal and
compute a graded minimal free resolution of $R/M$ we get the following Betti
diagram:

$\begin{array}[]{c|ccccccccccc}&0&1&2&3&4&5&6&7&8&9\\\ \hline\cr
0&1&0&0&0&0&0&0&0&0&0\\\ 1&0&2&0&0&0&0&0&0&0&0\\\ 2&0&6&3&0&0&0&0&0&0&0\\\
3&0&13&38&17&2&0&0&0&0&0\\\ 4&0&3&92&194&130&35&3&0&0&0\\\
5&0&0&83&599&1410&1621&1040&378&71&5\\\ 6&0&0&0&0&2&5&4&1&0&0\\\ \end{array}$

As one can observe, the Betti numbers of $R/M$ are smaller than those of
$R_{\Delta}$. Moreover, in this example the sequence $({\rm
min}\\{j\,|\,\beta_{i,j}\neq 0\\};1\leq i\leq 9)$ is given by
$(2,4,6,7,9,10,12,13,14)$ and it coincides with the GHW of $\mathcal{C}$.

###### Example 5

Let $\mathcal{C}$ be the binary non-degenerate $[10,7]$-code with generator
matrix

$G=\left(\begin{array}[]{cccccccccc}1&0&0&0&1&0&0&1&1&1\\\
1&1&1&1&0&1&1&1&1&1\\\ 1&0&0&1&0&1&1&0&0&0\\\ 0&0&1&1&1&0&1&0&0&1\\\
1&0&1&0&0&1&1&0&0&1\\\ 0&0&1&1&0&1&1&1&0&0\\\ 0&0&1&1&0&0&0&1&1&1\\\
\end{array}\right)\in\mathbb{F}_{2}^{\,7\times 10}$

Its Stanley-Reisner ring is $R_{\Delta}=R/I_{\Delta}$ where
$R=\mathbb{F}_{2}[x_{1},\ldots,x_{10}]$ and $I_{\Delta}$ is minimally
generated by $42$ monomials. If we compute a graded minimal free resolution of
$R_{\Delta}$ we get the following Betti diagram:

$\begin{array}[]{c|cccccccc}&0&1&2&3&4&5&6&7\\\ \hline\cr 0&1&0&0&0&0&0&0&0\\\
1&0&4&0&0&0&0&0&0\\\ 2&0&18&48&32&7&0&0&0\\\ 3&0&20&214&637&874&637&242&38\\\
\end{array}$

-/ Hence, by Theorem 2.1, we have $d_{1}(\mathcal{C})=2$, $d_{2}(\mathcal{C})=4$, $d_{3}(\mathcal{C})=5$, $d_{4}(\mathcal{C})=6$, $d_{5}(\mathcal{C})=8$, $d_{6}(\mathcal{C})=9$, $d_{7}(\mathcal{C})=10$.

If we compute a $\mathcal{G}_{\prec}$-test set $T$ with respect to the degree
reverse lexicographical order, we get that $T$ has only 10 elements. Computing
a minimal graded free resolution of $R/M$ being the ideal
$M:=\langle\\{X^{\mathbf{c}}\,|\,\mathbf{c}\in T\\}\rangle$, we get the
following Betti diagram.

$\begin{array}[]{c|ccccccc}&0&1&2&3&4&5&6\\\ \hline\cr 0&1&0&0&0&0&0&0\\\
1&0&4&0&0&0&0&0\\\ 2&0&4&14&5&0&0&0\\\ 3&0&2&23&56&48&17&2\\\ \end{array}$

Thus, we can recover the correct values of
$d_{1}(\mathcal{C}),d_{2}(\mathcal{C}),d_{3}(\mathcal{C})$ from this
resolution but not $d_{i}(\mathcal{C})$ for $i=4,5,6,7$.

Anyway, the experiments we dealt trying to prove that conjecture drove us the
results in Section 4 and to propose some further conjectures and future work
based on experimental evidences showed in Section 5.

##  4 Second GHW obtained from a $\mathcal{G}_{\prec}$-test set

In this section we will explain how to compute the second generalized Hamming
weight from a $\mathcal{G}_{\prec}$-test set. Throughout this section,
$\mathcal{C}$ will denote a binary linear code and let $\mathcal{G}_{\prec}$
be the reduced Gröbner basis of the ideal $I(\mathcal{C})$ with respect to
$\prec$, where we take $\prec$ to be any degree compatible ordering on
$\mathbb{K}[X]$. The following result is a technical lemma whose proof is just
an easy exercise in set theory. We will denote by $A\triangle B$ the
_symmetric difference_ of the subsets $A,B\subseteq[n]$, which is the set of
elements which are in the union of the two sets $A\cup B$, minus their
intersection $A\cap B$. Moreover, for a set $A$, its cardinality is denoted by
$|A|$.

###### Lemma 1

Let $A,B\subseteq X$ with $|A\cap B|>\frac{|A|}{2}$. Then $C=A\triangle B$
satisfies the following statements:

  1.   2. 1.

$A\cup B=A\cup C$

  3. 2.

$|A\cap C|<\frac{|A|}{2}$

  4. 3.

$|C|<|B|$.

Consider the set $\mathbf{M}$ of codewords in $\mathcal{C}$ belonging to a
linear subspace of dimension two of minimal support, more precisely,

$\mathbf{M}=\\{\mathbf{m}\in\mathcal{C}\,|\,\exists\mathbf{m}^{\prime}\in\mathcal{C}\text{
such that
}d_{2}(\mathcal{C})=\mathrm{supp}(\left\langle\mathbf{m},\mathbf{m}^{\prime}\right\rangle)\\},$
(4)

and define $\mathbf{m}_{1},\mathbf{m}_{2}\in\mathbf{M}$ as follows:

  * (a)

$\mathbf{m}_{1}:={\rm min}_{\prec}(\mathbf{M})$, i.e., $\mathbf{m}_{1}$ is the
smallest codeword with respect to $\prec$ in $\mathbf{M}$,

  * (b)

$\mathbf{m}_{2}:={\rm
min}_{\prec}\\{\mathbf{m}\in\mathbf{M}\,|\,d_{2}(\mathcal{C})=\mathrm{supp}(\left\langle\mathbf{m}_{1},\mathbf{m}\right)\rangle\\}$,
i.e, $\mathbf{m}_{2}$ is the smallest codeword with respect to $\prec$ such
that
$d_{2}(\mathcal{C})=\mathrm{supp}(\left\langle\mathbf{m}_{1},\mathbf{m}_{2}\right\rangle)$.

With these conditions we define:

$\begin{array}[]{ccc}I=\mathrm{supp}(\mathbf{m}_{1})&\hbox{ and
}&J=\mathrm{supp}(\mathbf{m}_{2}).\end{array}$

###### Remark 3

Since $\prec$ is degree compatible and, by Lemma 1, we have that

$|I\cap J|\leq\frac{|I|}{2}\leq\frac{|J|}{2}.$

###### Proposition 4

There exists a binomial $f\in\mathcal{G}_{\prec}\subseteq\mathbb{K}[X]$ such
that $\mathrm{supp}(f)=I$.

###### Proof

Consider $f=X^{I_{1}}-X^{I_{2}}\in I(\mathcal{C})$ any binomial with
$I=I_{1}\sqcup I_{2}$, $|I_{1}|-1\leq|I_{2}|\leq|I_{1}|$ and $X^{I_{1}}\succ
X^{I_{2}}$, we will show that $f\in\mathcal{G}_{\prec}$. For proving this, it
suffices to check that:

  * (a)

$X^{I_{1}}=\mathrm{LT}_{\prec}(f)$ is a minimal generator of
$\mathrm{in}_{\prec}(I(\mathcal{C}))$, and

  * (b)

$X^{I_{2}}\notin\mathrm{in}_{\prec}(I(\mathcal{C}))$.

Proof of (a). By contradiction suppose that there exists a binomial
$h=X^{K_{1}}-X^{K_{2}}\in\mathcal{G}_{\prec}$ with $K=K_{1}\sqcup K_{2}$,
$\mathrm{LT}_{\prec}(h)=X^{K_{1}}$ (and, in particular, $|K_{1}|\geq|K_{2}|$)
such that $K_{1}\neq I_{1}$ and $X^{I_{1}}$ is divisible by $X^{K_{1}}$, i.e.
$K_{1}\subsetneq I_{1}$.

_Claim:_ $d_{2}(\mathcal{C})\leq|I\cup K|-1$.

_Proof of the claim:_ We have that
$|K_{2}|\leq|K_{1}|\leq|I_{1}|-1\leq|I_{2}|$. Then $|K|<|I|$ and, in
particular, $X^{K}\prec X^{I}\prec X^{J}$. Hence, by the choice of
$\mathbf{m}_{2}$, we have that $|I\cup J|<|I\cup K|$ and the _Claim_ follows.

By the previous _Claim_ we have that

$\displaystyle|I\cup J|$ $\displaystyle=$ $\displaystyle
d_{2}(\mathcal{C})\leq|I\cup K|-1~{}~{}~{}~{}$ $\displaystyle=$
$\displaystyle|I\cup K_{2}|-1~{}~{}~{}~{}\hbox{(since }K_{1}\subsetneq
I\hbox{)}$ $\displaystyle=$ $\displaystyle|I|+|K_{2}|-|I\cap K_{2}|-1$
$\displaystyle\leq$ $\displaystyle|I|+|K_{1}|-|I\cap
K_{2}|-1~{}~{}~{}~{}\hbox{(since }X^{K_{1}}\succ X^{K_{2}}\hbox{)}.$

Thus,

$|I|+|J|-|I\cap J|=|I\cup J|\leq|I|+|K_{1}|-|I\cap K_{2}|-1$

or equivalently $|J|-|I\cap J|\leq|K_{1}|-|I\cap K_{2}|-1$. Thus,

$\displaystyle|I_{1}|-\frac{1}{2}$ $\displaystyle\leq$
$\displaystyle\frac{|I|}{2}\leq\frac{|J|}{2}\leq|J|-|I\cap
J|~{}~{}~{}~{}\hbox{ (by Remark \ref{remark1})}$ $\displaystyle\leq$
$\displaystyle|K_{1}|-|I\cap K_{2}|-1\leq|K_{1}|-1.$

Therefore, $|I_{1}|\leq|K_{1}|-\frac{1}{2}$ and $K_{1}\subsetneq I_{1}$, a
contradiction.

Proof of (b). By contradiction, we assume that there exists
$h=X^{K_{1}}-X^{K_{2}}\in\mathcal{G}_{\prec}$ such that
$\mathrm{LT}_{\prec}(h)=X^{K_{1}}$ divides $X^{I_{2}}$ or, equivalently,
$K_{1}\subseteq I_{2}$. Then $K_{2}\prec K_{1}\prec I_{2}\prec I_{1}$ and, in
particular, $X^{K}\prec X^{I}\prec X^{J}$. Hence, by the choice of
$\mathbf{m}_{2}$, we have that $|I\cup J|<|I\cup K|.$

$\displaystyle d_{2}(\mathcal{C})$ $\displaystyle=$ $\displaystyle|I\cup
J|\leq|I\cup K|-1=|I\cup K_{2}|-1=|I|+|K_{2}|-|I\cap K_{2}|-1$
$\displaystyle\leq$ $\displaystyle|I|+|K_{2}|-1\leq|I|+|K_{1}|-1.$

Therefore, $|J|-|I\cap J|\leq|K_{1}|-1$. And, by Remark 3 we deduce that

$|I_{2}|\leq\frac{|I|}{2}\leq\frac{|J|}{2}\leq|J|-|I\cap J|\leq|K_{1}|-1.$

Therefore, $|I_{2}|<|K_{1}|$ and $K_{1}\subseteq I_{2}$, which is a
contradiction. ∎

One can check that the same result (and the same proof) holds for all
$I^{\prime}$ such that $I^{\prime}={\rm supp}(\mathbf{m})$ with
$\mathbf{m}\in\mathbf{M}$ and $|I^{\prime}|=|I|$. As a consequence of this
observation we have that:

###### Corollary 1

If $|I|=|J|$. Then, we can always find binomials
$f,g\in\mathcal{G}_{\prec}\subseteq\mathbb{K}[X]$ such that
$\mathrm{supp}(f)=I$ and $\mathrm{supp}(g)=J$.

We just found that $I$ is always involved in the supports associated with
$\mathcal{G}_{\prec}$ and that if $J$ has the same cardinal as them, also the
second GHW can be derived from the Gröbner basis. Let us prove now the general
case.

###### Proposition 5

There exists a binomial $g\in\mathcal{G}_{\prec}\subseteq\mathbb{K}[X]$ such
that $\mathrm{supp}(g)=J$.

###### Proof

By Remark 3 we know that $|I\cap J|\leq|J|/2$, then one may consider

$g=X^{I\cap J}X^{J_{1}}-X^{J_{2}}\in I(\mathcal{C}),$

such that $J_{1}\cup J_{2}=J-I$, $J_{1}\cap J_{2}=\emptyset$ and
$|J_{2}|+1\geq|I\cap J|+|J_{1}|\geq|J_{2}|$. Our goal is to prove that $g$ (or
$-g$) is in $\mathcal{G}_{\prec}$. We split the proof in two:

Case ${\rm LT}_{\prec}(g)=X^{I\cap J}X^{J_{1}}$.

We are going to see that $g\in\mathcal{G}_{\prec}$. First, we show that
$X^{I\cap J}X^{J_{1}}$ is a minimal generator of ${\rm
in}_{\prec}(I(\mathcal{C}))$. By contradiction, suppose that there is a
binomial $h=X^{K_{1}}-X^{K_{2}}\in\mathcal{G}_{\prec}$ with
$\mathrm{LT}_{\prec}(h)=X^{K_{1}}$ such that $K_{1}\subsetneq(I\cap J)\cup
J_{1}$, and denote $K=K_{1}\sqcup K_{2}$. We have that $|J_{2}|\geq|I\cap
J|+|J_{1}|-1\geq|K_{1}|\geq|K_{2}|$ and, in particular, $|J|>|K|$.

However,

$\begin{array}[]{lcl}d_{2}(\mathcal{C})&=&|I\cup J|=|I|+|J|-|I\cap J|\\\
&=&|I|+|J_{1}|+|J_{2}|\\\ &\geq&|I|+|J_{1}|+|K_{2}|\\\ &\geq&|I\cup
K|,\end{array}$

which cannot happen by the choice of $J={\rm supp}(\mathbf{m}_{2})$.
Therefore, $X^{I\cap J}X^{J_{1}}$ is a minimal generator of ${\rm
in}_{\prec}(I(\mathcal{C}))$.

Now, we will show that $X^{J_{2}}\notin{\rm in}_{\prec}(I(\mathcal{C}))$.
Since $X^{I\cap J}X^{J_{1}}$ is a minimal generator of ${\rm
in}_{\prec}(I(\mathcal{C}))$, then there exists $h=X^{I\cap
J}X^{J_{1}}-X^{L}\in\mathcal{G}_{\prec}$. We have that $X^{L}\notin{\rm
in}_{\prec}(I(\mathcal{C}))$ and let us see that $L=J_{2}$. Suppose that
$L\neq J_{2}$, then $0\neq h-g=x^{J_{2}}-x^{L}\in I(\mathcal{C})$ and
$X^{J_{2}}\succ X^{L}$, because $X^{L}\notin{\rm in}_{\prec}(I(\mathcal{C})).$
However,

$|I\cup{\rm supp}(h)|\leq|I|+|J_{1}|+|L|\leq|I|+|J_{1}|+|J_{2}|=|I\cup
J|=d_{2}(\mathcal{C}),$

which is a contradiction.

Case ${\rm LT}_{\prec}(g)=X^{J_{2}}$.

We are going to see that $-g\in\mathcal{G}_{\prec}$. First, we show that
$X^{J_{2}}$ is a minimal generator of ${\rm in}_{\prec}(I(\mathcal{C}))$.
Suppose that there exists a binomial
$h=X^{K_{1}}-X^{K_{2}}\in\mathcal{G}_{\prec}$ with $X^{K_{1}}\succ X^{K_{2}}$
and $K_{1}\subsetneq J_{2}$. Consider $L:=J\triangle K$, we have that
$L\subseteq(J-K_{1})\cup K_{2}$ and, hence, $X^{L}\prec
X^{J-K_{1}}X^{K_{2}}\prec X^{J}$. However,

$|I\cup L|\leq|I\cup(J-K_{1})\cup K_{2}|\leq|I\cup
J|-|K_{1}|+|K_{2}|\leq|I\cup J|=d_{2}(\mathcal{C}),$

and again it is a contradiction.

We will show now that $X^{I\cap
J}X^{J_{1}}\notin\mathrm{in}_{\prec}(I(\mathcal{C}))$, by contradiction.
Suppose that $X^{I\cap J}X^{J_{1}}\in\mathrm{in}_{\prec}(I(\mathcal{C}))$,
then there exists a binomial $X^{I\cap J}X^{J_{1}}-X^{L}$ in $I(\mathcal{C})$.
Consider now $K=(I\cap J)\cup J_{1}\cup L$, again one can compute that $|I\cup
J|\geq|I\cup K|$ and that $K\prec J$, which again contradicts the choice of
$\mathbf{m}_{2}$.

∎

From the above results the main theorems of this paper will follow.

###### Theorem 4.1

Let $\prec$ a degree compatible order in $\mathbb{K}[X]$, then there are $f,g$
in the reduced Gröbner basis $\mathcal{G}_{\prec}$ of $I(\mathcal{C})$ with
respect to $\prec$, such that $d_{2}=|{\rm supp}(f)\cup{\rm supp}(g)|$.

###### Example 6

Let $\mathcal{C}$ be the binary non-degenerate $[6,3]$-code with generator
matrix

$G=\left(\begin{array}[]{cccccccccccccccccccccc}1&0&0&1&1&0\\\ 0&1&0&1&0&1\\\
0&0&1&0&1&1\end{array}\right)\in\mathbb{F}_{2}^{\,3\times 6}$

One can check that every codeword different from $0$ is a codeword of minimal
support and, hence, there are $7$ codewords of minimal support, namely
$w_{1}=111000,w_{2}=110011,w_{3}=101101,w_{4}=100110,w_{5}=101110,w_{6}=010101,w_{7}=001011$.
Moreover, we have that $d_{1}(\mathcal{C})=3$ and $d_{2}(\mathcal{C})=5$ and
$d_{3}(\mathcal{C})=6$ and the set $\mathbf{M}$ described in (4) coincides
with $\mathcal{C}-\\{0\\}$.

Consider the degree reverse lexicographic order $\prec_{1}$ with
$x_{6}\prec_{1}\cdots\prec_{1}x_{1}$, then

  * •

$\mathbf{m}_{1}:=\min_{\prec_{1}}(\mathbf{M})=001011=w_{7}$, and

  * •

$\mathbf{m}_{2}:={\rm
min}_{\prec_{1}}\\{\mathbf{m}\in\mathbf{M}\,|\,d_{2}(\mathcal{C})=\mathrm{supp}(\left\langle\mathbf{m}_{1},\mathbf{m}\right)\rangle\\}=010101=w_{6}.$

The reduced Gröbner basis $\mathcal{G}_{1}$ of $I(\mathcal{C})$ with respect
to $\prec_{1}$ has 20 elements. As proved in Proposition 4, the binomial
$f=x_{5}x_{6}-x_{3}$ belongs to $G_{1}$ and has ${\rm supp}(f)={\rm
supp}(w_{7})=\\{3,5,6\\}$ and, by Proposition 5, the binomial
$g=x_{4}x_{6}-x_{2}$ belongs to $\mathcal{G}_{1}$ and has ${\rm supp}(g)={\rm
supp}(w_{6})=\\{2,4,6\\}$. Moreover, $d_{2}=|{\rm supp}(f)\cup{\rm
supp}(g)|=|{\rm supp}(\langle w_{6},w_{7}\rangle)|=|\\{2,3,4,5,6\\}|=5$.

If we consider the degree reverse lexicographic order $\prec_{2}$ with
$x_{1}\prec_{2}\cdots\prec_{2}x_{6}$, then

  * •

$\mathbf{m}_{1}^{\prime}:=\min_{\prec_{2}}(\mathbf{M})=111000=w_{1}$, and

  * •

$\mathbf{m}_{2}^{\prime}:={\rm
min}_{\prec_{2}}\\{\mathbf{m}\in\mathbf{M}\,|\,d_{2}(\mathcal{C})=\mathrm{supp}(\left\langle\mathbf{m}_{1}^{\prime},\mathbf{m}\right)\rangle\\}=100110=w_{4}.$

The reduced Gröbner basis $\mathcal{G}_{2}$ of $I(\mathcal{C})$ with respect
to $\prec_{2}$ has 20 elements. As proved in Proposition 4, the binomial
$f^{\prime}=x_{1}x_{2}-x_{3}$ belongs to $\mathcal{G}_{2}$ and has ${\rm
supp}(f)={\rm supp}(w_{1})=\\{1,2,3\\}$ and, as proved in Proposition 5, the
binomial $g^{\prime}=x_{1}x_{4}-x_{5}$ belongs to $\mathcal{G}_{2}$ and has
${\rm supp}(g)={\rm supp}(w_{4})=\\{1,4,5\\}$. Moreover, $d_{2}=|{\rm
supp}(f^{\prime})\cup{\rm supp}(g^{\prime})|=|{\rm supp}(\langle
w_{1},w_{4}\rangle)|=|\\{1,2,3,4,5\\}|=5$.

As a consequence, we have that the two first GHWs $d_{1}(\mathcal{C})$ and
$d_{2}(\mathcal{C})$ can be obtained from the minimal graded free resolution
associated with the supports in the $\mathcal{G}_{\prec}$-test set. Moreover,
from this resolution one can also obtain upper bounds for all the
$d_{i}(\mathcal{C})$ where $i\in\left[k(\mathcal{C})\right]$. More precisely,
we have the following.

###### Theorem 4.2

Let $\mathcal{C}\subseteq\mathbb{F}_{2}^{n}$ be a binary code, $\prec$ a
degree compatible monomial order in $R$. Let $T$ denote the
$\mathcal{G}_{\prec}$-test, define the square-free monomial ideal

$M:=\langle\\{X^{\mathbf{c}}\,|\,\mathbf{c}\in T\\}\rangle\subseteq R,$

and consider a minimal graded free resolution of $R/M$:

$0\longrightarrow F_{p}\longrightarrow\cdots\longrightarrow
F_{2}{\longrightarrow}F_{1}\longrightarrow R\longrightarrow R/M\longrightarrow
0,$

where each $F_{i}$ is a graded free $R$-module of the form

$F_{i}=\bigoplus_{j\in\mathbb{N}}R(-j)^{\beta_{i,j}(R/M)}\hbox{ for
}i\in\left[p\right]_{0}.$

Then,

  * (a)

$p\leq k(\mathcal{C})$

  * (b)

$d_{i}(\mathcal{C})\leq{\rm min}\\{j\,|\,\beta_{i,j}(R/M)\neq 0\\}$ for all
$j\in\\{3,\ldots,p\\}$, and

  * (c)

$d_{i}(\mathcal{C})={\rm min}\\{j\,|\,\beta_{i,j}(R/M)\neq 0\\},$ for $i=1,2$.

###### Proof

By Proposition 3, every $\mathbf{c}\in T$ is a codeword of minimal support
and, hence, $\\{X^{\mathbf{c}}\,|\,\mathbf{c}\in T\\}$ is a subset of the
generators of the monomial ideal supported on all the codewords of minimal
support. Thus, (a) and (b) follow from Theorem 2.1.

The rest of the proof concerns (c). Again by Proposition 3,
$\\{X^{\mathbf{c}}\,|\,\mathbf{c}\in T\\}$ is the minimal monomial generating
set of $M$. Since $\beta_{1,i}(R/M)$ equals the number of minimal generators
of $M$ of degree $i$, then, $\beta_{1,i}(R/M)=|\\{\mathbf{c}\in
T\,|\,\mathrm{w}_{H}(c)=i\\}.|$ By Proposition 3, there is a $\mathbf{c}\in T$
such that $\mathrm{w}_{H}(\mathbf{c})=d_{1}$ and, thus,

$d_{1}(\mathcal{C})={\rm min}\\{\mathrm{w}_{H}(\mathbf{c})\,|\,\mathbf{c}\in
T\\}={\rm min}\\{i\,|\,\beta_{1,i}(R/M)\neq 0\\}.$

Assume now that $T=\\{\mathbf{c}_{1},\ldots,\mathbf{c}_{r}\\},$ and consider
$\mathcal{T}$ the Taylor resolution of $M=\langle
X^{\mathbf{c}_{1}},\ldots,X^{\mathbf{c}_{r}}\rangle$. The first steps of this
resolution are given by

$\mathcal{T}:\cdots\longrightarrow
F_{2}^{\prime}\overset{\varphi_{2}}{\longrightarrow}F_{1}^{\prime}\longrightarrow
R\longrightarrow R/M\longrightarrow 0,$

where $F_{i}^{\prime}:=\oplus_{|I|=i\atop I\subset[r]}R(-|{\rm
supp}\langle\mathbf{c}_{j}\,|\,j\in I\rangle|)$ and
$F_{1}^{\prime}:=\oplus_{1\leq i\leq r}R(-|{\rm supp}(\mathbf{c}_{i})|).$
Hence, the shifts in the second step of $\mathcal{T}$ are given by $|{\rm
supp}\langle\mathbf{c}_{i},\mathbf{c}_{j}\rangle|$ for $1\leq i<j\leq r$ and,
as a consequence, $d_{2}(\mathcal{C})\leq{\rm min}\\{|{\rm
supp}\langle\mathbf{c}_{i},\mathbf{c}_{j}\rangle|\,|\,1\leq i<j\leq r\\}.$
Moreover, by Theorem 4.1, this is indeed an inequality.

In general, the Taylor resolution is not minimal (it is usually very far from
minimal). However, it can be pruned to get a minimal one. Consider now

$\mathcal{F}:\cdots\longrightarrow F_{2}\longrightarrow F_{1}\longrightarrow
R\longrightarrow R/M\longrightarrow 0,$

a minimal graded free resolution of $R/M$ obtained after pruning the Taylor
one. As we said before, $\\{X^{\mathbf{c}_{1}},\ldots,X^{\mathbf{c}_{r}}\\}$
is the minimal monomial generating set of $M$ and, hence,
$F_{1}^{\prime}=F_{1}$. As a consequence, ${\rm
min}\\{i\,|\,\beta_{2,i}(R/M)\neq 0\\}={\rm min}\\{|{\rm
supp}\langle\mathbf{c}_{i},\mathbf{c}_{j}\rangle|\,|\,1\leq i<j\leq
r\\}=d_{2}(\mathcal{C}).$ ∎

##  5 Final remarks, future work & new conjectures

In this work, we build on the results of Borges-Quintana et al. [5] and
propose $\mathcal{G}_{\prec}$-test sets as a smaller structure from where one
can obtain the values of $d_{1}(\mathcal{C})$ and $d_{2}(\mathcal{C})$ for
binary codes. Several experiments with SageMath [31] suggest that Theorem 4.2
can also be extended for $i=3$. More precisely:

###### Question 1

Let $\mathcal{C}\subseteq\mathbb{F}_{2}^{n}$ be a binary code, $\prec$ a
degree compatible monomial order in $R$. Consider the
$\mathcal{G}_{\prec}$-test set $T$ and define the square-free monomial ideal

$M:=\langle\\{X^{\mathbf{c}}\,|\,\mathbf{c}\in T\\}\rangle\subseteq R.$

Is $d_{3}(\mathcal{C})={\rm min}\\{i\,|\,\beta_{3,i}(R/M)\neq 0\\}$?

In Example 5 one has that the projective dimension (i.e., the number of steps
of the resolution) of $R/M$ is ${\rm pd}(R/M)=6$, while the dimension of
$\mathcal{C}$ is $k(\mathcal{C})=7$. In all the counterexamples to the
original conjecture that we have found it turns out that ${\rm
pd}(R/M)<k(\mathcal{C})$. This motivates us to ask if the conjecture holds
provided that ${\rm pd}(R/M)=k(\mathcal{C})$. More precisely:

###### Question 2

Whenever ${\rm pd}(R/M)=k(\mathcal{C})$, is it true that

$d_{i}(\mathcal{C})={\rm min}\\{j\,|\,\beta_{i,j}(R/M)\neq 0\\}$

for all $i\in\\{1,\ldots,k(\mathcal{C})\\}$?

Also, the following natural questions arise.

###### Question 3

What is in between the test set and the complete set of codewords of minimal
support? i.e. Can we characterize a mid-way structure that provides the
complete set of GHWs?

A possible candidate fot that intermediate set could be the union of all
$\mathcal{G}_{\prec}$-test sets for all $\prec$ degree compatible orderings.
In general, this set can be smaller than the whole set of codewords of minimal
support and can be computed by the algorithm proposed in [6]. For example, for
the $[7,4]$ binary Hamming code, i.e., the code with generator matrix

$\left(\begin{matrix}1&0&0&0&0&1&1\\\ 0&1&0&0&1&0&1\\\ 0&0&1&0&1&1&0\\\
0&0&0&1&1&1&1\\\ \end{matrix}\right)\in\mathbb{F}_{2}^{\,4\times 7},$

has $14$ codewords of minimal support, half of them with weight $3$, and the
rest with weight $4$. Moreover, the sequence of GHW is
$(d_{1}(\mathcal{C}),\ldots,d_{4}(\mathcal{C}))=(3,5,6,7)$. One has that all
the $\mathcal{G}_{\prec}$-test sets when $\prec$ ranges over all degree
compatible orderings consists of the $7$ codewords of Hamming weight $3$. If
one computes the Betti diagram of the monomial ideal
$M\subset\mathbb{F}_{2}[x_{1},\ldots,x_{7}]$ corresponding to this set, one
gets the following

$\begin{array}[]{c|ccccccc}&0&1&2&3&4\\\ \hline\cr 0&1&0&0&0&0\\\
1&0&0&0&0&0\\\ 2&0&7&0&0&0\\\ 3&0&0&21&21&6\\\ \end{array}$

Hence, the sequence $({\rm min}\\{j\,|\,\beta_{i,j}\neq 0\\};1\leq i\leq
4)=(3,5,6,7)$ coincides with the sequence of GHWs of the code $\mathcal{C}$.

###### Question 4

Can we say something in the non-binary case?

In order to answer this question, one could try to apply the techniques in
[26], where a generalization of the ideal $I(\mathcal{C})$ for non-binary
codes is studied.

###### Question 5

Recently, E. Gorla and A. Ravagnani in [13] extend and generalize the results
of Johnsen and Verdure [17] to compute the generalized weights of a code with
respect to a different notions of weight. It would be interesting to see if
these generalized weights can be computed from a $\mathcal{G}_{\prec}$-test
set of $\mathcal{C}$.

## Acknowledgments

We would like to thank E. Gorla (University of Neuchatel, Switzerland) for her
valuable comments and suggestions.

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# Phase transition of holographic entanglement entropy in massive gravity

Xiao-Xiong Zeng1,2, Hongbao Zhang3,4, Li-Fang Li5
1School of Material Science and Engineering, Chongqing Jiaotong University,
Chongqing,
400074, China
2Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical
Physics,
Chinese Academy of Sciences, Beijing 100190, China
3 Department of Physics, Beijing Normal University, Beijing, 100875, China
4 Theoretische Natuurkunde, Vrije Universiteit Brussel, and The International
Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium
5State Key Laboratory of Space Weather, National Space Science Center,
Chinese Academy of Sciences, Beijing 100190, China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract:

The phase structure of holographic entanglement entropy is studied in massive
gravity for the quantum systems with finite and infinite volumes, which in the
bulk is dual to calculate the minimal surface area for a black hole and black
brane respectively. In the entanglement entropy$-$temperature plane, we find
for both the black hole and black brane there is a Van der Waals-like phase
transition as the case in thermal entropy$-$temperature plane. That is, there
is a first order phase transition for the small charge and a second order
phase transition at the critical charge. For the first order phase transition,
the equal area law is checked and for the second order phase transition, the
critical exponent of the heat capacity is obtained. All the results show that
the phase structure of holographic entanglement entropy is the same as that of
thermal entropy regardless of the volume of the spacetime on the boundary.

##  1 Introduction

As any ordinary thermodynamic system, black holes have entropy as well as
temperature, and obey the thermodynamical laws[1, 2, 3]. But different from
the thermal entropy for the ordinary system, which is proportional to the
volume, the entropy of a black hole is proportional to its area of horizon.
Among others, AdS/CFT correspondence offers us a natural explanation for such
a seemingly bizarre behavior because it maps a gravitational system to an
ordinary system on the boundary with one less dimension[4, 5, 6]. Moreover, in
the context of AdS/CFT, not only can the area of black hole horizon be
understood as the thermal entropy for its dual boundary system, but also the
area of the bulk co-dimension $2$ minimal surface anchored onto the entangling
surface on the boundary can be used to calculate the corresponding
entanglement entropy for the boundary system[7].

Since the entanglement entropy acquires a similar geometric description to the
thermal entropy in the bulk by holography, it is of interest to see whether
the entanglement entropy demonstrates the similar behavior as the thermal
entropy. Especially in the recent work [8], the authors showed that like the
black hole entropy, the entanglement entropy also undergoes a Van der Waals-
like phase transition at the same critical temperature in both the fixed
charge ensemble and chemical potential ensemble. They also found the second
order phase transition occurring for the entanglement entropy at the same
critical point as the thermal entropy with nearly the same value as that from
thermal phase transition.

In light of these interesting results, this work was then generalized to the
extended phase space where the cosmological constant is considered as a
thermodynamical variable, and it was found that the entanglement entropy has
the similar phase structure as that of the black hole entropy too[9]. Very
recently, Nguyen has investigated exclusively the equal area law of
holographic entanglement entropy and his result showed that, as black hole
entropy, the equal area law also holds for the entanglement entropy regardless
of the size of the entangling surface[10]. All of these works reinforce the
viewpoint that thermal phase transition can also been captured by the
entanglement entropy somehow.

In this paper, we intend to study the phase transition of holographic
entanglement entropy in massive gravity. The main motivation of this paper is
to explore whether the thermal phase transition in an infinite volume can also
be described by holographic entanglement entropy, which has not been reported
in the literature until now. Massive gravity, where the graviton is endowed
with mass, is one of the gravity theory that attends to modify general
relativity for it is not UV complete. An important black hole solution that is
ghost-free in massive gravity has been presented recently in [11]. It was
found that the graviton plays the same role as the lattice in the holographic
conductor model. In light of this interesting property, Ref.[12] generalized
this solution to more general cases and studied the corresponding thermal
phase transition in the grand canonical ensemble and canonical ensemble. The
$P-V$ criticality was also investigated in the extended phase space[13]. The
result shows that the phase transition depends not only on the curvature
parameter $k$ but also the graviton mass $m$. In particular, even for the case
$k=0$, the Van der Waals-like phase transition may also occur provided a
proper parameter related to the graviton mass is given. This interesting
character is the main motivation for us to study phase transition of
holographic entanglement entropy in this spacetime because it provides a
background with infinite volume on the boundary in which the Van der Waals-
like phase transition can take place.

The organization of this paper is as follows. In the next section, after a
brief review of the black hole solution in massive gravity, we will study the
thermal phase transitions and critical phenomena for a black hole and black
brane in the fixed charge ensemble,respectively. In both spacetimes, the equal
area laws are checked and the critical exponents are calculated. In Section 3,
we will concentrate mainly on studying the phase transition of holographic
entanglement entropy in the entanglement entropy$-$temperature plane. For both
the black hole and black brane, we find the equal area law is valid and the
critical exponent of the heat capacity in the neighborhood of the critical
point is the same as that obtained from the thermal phase transition. The last
section is devoted to our conclusions and discussions.

##  2 Black hole solutions in massive gravity and thermal phase transition

The action for an $(n+2)$-dimensional massive gravity is given by[11]

$S=\frac{1}{16\pi G_{n+2}}\int
d^{n+2}x\sqrt{-g}[R+\frac{n(n+1)}{l^{2}}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+m^{2}\sum^{4}_{i}c_{i}{\cal
U}_{i}(g,f)].$ (1)

Here $G_{n+2}$ is the Newton’s gravitational constant, which will be set to
$1$ later on. In addition, $m$ is the graviton mass, $l$ is the AdS radius,
$f$ is the reference metric. Here the reference metric is given by
$f_{\mu\nu}dx^{\mu}dx^{\nu}=h_{ij}dx^{i}dx^{j}$, where $h_{ij}dx^{i}dx^{j}$ is
the line element for a 2-dimensional Einstein space with constant curvature
2$k$, $k=-1,0,1$ corresponds to a sphere, flat, or hyperbolic horizon for the
spacetime, individually.
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, $c_{i}$ are
constants and ${\cal U}_{i}$ are symmetric polynomials of the eigenvalues of
the $(n+2)\times(n+2)$ matrix ${\cal K}^{\mu}_{\
\nu}\equiv\sqrt{g^{\mu\alpha}f_{\alpha\nu}}$, i.e.,

$\displaystyle{\cal U}_{1}=[{\cal K}],$ $\displaystyle{\cal U}_{2}=[{\cal
K}]^{2}-[{\cal K}^{2}],$ $\displaystyle{\cal U}_{3}=[{\cal K}]^{3}-3[{\cal
K}][{\cal K}^{2}]+2[{\cal K}^{3}],$ $\displaystyle{\cal U}_{4}=[{\cal
K}]^{4}-6[{\cal K}^{2}][{\cal K}]^{2}+8[{\cal K}^{3}][{\cal K}]+3[{\cal
K}^{2}]^{2}-6[{\cal K}^{4}],$ (2)

in which $[{\cal K}]={\cal K}^{\mu}_{\mu}$. Starting from the action in (1),
Ref.[12] has recently got the solution of black holes in such a massive
gravity as

$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}h_{ij}dx^{i}dx^{j},\ \ i,j=1,\ 2,\
3,\cdots,n$ (3)

with a chemical potential

$\mu=\frac{Q}{(n-1)r_{+}^{n-1}},$ (4)

in which

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
k+\frac{r^{2}}{l^{2}}-\frac{M}{r^{n-1}}+\frac{Q^{2}}{2n(n-1)r^{2(n-1)}}+\frac{c_{0}c_{1}m^{2}}{n}r+c_{0}^{2}c_{2}m^{2}$
(5)
$\displaystyle+\frac{(n-1)c_{0}^{3}c_{3}m^{2}}{r}+\frac{(n-1)(n-2)c_{0}^{4}c_{4}m^{2}}{r^{2}},$

$r_{+}$ is the event horizon which satisfies $f(r_{+})=0$, $M$ is the black
hole mass, and $Q$ is the black hole charge. In this paper, we only consider
the four$-$dimensional spacetime, namely $c_{3}=c_{4}=0$ as shown in [12]. For
convenience, we will relabel the remanent graviton mass dependent parameters,
$c_{0}c_{1}m^{2}/{2}$, $c_{0}^{2}c_{2}m^{2}$, as $a$, $b$ respectively later
in this paper.

The thermal phase structure of the spacetime in (3) has been investigated
extensively[12, 13]. In this paper, we are interested in the Van der Waals-
like phase transition111This behavior can be observed in both the $T-S$ plane
in general phase space and $P-V$ plane in extended phase space, where the
cosmological constant is viewed as a dynamical pressure of the black hole
system. It has been pointed out that these two perspectives are related to
each other by a duality similar to the T-duality of string theory[14].. To
compare the thermal entropy phase transition with entanglement entropy phase
transition directly, we focus only on the phase transition taking place at the
$T-S$ plane. In particular we will discuss the phase structure of the thermal
entropy and entanglement entropy for both the black hole and black brane,
namely $k=1$ and $k=0$ in (5)222In principle the case of $k=-1$ can also be
analysed, but in practice it is hard to choose a proper entangling surface to
implement the calculation..

###  2.1 The case of $k=1$

In this case, the metric in (5) can be simplified as

$f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{4r^{2}}+\frac{r^{2}}{l^{2}}+ar+b.$ (6)

The temperature and entropy of the black hole are

$T_{bh}=\frac{f(r)^{\prime}}{4\pi}\mid_{r_{+}}=\frac{12r_{+}^{4}+l^{2}\left(-Q^{2}+4r_{+}^{2}(1+b+2ar_{+})\right)}{16l^{2}\pi
r_{+}^{3}},$ (7) $S=\pi r_{+}^{2}.$ (8)

Substituting (8) into (7) and eliminating the parameter $r_{+}$, we can get
the relation between the temperature $T_{bh}$ and entropy $S$, namely

$T_{bh}=\frac{12S^{2}+l^{2}\left(-\pi^{2}Q^{2}+4(1+b)\pi
S+8a\sqrt{\pi}S^{3/2}\right)}{16l^{2}\pi^{3/2}S^{3/2}}.$ (9)

Based on this relation, one can plot the isocharge curves on the $T_{bh}-S$
plane for a given charge. The plot is presented in Figure 1, which is
obviously similar to that of the Van der Waals phase transition. As can be
seen from this plot, for different charges, the curves behave differently. For
the small charge in (a), we can see that the curve is not monotonic. According
to the definition of the specific heat capacity

(a) $Q=1.5/3$ (b) $Q=2/3$ (c) $Q=2.5/3$

Figure 1: Relation between the entropy and temperature for a black hole in a
fixed charge ensemble. The red dotted lines in $(a)$ and $(b)$ correspond to
the locations of first order phase transition and second order phase
transition.

$C_{Q}=T_{bh}\frac{\partial S}{\partial T_{bh}}\mid_{Q},$ (10)

we know that the specific heat capacity changes from positive to negative and
then to positive again. That is, for an intermediate range of event horizon
$r_{+}$ the black hole is thermodynamically unstable. In other words, there
are three holes competing thermodynamically. The smallest hole continues to
win until a temperature $T_{\star}$, above which the system jumps to the large
hole. As stressed in [15], this kind of transition is a first order phase
transition. When the charge increases to the critical charge, the smallest
hole and the largest hole merge into one and squeeze out the unstable black
hole. So there is an inflection point in the $T_{bh}-S$ plane, which is shown
in (b) of Figure 1. One can see that the specific heat capacity is divergent
at the inflection point and it has been shown that this phase transition is a
second order phase transition.

Figure 2: Relation between the free energy and temperature for a black hole at
$Q=1.5/3$. The intersection point between the red dotted line and horizontal
coordinate is the temperature of first order phase transition.

At the inflection point, the critical charge and critical entropy are
determined by the following equations

$(\frac{\partial T_{bh}}{\partial S})_{Q}=(\frac{\partial^{2}T_{bh}}{\partial
S^{2}})_{Q}=0.$ (11)

Substituting (9) into (11), we find the critical charge, critical entropy and
critical temperature

$Q_{c}=\frac{l(1+b)}{3},$ (12) $S_{c}=\frac{l^{2}(1+b)\pi}{6},$ (13)
$T_{c}=\frac{4+4b+\sqrt{6}a\sqrt{(1+b)l^{2}}}{2\sqrt{6}\sqrt{(1+b)l^{2}}\pi}.$
(14)

Obviously these critical parameters depend on the parameters associated with
the graviton mass333In this paper, we are not interested in the effect of the
graviton mass on the critical behavior, for more information about this topic
please see [12, 13]. . As $a$ and $b$ vanish, these values are the same as
that of the AdS-RN black hole. As $b>-1$, we can observe that there is always
a Van der Waals-like phase transition. In this paper, we choose $a=b=1$, and
for convenience, we also set the AdS radius $l=1$.

For the first order phase transition, we will check whether Maxwell’s equal
area law holds in this background. The first order transition temperature
$T_{\star}$ plays an important role on the equal area law. To find
$T_{\star}$, we first plot the curve about the free energy $F=M-TS$[12] with
respect to the temperature $T$. The plot is shown in Figure 2. We see that
there is a swallowtail structure, which corresponds to the unstable phase in
(a) of Figure (1). The non-smoothness of the junction indicates that the phase
transition is a first order. The critical temperature $T_{\star}$ is
apparently given by the horizontal coordinate of the junction. From Figure 2,
we find $T_{\star}=0.549$. Substituting this temperature into (9), we can get
three values of the entropy $S_{1}=0.210442$, $S_{2}=0.977147$,
$S_{3}=2.93091$. With these values, we can now check Maxwell’s equal area law,
which states

$T_{\star}(S_{3}-S_{1})=\int_{S_{1}}^{S_{3}}T_{bh}(S,Q)dS.$ (15)

After simplying the whole calculation, we find both the left and right of (15)
equal to 1.49354 exactly. Thus, Maxwell’s equal area law is verified in this
background.

For the second order phase transition, we are interested in the critical
exponent associated with the heat capacity defined in (10). Near the critical
point, we write the entropy as $S=S_{c}+\delta$, where $S_{c}$ is defined in
(13). Expanding the temperature in small $\delta$, we find

$\displaystyle T_{bh}=\frac{12S_{c}^{2}+l^{2}\left(-\pi^{2}Q^{2}+4(1+b)\pi
S_{c}+8a\sqrt{\pi}S_{c}^{3/2}\right)}{16l^{2}\pi^{3/2}S_{c}^{3/2}}+\frac{\left(12S_{c}^{2}+l^{2}\pi\left(3\pi
Q^{2}-4(1+b)S_{c}\right)\right)\delta}{32l^{2}\pi^{3/2}S_{c}^{5/2}}$
$\displaystyle+\frac{3\left(-4S_{c}^{2}+l^{2}\pi\left(-5\pi
Q^{2}+4(1+b)S_{c}\right)\right)\delta^{2}}{128l^{2}\pi^{3/2}S_{c}^{7/2}}+\frac{\left(12S_{c}^{2}+5l^{2}\pi\left(7\pi
Q^{2}-4(1+b)S_{c}\right)\right)\delta^{3}}{256l^{2}\pi^{3/2}S_{c}^{9/2}}.$

Near the critical point, the first term on the right is the critical
temperature, the second and third terms vanish according to (11). So we get

$\displaystyle T_{bh}-T_{c}=\frac{\left(12S_{c}^{2}+5l^{2}\pi\left(7\pi
Q^{2}-4(1+b)S_{c}\right)\right)}{256l^{2}\pi^{3/2}S_{c}^{9/2}}(S-S_{c})^{3}.$
(17)

With the definition of the heat capacity, we further find
$C_{q}\sim(T_{bh}-T_{c})^{-2/3}$, namely the critical exponent is $-2/3$,
which is the same as the one from the mean field theory.

###  2.2 The case of $k=0$

For the case $k=0$, the spacetime we are considered is a black brane. The
Hawking temperature is

$T_{bb}=-\frac{Q^{2}-4r_{+}^{2}(b+r_{+}(2a+3r_{+}))}{16\pi r_{+}^{3}}.$ (18)

Replacing the event horizon $r_{+}$ by the entropy $S$, (18) can be rewritten
as

$T_{bb}=\frac{-\pi^{2}Q^{2}+4b\pi
S+8a\sqrt{\pi}S^{3/2}+12S^{2}}{16\pi^{3/2}S^{3/2}},$ (19)

whereby, we can also observe the Van der Waals-like phase transition, which is
presented in Figure 3.

(a) $Q=0.8/3$ (b) $Q=1/3$ (c) $Q=1.2/3$

Figure 3: Relation between the entropy and temperature for a black brane in a
fixed charge ensemble. The red dotted lines in $(a)$ and $(b)$ correspond to
the locations of first order phase transition and second order phase
transition.

Based on (11), we can also get the critical charge, critical entropy and
critical temperature

$Q_{c}=b/3,$ (20) $S_{c}=\frac{b\pi}{6},$ (21)
$T_{c}=\frac{3a+2\sqrt{6}\sqrt{b}}{6\pi}.$ (22)

It is obvious that these critical values depend on the parameters associated
with the graviton mass.

Figure 4: Relation between the free energy and temperature for a black brane
at $Q=0.8/3$. The intersection point between the red dotted line and
horizontal ordinate is the temperature of first order phase transition.

As $a$ and $b$ vanish, the Van der Waals-like phase transition disappears.
That is, in Einstein gravity, this is no Van der Waals-like phase transition
for an AdS-RN black brane.

To check Maxwell’s equal area law, we should find the first order phase
transition temperature $T_{\star}$ from the $F-T_{bb}$ plot, which is shown in
Figure 4. Reading from the horizontal coordinate of the junction, we find
$T_{\star}=0.4316$. Substituting this temperature into (19), we get
$S_{1}=0.131235$, $S_{3}=1.32697$, where $S_{1}$, $S_{3}$ are the smallest and
largest roots of equation $T_{bb}=T_{\star}$. Substituting these values to
(15), we find the left equals 0.5161 and the right equals 0.5162. Obviously
both the left and the right are equal nearly within our numerical accuracy,
which indicates that Maxwell’s equal area law is satisfied in this background.

Similarly we can also get the critical exponent for the second phase
transition in (b) of Figure 3. Writing the entropy as $S=S_{c}+\delta$ and
expanding the temperature in small $\delta$, we find

$\displaystyle T_{bb}-T_{c}=\frac{\left(35\pi^{2}Q^{2}-20b\pi
S_{c}+12S_{c}^{2}\right)}{256\pi^{3/2}S_{c}^{9/2}}(S-S_{c})^{3},$ (23)

where we have used (11). This relation is similar to that of the black hole.
That is to say, the critical exponent is $-2/3$ too.

##  3 Holographic entanglement entropy and its phase structure

Having understood the phase structure of a black hole and black brane in
massive gravity from the viewpoint of thermodynamics, we will study
entanglement entropy in both spacetimes. The main motivation is to check
whether entanglement entropy has the similar phase structure as that of the
thermal entropy.

For a given quantum field theory described by a density matrix $\rho$,
entanglement entropy for a region $A$ and its complement $B$ is defined as

$\displaystyle S_{A}=-Tr_{A}(\rho_{A}\ln\rho_{A}),$ (24)

where $\rho_{A}=Tr_{B}(\rho)$ is the reduced density matrix. Usually it is not
easy to calculate this quantity in field theory. But from the viewpoint of
holography, Ref.[7] gave a very simple geometric description for computing
$S_{A}$ for static states in terms of the area of a bulk minimal surface
anchored on $\partial A$, which states that

$\displaystyle S_{A}=\frac{\text{Area}(\gamma)}{4},$ (25)

where $\gamma$ is the codimension-2 minimal surface with boundary condition
$\partial\gamma=\partial A$. Here we will still use (25) to calculate
entanglement entropy and study the corresponding phase structure for
preliminary study.

###  3.1 Phase transition of Entanglement entropy in a finite volume system

For a black hole, the space on the boundary is a round. The volume of the
space is finite in this case. To avoid the entanglement entropy to be affected
by the surface that wraps the horizon, we will choose a small region as $A$.
To be more precise, as done in [15, 16], we choose the region $A$ to be a
spherical cap on the boundary delimited by $\theta\leq\theta_{0}$. In this
case the area can be written as

$\displaystyle
A=2\pi\int_{0}^{\theta_{0}}\mathcal{L}(r(\theta),\theta)d\theta,\mathcal{L}=r\sin\theta\sqrt{\frac{(r^{\prime})^{2}}{f(r)}+r^{2}},$
(26)

in which $r^{\prime}=dr/d\theta$. Imagining $\theta$ as time, and treating
$\mathcal{L}$ as the Lagrangian, one can get the equation of motion for
$r(\theta)$ by making use of the Euler-Lagrange equation. It seems to be
impossible to get the analytical solution of $r(\theta)$, so we will solve it
numerically with the boundary conditions $r^{\prime}(0)=0$, $r(0)=r_{0}$. With
the numerical result of $r(\theta)$, we can easily obtain the area and further
entanglement entropy. Note that the area is divergent, so it should be
regularized by subtracting off the area in pure AdS with the same entangling
surface on the boundary.

(a) $Q=1.5/3$

(b) $Q=2/3$

(c) $Q=2.5/3$

Figure 5: Relation between the entanglement entropy and temperature for a
black hole in a fixed charge ensemble. The boundary is located at
$\theta_{0}=0.1$. The red dotted lines in $(a)$ and $(b)$ correspond to the
locations of first order phase transition and second order phase transition.

(a) $Q=1.5/3$

(b) $Q=2/3$

(c) $Q=2.5/3$

Figure 6: Relation between the entanglement entropy and temperature for a
black black hole in a fixed charge ensemble. The boundary is located at
$\theta_{0}=0.16$. The red dotted lines in $(a)$ and $(b)$ correspond to the
locations of first order phase transition and second order phase transition.

In Einstein gravity, one can often get an analytical solution of the minimal
surface in the pure AdS. In our massive gravity, we find that the spacetime
for the pure AdS is given by444Note that in this case, the space is not
strictly a pure AdS, but for convenience we also call it pure AdS.

$\displaystyle
dS^{2}=-(1+\frac{r^{2}}{l^{2}}+ar+b)dt^{2}+\frac{1}{(1+\frac{r^{2}}{l^{2}}+ar+b)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (27)

It is obvious that the parameters related to the graviton mass affect the
structure of the pure AdS. The analytical result for the pure AdS in Einstein
gravity thus can not be used and we obtain the corresponding minimal surface
for the pure AdS in our massive gravity by numeric strategy too. We label the
regularized entanglement entropy as $\delta S$. For the numerical computation,
we choose $\theta_{0}=0.1$ and $\theta_{0}=0.16$. The UV cutoff in the dual
field theory is set to be $r(0.099)$ and $r(0.159)$ by holography
respectively. To compare with the phase transition of thermal entropy, we will
study the relation between the entanglement entropy and Hawking temperature,
which is regarded as the temperature of the dual field theory. The numeric
results for $\theta_{0}=0.1$ and $\theta_{0}=0.16$ are shown in Figure 5 and
Figure 6 respectively. We can see that for a given charge, the relation
between the entanglement entropy and temperature is similar to that between
the black hole entropy and temperature. That is, there is a critical charge
and below it the first order phase transition occurs. In this case there is an
unstable hole interpolating between a small and a large stable hole. As the
temperature grows to the critical temperature $T_{\star}$, the small black
hole jumps to the large black hole. As the charge grows to the critical
charge, the small hole and the large hole merge so that the unstable hole
shrinks into an inflection point where the second order phase transition
occurs. For a large enough charge, a large stable black hole forms and the
entanglement entropy grows monotonically as the temperature increases.
Comparing Figure 5 with Figure 6, we find the profile for different sizes of
entangling surface seems to be the same, which indicates that the phase
structure of entanglement entropy does not depend on the size of entangling
surface.

For the first order phase transition of the entanglement entropy, we now check
whether Maxwell’s equal area law holds. For the case $\theta_{0}=0.1$, we
first construct an Interpolating Function of the temperature $T(\delta S,Q)$
using the data obtained numerically555Though the entanglement entropy
resembles the black hole entropy, they have different function dependence of
temperature.. At the first order phase transition point, we find the smallest
and largest roots for the equation $T=T_{\star}$ are $\delta
S_{1}=0.000136338$, $\delta S_{3}=0.00128997$. Substituting these values into
the equal area law

$\displaystyle T_{\star}(\delta S_{3}-\delta S_{1})=\int_{\delta
S_{1}}^{\delta S_{3}}T(\delta S,Q)d\delta S,$ (28)

we find the left equals 0.0006333 and the right equals 0.0006337. Obviously
both the left and the right are equal within our numerical accuracy.
Similarly, for the case $\theta_{0}=0.16$, we find the left equals 0.002625
and the right equals 0.002627, which also indicates that the equal area law is
satisfied.

Now let’s turn to the critical exponent of the second order phase transition
of the entanglement entropy. Similar to the definition of specific heat
capacity in thermodynamics, we also define an analogous specific heat capacity
for entanglement entropy

$\displaystyle\mathcal{C}_{Q}=T_{bh}\frac{\partial\delta S}{\partial
T_{bh}}\mid_{Q}.$ (29)

Provided a similar relation as that in (17) is satisfied, then with (29) we
can get the critical exponent of phase transition of entanglement entropy.
Here we are interested in the logarithm of the quantities $T_{bh}-T_{c}$,
$\delta S-\delta S_{c}$, in which $T_{c}$ is the critical temperature in (14)
and $\delta S_{c}$ is the critical entropy obtained numerically.

Figure 7: Plot of $\log\mid T_{bh}-T_{c}\mid$ versus $\log\mid\delta S-\delta
S_{c}\mid$ based on 45 points closest to the inflection point in Figure 5
$(b)$.

Taking the case $\theta=0.1$ as an example, the linear relation is shown in
Figure 7 and the analytical relation can be fitted as

$\displaystyle\log\mid T_{bh}-T_{c}\mid=15.0607+2.983\log\mid\delta S-\delta
S_{c}\mid.$ (30)

Obviously the slope is about 3, therefore the critical exponent for the second
order phase transition of entanglement entropy agrees with that of the thermal
entropy.

###  3.2 Phase transition of Entanglement entropy in an infinite volume system

As have seen in Section 2.2, for a black brane in massive gravity, there
exists a Van der Waals-like phase transition, which does’t exist in Einstein
gravity. So we will study entanglement entropy in this spacetime to check
whether the phase structure of entanglement entropy resembles that of the
thermal entropy.

To study the entanglement entropy, we should choose a proper region for $A$.
Here we choose a rectangular strip parametrized by the boundary coordinates
$(x_{1},x_{2})$ with the assumption that this infinite strip is invariant
under the $x_{2}$-direction. In order to make the notation as simple as
possible, we would like to rename this exceptional coordinate $x_{1}$ as $x$
and employ it to parameterize the minimal area surface such that the
entanglement entropy in this background can be written as

$\displaystyle
A=L\int_{-\ell/2}^{\ell/2}r\sqrt{\frac{(r^{\prime})^{2}}{f(r)}+r^{2}}dx,$ (31)

in which $L$ is length of the minimal surface along the $x_{2}$-direction
which is set to be 1 here, $\ell$ is the boundary separation connected by two
boundary points: $(t_{0},-\ell/2)$, $(t_{0},\ell/2)$, $r^{\prime}=dr/dx$ and

$f(r)=-\frac{2M}{r}+\frac{Q^{2}}{4r^{2}}+\frac{r^{2}}{l^{2}}+ar+b.$ (32)

The integrand in (31) can be thought as the Lagrangian $\cal{L}$ of a
fictitious system with $x$ the proper time. Since the Lagrangian does not
depend explicitly on $x$, there is an associated conserved quantity, which is
useful for us to get the following equation of motion of $r(x)$

$4r(x)^{2}f(r)^{2}-2f(r)r^{\prime}(x)^{2}+r(x)r^{\prime}(x)^{2}f^{\prime}(r)-2f(r)rr^{\prime\prime}(x)=0.$
(33)

To solve $r(x)$, we use the boundary conditions

$r(0)=r_{0},r^{\prime}(0)=0.$ (34)

(a) $Q=0.8/3$ (b) $Q=1/3$ (c) $Q=1.2/3$

Figure 8: Relation between the entanglement entropy and temperature for a
black black brane in a fixed charge ensemble. The boundary separation is
$\ell=1.2$. The red dotted lines in $(a)$ and $(b)$ correspond to the
locations of first order phase transition and second order phase transition.

(a) $Q=0.8/3$

(b) $Q=1/3$

(c) $Q=1.2/3$

Figure 9: Relation between the entanglement entropy and temperature for a
black black brane in a fixed charge ensemble. The boundary separation is
$\ell=2.2$. The red dotted lines in $(a)$ and $(b)$ correspond to the
locations of first order phase transition and second order phase transition.

Substituting the numerical result of $r(x)$ into (31), we can get the area and
further the entanglement entropy. We are also interested in the regularized
entanglement entropy $\delta S\equiv S-S_{0}$, where $S_{0}$ is entanglement
entropy in the pure AdS space of our massive gravity, which is not amenable to
analytical calculation but can be obtained by our numerical strategy. In what
follows, we are interested in the cases $\ell=1.2$, $\ell=2.2$. The UV cutoff
is set to be $r(1.19)$ and $r(2.19)$. The numerical results are shown in
Figure 8 and Figure 9. We can see that the behavior of the curves in Figure 8
or Figure 9 is reminiscent of that in Figure 3. Namely in the $T_{bb}-\delta
S$ phase space, the phase structure of entanglement entropy depends on the
charge of the black brane. As the charge grows from small to large enough, the
first order phase transition and second order phase transition occur
successively, and a stable black brane forms lastly.

We can also check whether Maxwell’s equal area law holds for the first order
phase transition in (a) of Figure 8 and Figure 9. For the case $\ell=1.2$, the
smallest and largest roots for the equation $T(\delta S,Q)=T_{\star}$ are
$\delta S_{1}=0.167805$, $\delta S_{3}=0.403158$, where $T(\delta S,Q)$ is an
Interpolating Function and $T_{\star}$ is the critical temperature of the
first order phase transition. Substituting these values into (28), we find the
left equals 0.10080 and the right equals 0.10076. Similarly, for the case
$\ell=2.2$, the left and right of (28) are found to be $0.20568$ and 0.20565
respectively. It is obvious that the equal area law is satisfied and its
validity is independent of the boundary separations.

We can also get the critical exponent of the heat capacity in the neighborhood
of the critical point for the second order phase transition, which is shown in
(b) of Figure 8 or Figure 9. Taking the case $\ell=1.2$ as an example.
According to the numeric result of the entanglement entropy$-$ temperature
curve, we first construct an Interpolating Function $T(\delta S,Q)$. At the
critical temperature, we obtain the critical entropy $\delta S_{c}$ by solving
the equation $T(\delta S,Q)=T_{c}$, where $T_{c}$ is defined in (18). The
relation between $\log\mid T_{bb}-T_{c}\mid$ and $\log\mid\delta S-\delta
S_{c}\mid$ is plotted in Figure 8.

Figure 10: Plot of $\log\mid T_{bh}-T_{c}\mid$ versus $\log\mid\delta S-\delta
S_{c}\mid$ based on 31 points closest to the inflection point in Figure 8
$(b)$.

This curve can be fitted as

$\displaystyle\log\mid T_{bh}-T_{c}\mid=6.29083+3.01454\log\mid\delta S-\delta
S_{c}\mid.$ (35)

It is obvious that the slope equals to 3 nearly. In other words, for
entanglement entropy in quantum system with infinite volume, the critical
exponent of the heat capacity also agrees with that of the thermal entropy.

##  4 Conclusions

In this paper, we have studied the phase structure of holographic entanglement
entropy in massive gravity. Our result shows that the phase structure of
holographic entanglement entropy is the same as that of the thermal entropy
regardless of the volume of a spacetime on the boundary. So one is suggested
to use the behavior of holographic entanglement entropy to identify the
thermal phase transition. But maybe more importantly, one should be encouraged
by these accumulated evidences to find a general proof for such a coincidence
between the holographic entanglement entropy and thermal entropy.

## Acknowledgements

Xiao-Xiong Zeng would like to thank Professor Rong-Gen Cai for his suggestions
and encouragement. He is also grateful to Phuc H. Nguyen, Kai Lin, and
Shanquan Lan for their valuable discussions on numerics. Hongbao Zhang is
grateful to Minyong Guo for his helpful discussion. He is supported in part by
the Belgian Federal Science Policy Office through the Interuniversity
Attraction Pole P7/37, by FWO-Vlaanderen through the project G020714N, and by
the Vrije Universiteit Brussel through the Strategic Research Program ”High-
Energy Physics”. He is also an individual FWO fellow supported by 12G3515N.
This work is also supported by the National Natural Science Foundation of
China (Grant No. 11365008, Grant No. 11205226, Grant No. 11575270), and
Natural Science Foundation of Education Committee of Chongqing (Grant No.
KJ1500530).

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