# Construction and Linearity of $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-Linear
Generalized Hadamard Codes ††thanks: This work has been partially supported by
the Spanish MINECO under Grant PID2019-104664GB-I00 (AEI /
10.13039/501100011033) and by Catalan AGAUR scholarship 2020 FI SDUR 00475.
††thanks: The material in this paper was presented in part at the 12th
International Workshop on Coding and Cryptography (WCC 2022), 7-11 March 2022
- Virtual.

Dipak K. Bhunia, Cristina Fernández-Córdoba, Mercè Villanueva The authors are
with the Department of Information and Communications Engineering, Universitat
Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Spain.

###### Abstract

The $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive codes are subgroups of
$\mathbb{Z}_{p}^{\alpha_{1}}\times\mathbb{Z}_{p^{2}}^{\alpha_{2}}$, and can be
seen as linear codes over $\mathbb{Z}_{p}$ when $\alpha_{2}=0$,
$\mathbb{Z}_{p^{2}}$-additive codes when $\alpha_{1}=0$, or
$\mathbb{Z}_{2}\mathbb{Z}_{4}$-additive codes when $p=2$. A
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear generalized Hadamard (GH) code is a
GH code over $\mathbb{Z}_{p}$ which is the Gray map image of a
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive code. In this paper, we generalize
some known results for $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes with
$p=2$ to any $p\geq 3$ prime when $\alpha_{1}\neq 0$. First, we give a
recursive construction of $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive GH codes
of type $(\alpha_{1},\alpha_{2};t_{1},t_{2})$ with $t_{1},t_{2}\geq 1$. Then,
we show for which types the corresponding
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes are non-linear over
$\mathbb{Z}_{p}$. Finally, according to some computational results, we see
that, unlike $\mathbb{Z}_{4}$-linear GH codes, when $p\geq 3$ prime, the
$\mathbb{Z}_{p^{2}}$-linear GH codes are not included in the family of
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes with $\alpha_{1}\not=0$.

##  1 Introduction

Let $\mathbb{Z}_{p}$ and $\mathbb{Z}_{p^{2}}$ be the ring of integers modulo
$p$ and $p^{2}$, respectively, where $p$ is a prime. Let $\mathbb{Z}_{p}^{n}$
and $\mathbb{Z}_{p^{2}}^{n}$ denote the set of all $n$-tuples over
$\mathbb{Z}_{p}$ and $\mathbb{Z}_{p^{2}}$, respectively. In this paper, the
elements of $\mathbb{Z}_{p}^{n}$ and $\mathbb{Z}^{n}_{p^{2}}$ will be called
vectors of length $n$. The order of a vector $\mathbf{u}$ over
$\mathbb{Z}_{p^{2}}$, denoted by $o(\mathbf{u})$, is the smallest positive
integer $m$ such that $m\mathbf{u}=(0,\ldots,0)$.

A code over $\mathbb{Z}_{p}$ of length $n$ is a nonempty subset of
$\mathbb{Z}_{p}^{n}$, and it is linear if it is a subspace of
$\mathbb{Z}_{p}^{n}$. Similarly, a nonempty subset of $\mathbb{Z}_{p^{2}}^{n}$
is a $\mathbb{Z}_{p^{2}}$-additive if it is a subgroup of
$\mathbb{Z}_{p^{2}}^{n}$. A $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive code
is a subgroup of
$\mathbb{Z}_{p}^{\alpha_{1}}\times\mathbb{Z}_{p^{2}}^{\alpha_{2}}$. Note that
a $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive code is a linear code over
$\mathbb{Z}_{p}$ when $\alpha_{2}=0$, a $\mathbb{Z}_{p^{2}}$-additive code
when $\alpha_{1}=0$, or a $\mathbb{Z}_{2}\mathbb{Z}_{4}$-additive code when
$p=2$.

The Hamming weight of a vector $\textbf{u}\in\mathbb{Z}_{p}^{n}$, denoted by
${\rm wt}_{H}(\textbf{u})$, is the number of nonzero coordinates of u. The
Hamming distance of two vectors $\textbf{u},\textbf{v}\in\mathbb{Z}_{p}^{n}$,
denoted by $d_{H}(\textbf{u},\textbf{v})$, is the number of coordinates in
which they differ. Note that $d_{H}(\textbf{u},\textbf{v})={\rm
wt}_{H}(\textbf{u}-\textbf{v})$. The minimum distance of a code $C$ over
$\mathbb{Z}_{p}$ is
$d(C)=\min\\{d_{H}(\textbf{u},\textbf{v}):\textbf{u},\textbf{v}\in
C,\textbf{u}\not=\textbf{v}\\}$.

In [1], a Gray map from $\mathbb{Z}_{4}$ to $\mathbb{Z}_{2}^{2}$ is defined as
$\phi(0)=(0,0)$, $\phi(1)=(0,1)$, $\phi(2)=(1,1)$ and $\phi(3)=(1,0)$. There
exist different generalizations of this Gray map, which go from
$\mathbb{Z}_{2^{s}}$ to $\mathbb{Z}_{2}^{2^{s-1}}$ [2, 3, 4, 5, 6].

The one given in [5] can be defined in terms of the elements of a Hadamard
code [6], and Carlet’s Gray map [2] is a particular case of the one given in
[6] satisfying $\sum\lambda_{i}\phi(2^{i})=\phi(\sum\lambda_{i}2^{i})$ [7]. In
this paper, we focus on a generalization of Carlet’s Gray map, from
$\mathbb{Z}_{p^{s}}$ to $\mathbb{Z}_{p}^{p^{s-1}}$, which is also a particular
case of the one given in [8]. Specifically,

$\begin{split}\phi:&\ \mathbb{Z}_{p^{2}}\longrightarrow\mathbb{Z}_{p}^{p}\\\
&\ u\mapsto(u_{0},u_{1})M,\end{split}$ (1)

where $u\in\mathbb{Z}_{p^{2}}$; $[u_{0},u_{1}]_{p}$ is the $p$-ary expansion
of $u$, that is $u=u_{0}+u_{1}p$ with $u_{0},u_{1}\in\mathbb{Z}_{p}$; and $M$
is the following matrix of size $2\times p$:

$\left(\begin{array}[]{ccccc}0&1&2&\cdots&p-1\\\ 1&1&1&\cdots&1\\\
\end{array}\right).$

Let
$\Phi:\mathbb{Z}_{p}^{\alpha_{1}}\times\mathbb{Z}_{p^{2}}^{\alpha_{2}}\rightarrow\mathbb{Z}_{p}^{n}$,
where $n=\alpha_{1}+p\alpha_{2}$, be an extension of the Gray map $\phi$ given
by

$\Phi(\mathbf{x}\mid\mathbf{y})=(\mathbf{x},\phi(y_{1}),\dots,\phi(y_{\alpha_{2}})),$

for any $\mathbf{x}\in\mathbb{Z}_{p}^{\alpha_{1}}$ and
$\mathbf{y}=(y_{1},\dots,y_{\alpha_{2}})\in\mathbb{Z}_{p^{2}}^{\alpha_{2}}$.

Let ${\cal C}$ be a $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive code over
$\mathbb{Z}_{p}^{\alpha_{1}}\times\mathbb{Z}_{p^{2}}^{\alpha_{2}}$. We say
that its Gray map image $C=\Phi({\cal C})$ is a
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear code of length
$\alpha_{1}+p\alpha_{2}$. Since ${\cal C}$ is isomorphic to a subgroup of
$\mathbb{Z}_{p^{2}}^{\alpha_{1}+\alpha_{2}}$, it is also isomorphic to an
abelian structure $\mathbb{Z}_{p^{2}}^{t_{1}}\times\mathbb{Z}_{p}^{t_{2}}$,
and we say that ${\cal C}$, or equivalently $C=\Phi({\cal C})$, is of type
$(\alpha_{1},\alpha_{2};t_{1},t_{2})$. Note that $|{\cal
C}|=p^{2t_{1}+t_{2}}$. Unlike linear codes over finite fields, linear codes
over rings do not have a basis, but there exists a generator matrix for these
codes having minimum number of rows, that is, $t_{1}+t_{2}$ rows.

A generalized Hadamard $(GH)$ matrix $H(p,\lambda)=(h_{ij})$ of order
$n=p\lambda$ over $\mathbb{Z}_{p}$ is a $p\lambda\times p\lambda$ matrix with
entries from $\mathbb{Z}_{p}$ with the property that for every $i,j$, $1\leq
i<j\leq p\lambda,$ each of the multisets $\\{h_{is}-h_{js}:1\leq s\leq
p\lambda\\}$ contains every element of $\mathbb{Z}_{p}$ exactly $\lambda$
times [9]. An ordinary Hadamard matrix of order $4\mu$ corresponds to a $GH$
matrix $H(2,\lambda)$ over $\mathbb{Z}_{2}$, where $\lambda=2\mu$ [10]. Two
$GH$ matrices $H_{1}$ and $H_{2}$ of order $n$ are said to be equivalent if
one can be obtained from the other by a permutation of the rows and columns
and adding the same element of $\mathbb{Z}_{p}$ to all the coordinates in a
row or in a column.

We can always change the first row and column of a $GH$ matrix into zeros and
we obtain an equivalent $GH$ matrix which is called normalized. From a
normalized GH matrix $H$, we denote by $F_{H}$ the code consisting of the rows
of $H$, and $C_{H}=\bigcup_{\alpha\in\mathbb{Z}_{p}}(F_{H}+\alpha\textbf{1})$,
where $F_{H}+\alpha\textbf{1}=\\{\textbf{h}+\alpha\textbf{1}:\textbf{h}\in
F_{H}\\}$ and 1 denotes the all-one vector. The code $C_{H}$ over
$\mathbb{Z}_{p}$ is called generalized Hadamard $(GH)$ code [11]. Note that
$C_{H}$ is generally a non-linear code over $\mathbb{Z}_{p}$. Moreover, if it
is of length $N$, it has $pN$ codewords and minimum distance $N(p-1)/p$.

The $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive codes such that after the Gray
map $\Phi$ give GH codes are called
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive GH codes and the corresponding
images are called $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes. The
classification of $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear GH codes of length
$2^{t}$ with $\alpha_{1}=0$ and $\alpha_{1}\not=0$ is given in [12, 13],
showing that there are $\lfloor(t-1)/2\rfloor$ and $\lfloor t/2\rfloor$ such
non-equivalent codes, respectively.

Moreover, in [14], it is shown that each $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear
GH code with $\alpha_{1}=0$ is equivalent to a
$\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear GH code with $\alpha_{1}\not=0$, so
indeed there are only $\lfloor t/2\rfloor$ non-equivalent
$\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear GH codes of length $2^{t}$. Later, in
[7, 15, 16, 17], an iterative construction for $\mathbb{Z}_{p^{s}}$-linear GH
codes is described, the linearity is established, and a partial classification
is obtained, giving the exact amount of non-equivalent non-linear such codes
for some parameters.

This paper is focused on $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes
with $\alpha_{1}\not=0$ and $p\geq 3$ prime, generalizing some results given
for $p=2$ in [13, 18] related to the construction and linearity of such codes.
For $p=3$ and $2\leq t\leq 8$, these codes are compared with the
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes of length $p^{t}$ with
$\alpha_{1}=0$ studied in [15]. This paper is organized as follows. In Section
2, we recall the definition of the Gray map considered in this paper and some
of its properties. In Section 3, we describe the construction of
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes of type
$(\alpha_{1},\alpha_{2};t_{1},t_{2})$ with $\alpha_{1}\not=0$. In Section 5,
we establish for which types these codes are linear. Finally, in Section 6, we
show some computational results for $p=3$ and $p=5$, which point out that,
unlike $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear GH codes, when $p\geq 3$ prime,
the $\mathbb{Z}_{p^{2}}$-linear GH codes are not included in the family of
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes with $\alpha_{1}\not=0$.
Moreover, we also observe that they are not equivalent to any of the
$\mathbb{Z}_{p^{s}}$-linear GH codes considered in [7, 15] by using the same
Gray map.

##  2 Preliminary results

In this section, we give the definition of the Gray map considered in this
paper for elements of $\mathbb{Z}_{p^{2}}$. We also include some of its
properties used in the paper.

We consider the following Gray map $\phi$, given in [2, 19], for $s=2$:

$\begin{split}\phi:&\ \mathbb{Z}_{p^{2}}\longrightarrow\mathbb{Z}_{p}^{p}\\\
&\ u\mapsto(u_{0},u_{1})M,\end{split}$ (2)

where $u\in\mathbb{Z}_{p^{2}}$; $[u_{0},u_{1}]_{p}$ is the $p$-ary expansion
of $u$, that is, $u=u_{0}+u_{1}p$ with $u_{0},u_{1}\in\mathbb{Z}_{p}$; and $M$
is the following matrix of size $2\times p$:

$\left(\begin{array}[]{ccccc}0&1&2&\cdots&p-1\\\ 1&1&1&\cdots&1\\\
\end{array}\right).$

Let $u,v\in\mathbb{Z}_{p^{2}}$ and $[u_{0},u_{1}]_{p}$, $[v_{0},v_{1}]_{p}$ be
the $p$-ary expansions of $u$ and $v$, respectively, i.e. $u=u_{0}+u_{1}p$ and
$v=v_{0}+v_{1}p$. We define the operation “$\odot_{p}$” between elements in
$\mathbb{Z}_{p^{2}}$ as $u\odot_{p}v=t_{0}+t_{1}p$, where

$t_{i}=\left\\{\begin{array}[]{ll}1,&\textrm{if }\quad u_{i}+v_{i}\geq p,\\\
0,&\textrm{otherwise}.\\\ \end{array}\right.$

Note that the $p$-ary expansion of $u\odot_{p}v$ is $[t_{0},t_{1}]_{p}$, where
$t_{0},t_{1}\in\\{0,1\\}$. We denote in the same way, “$\odot_{p}$”, the
component-wise operation. For $\mathbf{u}=(u\mid u^{\prime}),\mathbf{v}=(v\mid
v^{\prime})\in\mathbb{Z}_{p}^{\alpha_{1}}\times\mathbb{Z}_{p^{2}}^{\alpha_{2}}$,
we denote $\mathbf{u}\odot_{p}\mathbf{v}=({\mathbf{0}}\mid
u^{\prime}\odot_{p}v^{\prime})$.

From [15], we have the following results:

###### Lemma 2.1

[15] Let $u\in\mathbb{Z}_{p^{2}}$ and $\lambda\in\mathbb{Z}_{p}$. Then,
$\phi(u+\lambda p)=\phi(u)+(\lambda,\lambda,\dots,\lambda)$.

###### Corollary 2.1

[15] Let $\lambda,\mu\in\mathbb{Z}_{p}$. Then, $\phi(\lambda\mu
p)=\lambda\phi(\mu p)=\lambda\mu\phi(p)$.

###### Corollary 2.2

[15] Let $u,v$ $\in\mathbb{Z}_{p^{2}}$. Then,
$\phi(u)+\phi(v)=\phi(u+v-p(u\odot_{p}v))$.

###### Corollary 2.3

[15] Let $u,v$ $\in\mathbb{Z}_{p^{2}}$. Then, $\phi(pu+v)=\phi(pu)+\phi(v)$.

###### Corollary 2.4

[15] For $u,v$ $\in\mathbb{Z}_{p^{2}}$,
$\phi(u+v)=\phi(u)+\phi(v)+(t_{0},t_{0},\dots,t_{0})$, where $t_{0}=1$ if
$u_{0}+v_{0}\geq p$ and $0$ otherwise.

###### Proposition 2.1

[15] Let $u,v\in\mathbb{Z}_{p^{2}}$ be two distinct elements. Then,
$\phi(u)-\phi(v)=\phi(u-v)=(\lambda,\ldots,\lambda)$ if $u-v=\lambda p\in
p\mathbb{Z}_{p^{2}}$, and $\phi(u)-\phi(v)$ contains every element of
$\mathbb{Z}_{p}$ exactly once if $u-v\in\mathbb{Z}_{p^{2}}\setminus
p\mathbb{Z}_{p^{2}}$.

###### Proposition 2.2

[15] Let $u,v\in\mathbb{Z}_{p^{2}}$. Then, $d_{H}(\phi(u),\phi(v))={\rm
wt}_{H}(\phi(u-v))$.

From [20], the homogeneous weight of an element $u\in\mathbb{Z}_{p^{2}}$ is
defined by

${\rm wt}^{*}(u)=\left\\{\begin{array}[]{ll}0&\textrm{if}\quad u=0,\\\
p&\textrm{if}\quad u\in p\mathbb{Z}_{p^{2}}\setminus\\{0\\},\\\
p-1&\textrm{otherwise},\\\ \end{array}\right.$ (3)

and the homogeneous weight of a vector
$\mathbf{u}=(u_{1},\dots,u_{n})\in\mathbb{Z}_{p^{2}}^{n}$ is ${\rm
wt}^{*}(\mathbf{u})=\sum_{i=1}^{n}{\rm wt}^{*}(u_{i})$. The corresponding
homogeneous distance of $\mathbf{u}=(u_{1},\dots,u_{n})$ and
$\mathbf{v}=(v_{1},\dots,v_{n})\in\mathbb{Z}_{p^{2}}^{n}$ is defined as
follows:

$d^{*}(\mathbf{u},\mathbf{v})=\sum_{i=1}^{n}{\rm wt}^{*}(u_{i}-v_{i}).$ (4)

The Gray map $\Phi$ over $\mathbb{Z}_{p^{2}}^{n}$ is an isometry which
transforms homogeneous distances defined in $\mathbb{Z}_{p^{2}}^{n}$ to
Hamming distances defined in $\mathbb{Z}_{p}^{np}$ [19].

Then, we define the homogeneous weight of $\mathbf{u}=(u\mid
u^{\prime})=(u_{1},\dots,u_{\alpha_{1}}\mid
u^{\prime}_{1},\dots,u^{\prime}_{\alpha_{2}})\in\mathbb{Z}_{p}^{\alpha_{1}}\times\mathbb{Z}_{p^{2}}^{\alpha_{2}}$
as ${\rm wt}^{*}(\mathbf{u})={\rm wt}_{H}(u)+{\rm wt}^{*}(u^{\prime})$. From
(4), the corresponding homogeneous distance of $\mathbf{u}=(u\mid
u^{\prime})=(u_{1},\dots,u_{\alpha_{1}}\mid
u^{\prime}_{1},\dots,u^{\prime}_{\alpha_{2}})$ and $\mathbf{v}=(v\mid
v^{\prime})=(v_{1},\dots,v_{\alpha_{1}}\mid
v^{\prime}_{1},\dots,v^{\prime}_{\alpha_{2}})\in\mathbb{Z}_{p}^{\alpha_{1}}\times\mathbb{Z}_{p^{2}}^{\alpha_{2}}$
is defined as follows:

$d^{*}(\mathbf{u},\mathbf{v})={\rm wt}_{H}(u-v)+{\rm
wt}^{*}(u^{\prime}-v^{\prime}).$ (5)

Note that the extension of the Gray map $\Phi$ over
$\mathbb{Z}_{p}^{\alpha_{1}}\times\mathbb{Z}_{p^{2}}^{\alpha_{2}}$ is also an
isometry by using this homogeneous metric, that is,
$d^{*}(\mathbf{u},\mathbf{v})=d_{H}(\Phi(\mathbf{u}),\Phi(\mathbf{v}))$ for
all
$\mathbf{u},\mathbf{v}\in\mathbb{Z}_{p}^{\alpha_{1}}\times\mathbb{Z}_{p^{2}}^{\alpha_{2}}$.
Moreover, the $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear codes obtained from
this Gray map $\Phi$ are distance invariant by Proposition 2.2.

##  3 Construction of $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive GH codes

The description of a generator matrix having minimum number of rows for
$\mathbb{Z}_{2}\mathbb{Z}_{4}$-additive GH codes with $\alpha_{1}\not=0$, as
long as an iterative construction of these matrices, are given in [13, 18]. In
this section, we generalize these results for
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive GH codes with $\alpha_{1}\not=0$
and any $p\geq 3$ prime. Specifically, we define an iterative construction for
the generator matrices and establish that they generate
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive GH codes. The proof that the codes
are GH is completely different from the binary case.

Let $\mathbf{0},\mathbf{1},\mathbf{2},\ldots,\mathbf{p^{2}-1}$ be the vectors
having the elements $0,1,2,\ldots,p^{2}-1$ repeated in each coordinate,
respectively. Let

$A_{p}^{1,1}=\left(\begin{array}[]{cccc|ccccc}1&1&\cdots&1&p&p&\cdots&p\\\
0&1&\cdots&p-1&1&2&\cdots&p-1\\\ \end{array}\right).$ (6)

Any matrix $A_{p}^{t_{1},t_{2}}$ with $t_{1}\geq 1,t_{2}\geq 2$ or $t_{1}\geq
2,t_{2}\geq 1$ can be obtained by applying the following iterative
construction. First, if $A$ is a generator matrix of a
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive code, that is, a subgroup of
$\mathbb{Z}_{p}^{\alpha_{1}}\times\mathbb{Z}_{p^{2}}^{\alpha_{2}}$, then we
denote by $A_{1}$ the submatrix of $A$ with the first $\alpha_{1}$ columns
over $\mathbb{Z}_{p}$, and $A_{2}$ the submatrix with the last $\alpha_{2}$
columns over $\mathbb{Z}_{p^{2}}$. We start with $A_{p}^{1,1}$. Then, if we
have a matrix $A=A_{p}^{t_{1},t_{2}}$, we may construct the matrices

$A_{p}^{t_{1},t_{2}+1}=\left(\begin{array}[]{cccc|cccc}A_{1}&A_{1}&\cdots&A_{1}&A_{2}&A_{2}&\cdots&A_{2}\\\
\mathbf{0}&\mathbf{1}&\cdots&\mathbf{p-1}&p\cdot\mathbf{0}&p\cdot\mathbf{1}&\cdots&p\cdot\mathbf{(p-1)}\\\
\end{array}\right)$ (7)

and

$\footnotesize
A_{p}^{t_{1}+1,t_{2}}=\left(\begin{array}[]{cccc|ccccccc}A_{1}&A_{1}&\cdots&A_{1}&pA_{1}&\cdots&pA_{1}&A_{2}&A_{2}&\cdots&A_{2}\\\
\mathbf{0}&\mathbf{1}&\cdots&\mathbf{p-1}&\mathbf{1}&\cdots&\mathbf{p-1}&\mathbf{0}&\mathbf{1}&\cdots&\mathbf{p^{2}-1}\\\
\end{array}\right).$ (8)

###### Example 3.1

Let

$A_{3}^{1,1}=\left(\begin{array}[]{ccc|cc}1&1&1&3&3\\\ 0&1&2&1&2\\\
\end{array}\right)$

be the matrix described in (6) for $p=3$. By using the constructions described
in (7) and (8), we obtain $A_{3}^{1,2}$ and $A_{3}^{2,1}$, respectively, as
follows:

$A_{3}^{1,2}=\left(\begin{array}[]{ccccccccc|cccccc}1&1&1&1&1&1&1&1&1&3&3&3&3&3&3\\\
0&1&2&0&1&2&0&1&2&1&2&1&2&1&2\\\ 0&0&0&1&1&1&2&2&2&0&0&3&3&6&6\\\
\end{array}\right)$ $\footnotesize
A_{3}^{2,1}=\left(\begin{array}[]{ccccccccc|ccccccccccccc}1&1&1&1&1&1&1&1&1&3&3&3&3&3&3&3&3&3&3&\cdots&3&3\\\
0&1&2&0&1&2&0&1&2&0&3&6&0&3&6&1&2&1&2&\cdots&1&2\\\
0&0&0&1&1&1&2&2&2&1&1&1&2&2&2&0&0&1&1&\cdots&8&8\\\ \end{array}\right).$

In this paper, we consider that the matrices $A_{p}^{t_{1},t_{2}}$ are
constructed recursively starting from $A_{p}^{1,1}$ in the following way.
First, we add $t_{1}-1$ rows of order $p^{2}$, up to obtain $A_{p}^{t_{1},1}$;
and then we add $t_{2}-1$ rows of order $p$ up to achieve
$A_{p}^{t_{1},t_{2}}$. Note that in the first row there is always the row
$({\mathbf{1}}\mid{\mathbf{p}})$.

The $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive code generated by
$A_{p}^{t_{1},t_{2}}$ is denoted by ${{\cal H}}_{p}^{t_{1},t_{2}}$, and the
corresponding $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear code $\Phi({{\cal
H}}_{p}^{t_{1},t_{2}})$ by $H_{p}^{t_{1},t_{2}}$. We also write
$A^{t_{1},t_{2}}$, ${{\cal H}}^{t_{1},t_{2}}$, and $H^{t_{1},t_{2}}$ instead
of $A_{p}^{t_{1},t_{2}}$, ${{\cal H}}_{p}^{t_{1},t_{2}}$, and
$H_{p}^{t_{1},t_{2}}$, respectively, when the value of $p$ is clear by the
context.

###### Proposition 3.1

Let $t_{1},t_{2}\geq 1$ and $p$ prime. Then, ${{\cal H}}_{p}^{t_{1},t_{2}}$ is
a $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive code of type

$(p^{t_{1}+t_{2}-1},(p-1)\displaystyle\sum_{i=1}^{t_{1}}p^{t_{1}+t_{2}+i-3};t_{1},t_{2}).$
(9)

###### Theorem 3.1

The $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive code ${{\cal H}}_{p}^{1,1}$
generated by the matrix

$A_{p}^{1,1}=\left(\begin{array}[]{cccc|ccccc}1&1&\cdots&1&p&p&\cdots&p\\\
0&1&\cdots&p-1&1&2&\cdots&p-1\\\ \end{array}\right)$

is a $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive GH code of type
$(p,p-1;1,1)$.

###### Example 3.2

The $\mathbb{Z}_{3}\mathbb{Z}_{9}$-additive code ${{\cal H}}_{3}^{1,1}$
generated by the matrix $A_{3}^{1,1}$, given in Example 3.1, is a
$\mathbb{Z}_{3}\mathbb{Z}_{9}$-additive GH code of type $(3,2;1,1)$. Indeed,
we have that $H_{3}^{1,1}=\Phi({{\cal
H}}_{3}^{1,1})=\bigcup_{\lambda\in\mathbb{Z}_{3}}(\Phi(A_{0})+\lambda\textbf{1})$,
where $A_{0}=\\{\lambda(0,1,2\mid 1,2):\lambda\in\mathbb{Z}_{9}\\}$, and then
$\Phi(A_{0})$ consists of all the rows of the GH matrix

$H(3,3)=\left(\begin{array}[]{ccccccccc}0&0&0&0&0&0&0&0&0\\\
0&1&2&0&1&2&0&2&1\\\ 0&2&1&0&2&1&1&2&0\\\ 0&0&0&1&1&1&2&2&2\\\
0&1&2&1&2&0&2&1&0\\\ 0&2&1&1&0&2&0&1&2\\\ 0&0&0&2&2&2&1&1&1\\\
0&1&2&2&0&1&1&0&2\\\ 0&2&1&2&1&0&2&0&1\\\ \end{array}\right).$ (10)

The $\mathbb{Z}_{3}\mathbb{Z}_{9}$-linear code $H_{3}^{1,1}$ has length $N=9$,
$pN=3\cdot 9=27$ codewords and minimum distance $N(p-1)/p=9(3-1)/3=6$.

###### Theorem 3.2

The $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive code
$\mathcal{H}_{p}^{t_{1},t_{2}}$ generated by the matrix $A_{p}^{t_{1},t_{2}}$,
with $t_{1},t_{2}\geq 1$ and $p$ prime, is a
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive GH code.

###### Proposition 3.2

Let $\mathcal{H}_{p}^{t_{1},t_{2}}$ be a
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive GH code of type
$(\alpha_{1},\alpha_{2};t_{1},t_{2})$ with $t_{1},t_{2}\geq 1$ and $p$ prime.
Let $H_{p}^{t_{1},t_{2}}$ be the corresponding
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH code of length $p^{t}$, with
$t\geq 2$. Then, $\alpha_{1}=p^{t-t_{1}}$, $\alpha_{2}=p^{t-1}-p^{t-t_{1}-1}$
and $t=2t_{1}+t_{2}-1$.

###### Example 3.3

Let ${{\cal H}}_{3}^{1,2}$ be the $\mathbb{Z}_{3}\mathbb{Z}_{9}$-additive code
generated by the matrix $A_{3}^{1,2}$ given in Example 3.1. By Theorem 3.2,
$H_{3}^{1,2}=\Phi({{\cal H}}_{3}^{1,2})$ is a
$\mathbb{Z}_{3}\mathbb{Z}_{9}$-linear GH code of type $(9,6;1,2)$. Actually,
we can write
$H_{3}^{1,2}=\cup_{\lambda\in\mathbb{Z}_{3}}(F_{H}+\lambda\textbf{1})$, where
$F_{H}$ consists of all the rows of a GH matrix $H(3,9)$. Also, note that
$H_{3}^{1,2}$ has length $N=27$, $pN=3\cdot 27=81$ codewords and minimum
distance $N(p-1)/p=27(3-1)/3=18$.

###### Example 3.4

Let ${{\cal H}}_{3}^{2,1}$ be the $\mathbb{Z}_{3}\mathbb{Z}_{9}$-additive code
generated by the matrix $A_{3}^{2,1}$ given in Example 3.1. By Theorem 3.2,
$H_{3}^{2,1}=\Phi({{\cal H}}_{3}^{2,1})$ is a
$\mathbb{Z}_{3}\mathbb{Z}_{9}$-linear GH code of type $(9,24;2,1)$, which has
length $N=81$, $pN=3\cdot 81=243$ codewords and minimum distance
$N(p-1)/p=81(3-1)/3=54$.

Let $\mathcal{H}$ be a $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive code of
type $(\alpha_{1},\alpha_{2};t_{1},t_{2})$ with $p$ prime. Let
$\mathcal{H}_{1}$ (respectively, $\mathcal{H}_{2}$) be the punctured code of
$\mathcal{H}$ by deleting the last $\alpha_{2}$ coordinates over
$\mathbb{Z}_{p^{2}}$ (respectively, the first $\alpha_{1}$ coordinates over
$\mathbb{Z}_{p}$). Let $H_{2}=\Phi(\mathcal{H}_{2})$.

###### Remark 3.1

Let $\mathcal{H}=\mathcal{H}_{p}^{t_{1},t_{2}}$ be a
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive GH code of type
$(\alpha_{1},\alpha_{2};t_{1},t_{2})$ with $t_{1},t_{2}\geq 1$ and $p$ prime.
Let $H=\Phi(\mathcal{H}_{p}^{t_{1},t_{2}})$ be the corresponding
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH code of length
$\alpha_{1}+p\alpha_{2}$. Then, since $H$ is a GH code, its minimum distance
is

$\frac{(p-1)(\alpha_{1}+p\alpha_{2})}{p}.$

Note that, by construction, $\mathcal{H}_{1}$ is a GH code over
$\mathbb{Z}_{p}$ of length $\alpha_{1}$ and minimum distance
$(p-1)\alpha_{1}/p$. Therefore, $H_{2}=\Phi(\mathcal{H}_{2})$ has minimum
distance $(p-1)\alpha_{2}$.

###### Remark 3.2

Since the length of the $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH code
$\Phi(\mathcal{H}_{p}^{1,1})$ is $p^{2}$, its minimum distance is
$(p-1)p^{2}/p=p(p-1)$ by Remark 3.1.

###### Remark 3.3

The above constructions (7) and (8) give always
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes with $\alpha_{2}\not=0$
since the starting matrix $A_{p}^{1,1}$ has $\alpha_{2}\not=0$. If
$\alpha_{2}=0$, the $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes
coincide with the codes obtained from a Sylvester GH matrix, so they are
always linear and of type $(p^{t_{2}-1},0;0,t_{2})$ [11]. Therefore, we only
focus on the ones with $\alpha_{2}\not=0$ to study whether they are linear or
not.

##  4 Other equivalent $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes

In this section, we see that if we consider other starting matrices, instead
of the matrix $A_{p}^{1,1}$ given in (6), and apply the same recursive
constructions $(\ref{eq:ZpZp2recGenMatrix2})$ and
$(\ref{eq:ZpZp2recGenMatrix1})$, we also obtain
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive GH codes with $\alpha_{2}\not=0$.
Indeed, the corresponding $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes,
after applying the Gray map $\Phi$, are permutation equivalent to the codes
$\Phi(\mathcal{H}_{p}^{t_{1},t_{2}})$ constructed in Section 3.

Let ${\cal S}_{n}$ be the symmetric group of permutations on the set
$\\{1,\dots,n\\}$. A permutation $\pi\in{\cal S}_{n}$ acts linearly on vectors
$(c_{1},\ldots,c_{n})\in\mathbb{Z}_{p}^{n}$ by permuting their coordinates as
follows: $\pi(c_{1},\ldots,c_{n})=(c_{\pi^{-1}(1)},\ldots,c_{\pi^{-1}(n)})$.
Given two permutations $\pi_{1}\in{\cal S}_{n}$ and $\pi_{2}\in{\cal S}_{m}$,
we define $(\pi_{1}|\pi_{2})\in{\cal S}_{n+m}$, where $\pi_{1}$ acts on the
coordinates $\\{1,\dots,n\\}$ and $\pi_{2}$ on $\\{n+1,\dots,n+m\\}$.

Two codes $C_{1}$ and $C_{2}$ over $\mathbb{Z}_{p}$ of length $n$ are said to
be monomially equivalent (or just equivalent) provided there is a monomial
matrix $M$ such that $C_{2}=\\{\textbf{c}M:\textbf{c}\in C_{1}\\}$. Recall
that a monomial matrix is a square matrix with exactly one nonzero entry in
each row and column. They are said to be permutation equivalent if there is a
permutation matrix $P$ such that $C_{2}=\\{\textbf{c}P:\textbf{c}\in
C_{1}\\}$. Recall that a permutation matrix is a square matrix with exactly
one 1 in each row and column and $0$s elsewhere. A permutation matrix
represents a permutation of coordinates, so we can also say that they are
permutation equivalent if there is a permutation of coordinates $\pi\in{\cal
S}_{n}$ such that $C_{2}=\\{\pi(\textbf{c}):\textbf{c}\in C_{1}\\}$.

We denote by $N_{p}$ the set $\\{0,1,\dots,p-1\\}\subset\mathbb{Z}_{p^{2}}$
and $N_{p}^{-}=N_{p}\setminus\\{0\\}$. When including all the elements in
those sets as coordinates of a vector, we place them in increasing order. For
example, $N_{3}=\\{0,1,2\\}\subset\mathbb{Z}_{9}$,
$N_{3}^{-}=\\{1,2\\}\subset\mathbb{Z}_{9}$.

###### Proposition 4.1

Let $a=(a_{1},\ldots,a_{p-1})\in\mathbb{Z}_{p^{2}}^{p-1}$ such that
$\\{pa_{1},pa_{2},$ $\dots,$
$pa_{p-1}\\}=p\mathbb{Z}_{p^{2}}\setminus\\{0\\}$. Consider the matrix

$A_{p,a}^{1,1}=\left(\begin{array}[]{cccc|ccccc}1&1&\cdots&1&p&p&\cdots&p\\\
0&1&\cdots&p-1&a_{1}&a_{2}&\cdots&a_{p-1}\\\ \end{array}\right).$ (11)

The code generated by $A_{p,a}^{1,1}$, denoted by $\mathcal{H}_{p,a}^{1,1}$,
is a $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive GH code of type
$(p,p-1;1,1)$.

Moreover, the corresponding $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear code
after the Gray map is permutation equivalent to
$\Phi(\mathcal{H}_{p}^{1,1})=\Phi(\mathcal{H}_{p,N_{p}^{-}}^{1,1})$.

###### Theorem 4.1

Let $a=(a_{1},\ldots,a_{p-1})\in\mathbb{Z}_{p^{2}}^{p-1}$ such that
$\\{pa_{1},pa_{2},$ $\dots,$
$pa_{p-1}\\}=p\mathbb{Z}_{p^{2}}\setminus\\{0\\}$. Let $A_{p,a}^{t_{1},t_{2}}$
be the matrix obtained by using constructions $(\ref{eq:ZpZp2recGenMatrix2})$
and $(\ref{eq:ZpZp2recGenMatrix1})$, starting with the following matrix

$A_{p,a}^{1,1}=\left(\begin{array}[]{cccc|ccccc}1&1&\cdots&1&p&p&\cdots&p\\\
0&1&\cdots&p-1&a_{1}&a_{2}&\cdots&a_{p-1}\\\ \end{array}\right),$ (12)

instead of $A_{p}^{1,1}$. Then, the codes generated by
$A_{p,a}^{t_{1},t_{2}}$, denoted by $\mathcal{H}_{p,a}^{t_{1},t_{2}}$, are
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive GH codes of type
$(\alpha_{1},\alpha_{2};t_{1},t_{2})$.

Moreover, the corresponding $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear codes
after the Gray map are permutation equivalent to
$\Phi(\mathcal{H}_{p}^{t_{1},t_{2}})=\Phi(\mathcal{H}_{p,N_{p}^{-}}^{t_{1},t_{2}})$.

###### Example 4.1

Let $p=3$. Let
$S=\\{(a_{1},a_{2}):(3a_{1},3a_{2})=(3,6)=3\mathbb{Z}_{9}\setminus\\{0\\}\\}$.
Note that $S=\\{(1,2),(1,5),(1,8),(4,2),(4,5),(4,8),(7,2),(7,5),(7,8)\\}$ and
it can also be written as $\\{(1+3x,2+3y):x,y\in N_{3}\\}$. Therefore, in this
case, there are $9$ different starting matrices

$A_{3,a}^{1,1}=\left(\begin{array}[]{ccc|cc}1&1&1&3&3\\\ 0&1&2&a_{1}&a_{2}\\\
\end{array}\right),$

where $a=(a_{1},a_{2})\in S$. If $a=(1,2)$, we obtain the matrix $A_{3}^{1,1}$
given in Example 3.1. These $9$ matrices generate $9$ different
$\mathbb{Z}_{3}\mathbb{Z}_{9}$-additive codes $\mathcal{H}_{3,a}^{1,1}$ whose
corresponding $\mathbb{Z}_{3}\mathbb{Z}_{9}$-linear codes,
$\Phi(\mathcal{H}_{3,a}^{1,1})$, are permutation equivalent to each other.
Moreover, note that if we permute the coordinates of $a$, we also obtain
$\mathbb{Z}_{3}\mathbb{Z}_{9}$-linear codes which are permutation equivalent
to the previous ones.

##  5 Linearity of $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes

In [13], it is shown that the $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear GH codes
of type $(\alpha_{1},\alpha_{2};1,t_{2})$ are the only ones which are linear,
when $\alpha_{1}\not=0$. The next result shows that there are no
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes of type
$(\alpha_{1},\alpha_{2};t_{1},t_{2})$, with $\alpha_{1}\not=0$,
$t_{1},t_{2}\geq 1$ and $p\geq 3$ prime, which are linear. Note that this
result for $p\geq 3$ does not coincide with the known result for $p=2$ if
$t_{1}=1$. As it is mentioned in Remark 3.3, we only need to focus on codes
with $\alpha_{2}\not=0$.

###### Theorem 5.1

Let $\mathcal{H}_{p}^{t_{1},t_{2}}$ be the
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-additive GH code of type
$(\alpha_{1},\alpha_{2};t_{1},t_{2})$ with $\alpha_{1}\not=0$,
$t_{1},t_{2}\geq 1$ and $p\geq 3$ prime. Then,
$H_{p}^{t_{1},t_{2}}=\Phi(\mathcal{H}_{p}^{t_{1},t_{2}})$ is non-linear.

###### Example 5.1

z Let ${\cal H}_{3}^{1,1}$ be the $\mathbb{Z}_{3}\mathbb{Z}_{9}$-additive GH
code of type $(3,2;1,1)$ considered in Example 3.2. For example, we have that
$\Phi(0,1,2\mid 1,2)+\Phi(0,2,1\mid
2,4)=(0,1,2,0,1,2,0,2,1)+(0,2,1,0,2,1,1,2,0)=(0,0,0,0,0,0,1,1,1)\notin\Phi({\cal
H}_{3}^{1,1})$, so $H_{3}^{1,1}=\Phi({\cal H}_{3}^{1,1})$ is a non-linear
code.

###### Example 5.2

Considering all the integer solutions with $t_{1},t_{2}\geq 1$ of the equation
$5=2t_{1}+t_{2}-1$, we have that the $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear
GH codes with $\alpha_{1}\not=0$ of length $p^{5}$, $p\geq 3$ prime, are the
following: $H_{p}^{1,4}$ and $H_{p}^{2,2}$. By Theorem 5.1, we have that
$H_{p}^{1,4}$ and $H_{p}^{2,2}$ are non-linear.

###### Example 5.3

Considering all the integer solutions with $t_{1},t_{2}\geq 1$ of the equation
$6=2t_{1}+t_{2}-1$, we have that the $\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear
GH codes with $\alpha_{1}\not=0$ of length $p^{6}$, $p\geq 3$ prime, are the
following: $H_{p}^{1,5}$, $H_{p}^{2,3}$ and $H_{p}^{3,1}$. By Theorem 5.1, we
have that $H_{p}^{1,5}$, $H_{p}^{2,3}$ and $H_{p}^{3,1}$ are non-linear.

##  6 Conclusions

Two structural properties of codes over $\mathbb{Z}_{p}$ are the rank and
dimension of the kernel. The rank of a code $C$ over $\mathbb{Z}_{p}$ is
simply the dimension of the linear span, $\langle C\rangle$, of $C$. The
kernel of a code $C$ over $\mathbb{Z}_{p}$ is defined as
$\mathrm{K}(C)=\\{\textbf{x}\in\mathbb{Z}_{p}^{n}:\textbf{x}+C=C\\}$ [21, 22].
If the all-zero vector belongs to $C$, then $\mathrm{K}(C)$ is a linear
subcode of $C$. Note also that if $C$ is linear, then $K(C)=C=\langle
C\rangle$. We denote the rank of $C$ as $r$ and the dimension of the kernel as
$k$. The parameters $(r,k)$ can be used to distinguish between non-equivalent
codes, since equivalent ones have the same value of rank and dimension of the
kernel.

In [12, 13], the rank and dimension of the kernel are used to classify
$\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear Hadamard codes with $\alpha_{1}=0$ and
$\alpha_{1}\not=0$, respectively. Moreover, it is also known that the family
of $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear Hadamard codes with
$\alpha_{1}\not=0$ includes the family of
$\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear Hadamard codes with $\alpha_{1}=0$ [14],
since each $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear Hadamard code with
$\alpha_{1}=0$ is equivalent to a $\mathbb{Z}_{2}\mathbb{Z}_{4}$-linear
Hadamard code with $\alpha_{1}\not=0$. The rank and dimension of the kernel
have also been used to classify $\mathbb{Z}_{p^{s}}$-linear GH codes of length
$p^{t}$, with $p$ prime [7, 16, 17, 15].

Table 1 shows the type $(\alpha_{1},\alpha_{2};t_{1},t_{2})$ and parameters
$(r,k)$ for all $\mathbb{Z}_{3}\mathbb{Z}_{9}$-linear GH codes of length
$3^{t}$, with $\alpha_{1}\not=0$ and $2\leq t\leq 8$, considered in this
paper. It also includes the type $(0,\alpha_{2};t_{1},t_{2})$ and parameters
$(r,k)$ for all $\mathbb{Z}_{9}$-linear GH codes of the same length.

By looking at Table 1, we have that all $\mathbb{Z}_{3}\mathbb{Z}_{9}$-linear
GH codes of length $3^{t}$, with $\alpha_{1}\not=0$ and $2\leq t\leq 8$, are
pairwise non-equivalent since all of them have a different value of the
kernel. This means that there are at least $\lfloor t/2\rfloor+1$ such codes
for $2\leq t\leq 8$. Moreover, we can see that these non-linear codes are also
non-equivalent to any $\mathbb{Z}_{9}$-linear GH code of the same length.
Similar results can be obtained computationally for $p=5$ and $p=7$. This
means that, unlike for $p=2$, in general, for $p\geq 3$ prime, the
$\mathbb{Z}_{p^{2}}$-linear GH codes are not included in the family of
$\mathbb{Z}_{p}\mathbb{Z}_{p^{2}}$-linear GH codes with $\alpha_{1}\not=0$.
Finally, we can also observe that the $\mathbb{Z}_{3}\mathbb{Z}_{9}$-linear GH
codes with $\alpha_{1}\not=0$ are not equivalent to any
$\mathbb{Z}_{3^{s}}$-linear code of the same length, by comparing the values
of $(r,k)$ given in Table 1 with the ones given in [15, Tables 4 and 5].

$p$ | $t$ | $\mathbb{Z}_{9}$ | $\mathbb{Z}_{3}\mathbb{Z}_{9}$
---|---|---|---
|  | $(0,\alpha_{2};t_{1},t_{2})$ | $(r,k)$ | $(\alpha_{1},\alpha_{2};t_{1},t_{2})$ | $(r,k)$
$3$ | $2$ | (0,3;1,1) | (3,3) | (9,0;0,3) | $(3,3)$
|  |  |  | (3,2;1,1) | $(4,2)$
| $3$ | $(0,9;1,2)$ | (4,4) | $(27,0;0,4)$ | (4,4)
|  | $(0,9;2,0)$ | (5,2) | $(9,6;1,2)$ | (5,3)
| $4$ | $(0,27;1,3)$ | (5,5) | $(81,0;0,5)$ | (5,5)
|  | $(0,27;2,1)$ | (6,3) | $(27,18;1,3)$ | (6,4)
|  |  |  | $(9,24;2,1)$ | (10,3)
| $5$ | $(0,81;1,4)$ | (6,6) | $(243,0;0,6)$ | (6,6)
|  | $(0,81;2,2)$ | (7,4) | $(81,54;1,4)$ | (7,5)
|  | $(0,81;3,0)$ | (11,3) | $(27,72;2,2)$ | (11,4)
| $6$ | $(0,243;1,5)$ | (7,7) | $(729,0;0,7)$ | (7,7)
|  | $(0,243;2,3)$ | (8,5) | $(243,162;1,5)$ | (8,6)
|  | $(0,243;3,1)$ | (12,4) | $(81,216;2,3)$ | (12,5)
|  |  |  | $(27,234;3,1)$ | (20,4)
| $7$ | $(0,729;1,6)$ | (8,8) | $(2187,0;0,8)$ | (8,8)
|  | $(0,729;2,4)$ | (9,6) | $(729,486;1,6)$ | (9,7)
|  | $(0,729;3,2)$ | (13,5) | $(243,648;2,4)$ | (13,6)
|  | $(0,729;4,0)$ | (21,4) | $(81,702;3,2)$ | (21,5)
| $8$ | $(0,2187;1,7)$ | (9,9) | $(6561,0;0,9)$ | (9,9)
|  | $(0,2187;2,5)$ | (10,7) | $(2187,1458;1,7)$ | (10,8)
|  | $(0,2187;3,3)$ | (14,6) | $(729,1944;2,5)$ | (14,7)
|  | $(0,2187;4,1)$ | (22,5) | $(243,2106;3,3)$ | (22,6)
|  |  |  | $(81,2160;4,1)$ | (35,5)
Table 1: Type and parameters $(r,k)$ of $\mathbb{Z}_{9}$-linear and
$\mathbb{Z}_{3}\mathbb{Z}_{9}$-linear GH codes.

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# The latitudinal migration of the sunspot groups

Jouni Takalo 1 Space physics and astronomy research unit, University of Oulu,
POB 3000, FIN-90014, Oulu, Finland
11email<EMAIL_ADDRESS>
###### keywords:

Sun: Sunspot cycle, Sun: sunspot latitude, Sun: sunspot groups,
Method: Regression analysis, Method: Statistical analysis

##  1 Introduction

The most striking pattern in solar research was discovered by Maunder in
studying the occurrence of the sunspots 1874 – 1902 (Maunder, 1904).
Especially, when plotting the distribution of the sunspots during 1879 – 1889
(Solar Cycle 12) he noticed how the sunspots systemically approached the
Equator of the Sun during the course of the years. S.Carrington had noted
already in the transition of the Cycle 9 to Cycle 10 that sunspots disappeared
from the low latitudes and soon appeared at latitudes between 20 – 40 degrees
of heliographic latitudes. He reported this phenomenon in his article
(Carrington, 1858). The shape of the diagram, resembling butterfly, was first
drawn by Maunder for the Cycle 12 and 13, but he did not use that name in his
original paper (Maunder, 1904). Gleissberg and Damboldt (1971) mention that it
is not known who used the name ”‘Butterfly diagram”’ for the first time, but
at least Maunder used the name by 1922. In this same article Gleissberg and
Damboldt (1971) find that the mean latitude of sunspots decrease 0.012
$\times\,sin(2\phi_{m})$ degrees/day, where $\phi_{m}$ is the heliographic
latitude..

Later there have been some additional studies to find the formula for the
average drift and velocity of the migration for the sunspot and sunspot
groups. Li, Yun, and Gu (2001) found the mean latitude for the Royal Greenwich
Observatory sunspot group data for Cycles 12 – 22. They used quarterly year
data and fitted second-order polynomial to the overlapped data. The equations
of the fitting curves were $l$=0.0893$\times t^{2}$-2.8$\times$t+27.24 and
$l$=-0.0767$\times t^{2}$+2.6$\times$t-26.72 and correlation coefficients
0.9753 and 0.9749 for the fits of the northern and southern hemisphere,
respectively. They state that the migration velocity is largest in the
beginning of the cycle and the latitude decreases with an average of 1.6
degrees and 1.56 degrees in the northern and southern hemisphere,
respectively.

Hathaway (2011) used the same data but for the Cycles 12 – 23 in his analysis.
He stated that there is no dependence of the migration path on the strength of
the cycle and found a standard exponential equation for the trajectory of the
mean latitude, i.e. $\overline{\lambda}=28^{o}\,exp(-(t\,-\,t_{0})/90)$ ,
where t is measured in months.

Zhang, Li, and Feng (2018) studied the Extended time series of Solar Activity
Indices (ESAI) for the years 1854-1985. They used yearly averaged data and
were interested if the migration of sunspots is linear or nonlinear. They
found that second-order polynomial fit outshines the linear (first-order) fit
in their analysis. They found that migration velocity is about 2.2
degrees/year in the start of the cycle and decreases to 1.8 degrees/year in
the maximum phase and is 0.7 degrees/year at the end of the average cycle.

In this study we analyse the average drift of the sunspot groups for Solar
Cycles (SC) 8 – 23 from three different databases and find the best-fit
functions for the evolution of the drift. This article is organized as
follows. Section 2 presents the used data. In Section 3 we study and discuss
different scenarios of the migration and give our conclusion in Section 4.

##  2 Data and Methods

###  2.1 Sunspot Group Data

For the analyses of solar cycles we use two different sunspot group datasets
used also earlier by Takalo (2020a, b). The newest one is recently published
catalogue of sunspot groups by Mandal et al. (2020) (hereafter M set). The
main part of this dataset consists of data from the Royal Greenwich
observatory (RGO) measurements between 1874-1976. In this time series the
composite data sets from Kislovodsk, Pulkova and Debrecen observatories were
used for the years from 1976 to October 2019, i.e., containing (almost) solar
cycle 24. We, however, omit the Cycle 24 here, because it is incomplete and
study Cycles C12 – 23. The other dataset, which we use is the sunspot group
data for SC8 – SC23 by Leussu et al. (2017b, a) (hereafter L set). The years
1874 – 1976 for this data are also from the RGO dataset, but the solar cycles
SC21 – SC23 are constructed from SOON (Solar Observatory Optical Network) data
for the period 1976-2015 (Neidig et al., 1998). Again the Cycle 24 is
incomplete, and we do not include it to the analyses. This dataset has been
completed using digitized observations of S. H. Schwabe (Arlt, 2011; Arlt et
al., 2013), which provide an extension of the data to the years 1825 – –1867,
covering total Solar Cycles 8 – 10. Later this dataset has been filled with
Cycle 11 using newly digitized Spörer data (Diercke, Arlt, and Denker, 2015)
for the years 1868 – –1874 (Leussu et al., 2017b). There are differences in
the datasets such that the L set does not contain the area data of the sunspot
groups, while M set includes also the areas. In addition, the number of groups
in L set is much smaller, because it contains only the first appearance of the
sunspot group in its record, while M set contains daily appearance of the
groups. Thus the lifetime of each group also affects the number of the groups
in M set. When comparing the sunspot group data to separate sunspot data, we
use also recently published Kodaikanal dataset for the Cycles C16 – 23
(hereafter K set) (Jha, Mandal, and Banerjee, 2019; Jha et al., 2022). The
minima and lengths for the Solar Cycles 12 – 23 used in this study are listed
in Table 1. Note, that the L set database already contains northern and
southern wings for separate cycles.

Table 1.: Sunspot-cycle lengths and dates [fractional years, and year and month] of (starting) sunspot minima for Solar Cycles 12 – 23 (NGDC, 2013). Sunspot cycle | Fractional | Year and month | Cycle length
---|---|---|---
number | year of minimum | of minimum | [years]
12 | 1879.0 | 1878 December | 10.6
13 | 1889.6 | 1889 August | 12.1
14 | 1901.7 | 1901 September | 11.8
15 | 1913.5 | 1913 July | 10.1
16 | 1923.6 | 1923 August | 10.1
17 | 1933.7 | 1933 September | 10.4
18 | 1944.1 | 1944 February | 10.2
19 | 1954.3 | 1954 April | 10.5
20 | 1964.8 | 1964 October | 11.7
21 | 1976.5 | 1976 June | 10.2
22 | 1986.7 | 1986 September | 10.1
23 | 1996.8 | 1996 October | 12.2
24 | 2009.0 | 2008 December |

###  2.2 Resampled Cycles

To be able to add up cycles with different lengths in order to get more data
for solid statistics, we resample the monthly sunspot group (SG) and sunspot
data such that all cycles have the same length of 132 time steps (months),
i.e. 11 years. This is near the average length of the Cycles 8 – 23.

##  3 Results and Discussion

We start with the analysis with the Lset, which is the longest dataset and
gives thus the most solid statistics. Figure 1 shows the pattern of the
butterfly diagram of the whole L set. The black curves are scarce averages of
the sunspot groups calculated for the L set. Note, that in this set the Cycles
21 – 23 have only integer values as latitudes. It is, however, evident that
the drift of the sunspot groups during the evolution of the cycle is not
linear.

Figure 1.: Butterfly pattern of the sunspot groups for L set used in the
analysis. The black curves are averages of the sunspot groups calculated for
the set. The blue vertical lines are the maxima of the cycles.

In order to study the drift of the average latitude of the SGs in more detail,
we plot the monthly latitudes of the average SGs in Figure 2a and b for
northern and southern hemisphere (NH and SH wings) of the L set, respectively.
We made three different fits for the points of the average latitudes;
exponential, bi-linear (two first order polynomials) and second order
polynomial. The exponential fits are slightly better than the second order
fits. The R2s are 0.992 and 0.988 for exponential fits of northern and
southern hemispheres, respectively. The second order polynomial fits have R2s
0.991 and 0.987 for northern and southern hemispheres, respectively. In Figure
2 we show the exponential fits with 95 % confidence limits. The equations of
the exponential fits are

$\overline{\lambda}=27.55\times e^{-0.01092\times t}$ (1)

and

$\overline{\lambda}=-26.25\times e^{-0.0104\times t}$ (2)

for the northern and southern hemisphere, respectively. Here
$\overline{\lambda}$ is the estimated average latitude of the SGs and t is
time in months. Note, that we do not make any area weighting, because L
dataset does not contain area information.

Figure 2.: a) The latitudinal drift of the average sunspot group of the
northern hemisphere for the L set (Cycles 8 – 23) with exponential fit. b)
Same as a) but for the southern hemisphere.

Figures 3a and b show the same data point as Figures 2a and b, but now with
linear fits which is made in two fragments (bi-linear fit). We have also
determined the knee (crossing point) such that the difference of the slopes is
largest. The equation of the lines are also shown in the figures with the
lines of the 95 % confidence limits. Notice, that the confidence limits are in
this case narrower towards the higher latitudes. The overall R2s are 0.961 and
0.966 for the first and second parts of the northern hemisphere fits and 0.986
and 0.944 for the first and second parts of the southern hemisphere fits,
respectively. However, the advantage of the bi-linear fit is that we can
estimate the crossing point where the slope radically changes. According to
our earlier studies we assumed this to be near the latitude of 15 degrees, and
it indeed turned out to be near of that, i.e. the coordinates of the crossing
points are (50 months, 15.4 degrees) and (54 months, -14.4 degrees) for the
northern and southern hemisphere, respectively. The so-called Gnevyshev gap is
also located at the time when the mean latitude of the sunspots crosses about
latitude of 15 degrees and this point is kind of a separatrix that divides the
solar cycle into two parts, the ascending phase and the declining phase
(Takalo and Mursula, 2018; Takalo, 2020a, b).The drift velocities can be
estimated the slopes of the fitting lines, i.e 0.233 and 0.216 degrees/month
(meridional speed$\approx$ 1.10 and 1.02 m/s) before the crossing point
(ascending phase of the cycle) and 0.118 and 0.101 degrees/month ($\approx$
0.55 and 0.47 m/s) after the crossing point (descending phase of the cycle).
On the other hand, the advantage of the exponential fit is, that we can
calculate the drift velocity exactly (point-by-point) as a derivative of the
equation of the exponential fit. Figure 4 shows the average momentary drift
velocities. The equations for the velocities are
$v_{\overline{\lambda}}=0.301\times exp(-0.01092\times t)$ and
$v_{\overline{\lambda}}=0.273\times exp(-0.01041\times t)$ (where t is in
months) for the northern and southern hemisphere, respectively.The numerical
values for the velocities are 0.30 and 0.27 degrees/month ($\approx$ 1.41 and
1.27 m/s) and at the beginning and 0.071 and 0.069 degrees/month ($\approx$
0.33 and 0.32 m/s) at the end of the average cycle for northern and southern
hemisphere, respectively. Note, that the velocity curves of exponential fits
are also exponential. On the other hand, in the second-order polynomial fits
the velocity curves would be linear. Because exponential fits turned to be
slightly better than other fits, we use mainly this fitting method in the rest
of this study.

Figure 3.: Same data points as in the Figure 2 but now with bi-linear fits. a)
Northern hemisphere b) Southern hemisphere. Figure 4.: The momentary drift
velocities of the average SGs on the northern and southern hemisphere of the
Sun.

We also wanted to know if the strength of the cycle affects to the path of the
drift of the SGs. The categories of the large, medium and small cycles are
defined similarly to Hathaway (2011); large cycles: 18, 19, 21 and 22, medium
cycles: 15, 17 20 and 23, small cycles: 12, 13, 14 and 16. Figure 5a and b
show the exponential fits and points of the monthly average latitudes of the L
set SGs for the northern and southern hemispheres, respectively. It is evident
that the large cycles have higher latitudes and small cycles lowest latitudes
through the evolution of the average cycle. This is, however, at least partly
due to the fact that the small cycles have on average starting times about
seven months earlier than the minimum, while medium and large cycles have
starting times about equal to minimum (Hathaway, 2011). There seems to be,
however, a slight difference between the drift paths of the large and medium
sized cycles such that the path of the average large cycle is about one degree
higher than the average medium cycle, at least at the descending phase of the
cycle. This applies for both northern and southern hemisphere of the average
cycle. It seems that the drift path of the stronger cycles follow slightly
higher latitudes than those of the weaker cycles.

Figure 5.: a) The exponential fits of the average drift for large, medium and
small cycles of the northern hemisphere. b) Same as a but for the southern
hemisphere.

Another interesting question is, do the areas of the SGs affect to the paths
of the groups. While L set has a shortage that it does not contain area
information, we must analyse in this respect M set for the Cycles 12 – 23. Let
us, however, first compare the average NH cycle between L set and M set.
Figure 6a and b show the L set and M set drift path for the average northern
and southern hemispheres, respectively. It seems that the path of the M set is
about a half degree higher latitudes than the L set set for the southern
hemisphere. (However, when doing an area weighted analysis for the M set, the
paths of L set and M set were almost identical). It is evident, that there is
no big difference between these datasets, although they differ in the way the
the SGs have been recorded, i.e that L set contains only the first appearance
of each SG and M set includes daily appearances of the same SGs.

Figure 6.: a) The exponential fits of the average drift for the average L and
M set cycles of the northern hemisphere. b) Same than a but for the southern
hemisphere.

It is now clear that we can study the drift paths of small and large area
sunspot groups using only the M dataset. We divide the SGs into two
categories; small SGs with area smaller or equal to 200 MH (millionths of
hemisphere), and large SGs with area larger than 200 MH. Figure 7 shows the
drift paths for both area categories of the M set for northern hemisphere. We
have used here are area weighted calculation of the mean latitude, i.e.
$\overline{\lambda}=\sum\,A(\lambda_{i})\lambda_{i})/\sum\,A(\lambda_{i})$. It
seems that the area does not affect very much to the drift path of the SGs.
There is, however, difference in the northern hemisphere such that small area
SGs are located at slightly higher latitudes in the ascending phase and at
lower latitudes in the descending phase of the average cycle. This is because
the large area SGs are spread mainly between latitudes 10 – 25 degrees, but
small area SGs more widely between 0 – 40 degrees (Takalo, 2020a). In the
southern hemisphere the drift paths are, however, more similar. Note also,
that there are quite a lot variations in the data points at the beginning and
in the end of the cycles, especially for large category SGs, due to lack of
data and thus worse statistics near cycle minima.

Figure 7.: a) The exponential fits of the average drift for small ($<=$200 MH)
and large ($>200$ MH) sunspot groups for the northern hemisphere. b) Same as a
but for the southern hemisphere.

Figure 8a and b show the exponential fits of L (C8 – C23), M (C12 – 23) and K
(C16 – C23) sets with data points of K set for the northern and southern
hemispheres, respectively. The points marked with ’x’ in the figure and
confidence limits (red dashed curves) are those for the K set. Notice the
quite wide variation in the beginning and at the end of the average cycle.
This is also seen as wider 95 % confidence limits than earlier for the L set.
These are due to less amount of data compared to L and M datasets. Although
the K set (Kodaikanal) data contains separate sunspot data and for a shorter
interval of cycles, its seems to scale similarly to the sunspot group
datasets. Especially, M set and K set exponential fits are very similar,
except for the descending phase of the northern hemisphere.

Figure 8.: a) The exponential fits of the average drift of L and M sunspot
groups and of K set sunspots for the northern hemisphere. b) Same as a but for
the southern hemisphere. The data points and 95 % confidence limits of K set
are also shown in both figures.

##  4 Conclusion

We have analysed the monthly sunspot group (SG) data for Solar Cycles 8 – 23
and calculated the average latitude of the migration for the northern and
southern hemisphere. We find that exponential function fits best to the
average drift of the SGs. The R2s for the exponential fits are 0.992 and
0.988, but second-order polynomial fits are only slightly worse. The drift
velocities are are 0.30 and 0.27 degrees/month (meridional speed$\approx$ 1.41
and 1.27 m/s) in the beginning and 0.071 and 0.069 degrees/month ($\approx$
0.33 and 0.32 m/s) at the end of the average cycle for northern and southern
hemisphere, respectively.

We do also bi-linear fits (first-order polynomial fits in two fractions)
because they tell us the crossing point (knee) were the drift velocity
decreases the most. Furthermore, we find the crossing point such that the
difference of the slopes is largest. This crossing point for the Cycles 8- 23
is at 50 months and 15.4 degrees of latitude in the northern hemisphere. The
crossing point for the southern hemisphere is somewhat further from the start
of the cycle, i.e. 54 months and -14.4 degrees of southern latitude. The drift
velocities calculated from the slopes of the bi-linear fits are 0.233 and
0.216 degrees/month ($\approx$ 1.10 and 1.02 m/s) before the crossing point
and 0.118 and 0.101 degrees/month ($\approx$ 0.55 and 0.47 m/s) after the
crossing point. The bi-linear fit is interesting, because the possible abrupt
change in the drift velocity and thus in meridional speed near 15 degrees of
latitude could be related to Gnevyshev gap. This issue is really worth
studying in a forthcoming project.

We also found that there is a slight dependence of the migration on the
strength of the cycle such that the drift path of the stronger cycles follow
slightly higher latitudes than those of the weaker cycles. The areas of the
sunspot groups affect also slightly to the drift path such that the path of
large area SGs exists at somewhat lower latitudes in the ascending phase and
at higher latitudes in the descending phase of the cycle than those for the
smaller area SGs. It is also evident, that the form of the drift path is
almost similar for all three dataset (L set, M set and K set), which we have
investigated.

#### Acknowledgements

The sunspot group data are downloaded from VizieR database
(https://vizier.u-strasbg.fr/viz-bin/VizieR?-source=J/A+A/599/
A131&-to=3). The RGO sunspot area data are from RGO-USAF/NOAA
(https://solarscience.msfc.nasa.gov/greenwch.shtml) and Kodaikanal data from
https://github.com/bibhuraushan/KoSoDigitalArchive. The dates of cycle maxima
were obtained from from the National Geophysical Data Center (NGDC), Boulder,
Colorado, USA (ftp.ngdc.noaa.gov). I am grateful to J.-G. Richard for useful
communication.

Disclosure of Potential Conflicts of Interest

The author declares that there are no conflicts of interest.

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