On the number of ${{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}$ and ${{\mathbb {Z}}}_{p}{{\mathbb {Z}}}_{p^{2}}$-additive cyclic codes

Abstract

In this paper, we give the exact number of ${{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}$-additive cyclic codes of length $n=r+s,$ for any positive integer r and any positive odd integer s. We will provide a formula for the the number of separable ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes of length n and then a formula for the number of non-separable ${{\mathbb {Z}} _{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes of length n. Then, we have generalized our approach to give the exact number of ${{\mathbb {Z}}_{p}{\mathbb { Z}_{p^{2}}}}$-additive cyclic codes of length $n=r+s,$ for any prime p,  any positive integer r and any positive integer s where $\gcd \left( p,s\right) =1.$ Moreover, we will provide examples of the number of these codes with different lengths $n=r+s$.

1 Introduction

In coding theory, the class of linear codes is one of the most studied codes because of their rich algebraic structure and their well-defined mathematical properties. A linear code of length n over a finite field $\mathbb {F}_{q}$ is a subspace of $\mathbb {F}_{q}^{n}.$ In the early history of coding theory, researchers mainly studied linear codes over finite fields, especially over ${{\mathbb {Z}}}_{2}.$ Later, codes over rings have been considered by many researchers since the early seventies. However, they became a very popular research area with the work of Hammons et al. [9]. In [9], Hammons and coauthors showed that some well-known non-linear codes such as the Kerdock and Preparata codes, are actually Gray images of linear codes over ${{\mathbb {Z}}}_{4}.$ This work has led researchers to study codes over different rings, such as ${{\mathbb {Z}} }_{2^{k}}$, ${{\mathbb {Z}}}_{p^{k}}$ and $\mathbb {F}_{q}+u\mathbb {F}_{q}.$ The reader may find some of such studies in [6, 8, 10].

In 2010, Borges et. al. introduced a new class of codes over rings, called ${ {\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive codes [3]. They defined ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive codes as subgroups of ${{\mathbb {Z}} _{2}^{r}\times {{\mathbb {Z}}_{4}^{s}}}.$ In fact, ${{\mathbb {Z}}_{2}{{\mathbb {Z}} _{4}}}$-additive codes are generalization of binary linear codes and quaternary linear codes. If we take $s=0,$ then we have the binary linear codes of length r and if $r=0,$ then ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive codes are quaternary linear codes over ${{\mathbb {Z}}}_{4}$ of length s. Although the class of ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive codes is a very new family of codes, they have some applications in the field of Steganography [11]. In [1], a number of optimal binary linear codes were constructed as images of ${{\mathbb {Z}}_{2}{ {\mathbb {Z}}_{4}}}$-additive cyclic codes using the Gray map. In [5], Borges et. al. generalized the study of ${{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}$-additive cyclic codes to ${{\mathbb {Z}}}_{p^{r}}{{\mathbb {Z}}}_{p^{s}}$-additive cyclic codes where p is a prime number and, r and s are coprimes with p.

The class of ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive codes is a very huge class. This implies that the number of distinct ${{\mathbb {Z}}_{2}{ {\mathbb {Z}}_{4}}}$-additive codes is huge compared to the number of linear codes over ${{\mathbb {Z}}}_{2}$ or the number of linear codes over ${{\mathbb {Z }_{4}.}}$ In [7], Steven Dougherty et. al. studied the number of ${ {\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive codes. Moreover, Siap and Aydogdu studied counting the number of generator matrices of ${{\mathbb {Z}}_{2}{ {\mathbb {Z}}_{8}}}$-additive codes in [12].

In this paper, we are interested in finding the exact number of distinct ${ {\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes of length $n=r+s,$ for any positive integer r and any positive odd integer s. If s is any positive odd integer, then the ring ${{{\mathbb {Z}}_{4}}}\left[ x\right] {/}\left\langle x^{s}-1\right\rangle$ is a principal ideal ring and hence cyclic codes of length s over ${{{\mathbb {Z}}_{4}}}$ are principal ideals. We will provide a formula for the number of separable ${{\mathbb {Z}}_{2}{ {\mathbb {Z}}_{4}}}$-additive cyclic codes of length n and another formula for the number of non-separable ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes of length n. Then, we have generalized our approach to provide the exact number of ${{\mathbb {Z}}_{p}{{\mathbb {Z}}_{p^{2}}}}$-additive cyclic codes of length $n=r+s,$ for any prime p,  any positive integer r and any positive integer s where $\gcd \left( p,s\right) =1.$ The condition that $\gcd \left( p,s\right) =1$ will guarantee that the ring ${{{\mathbb {Z}}_{p^{2}} }}\left[ x\right] {/}\left\langle x^{s}-1\right\rangle$ is a principal ideal ring and hence cyclic codes of length s over ${{{\mathbb {Z}}_{p^{2}}}}$ are principal ideals. As an application of our study, we will provide examples of the exact number of ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes and ${{\mathbb {Z}}}_{3}{{\mathbb {Z}}}_{9}$-additive cyclic codes of different lengths.

2 ${{\mathbb {Z}}_2{{\mathbb {Z}}_4}}$-additive and ${{\mathbb {Z}}_2{{\mathbb {Z}}_4}}$-cyclic codes

In this section, we give the definitions of ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive and ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes, and we also give some properties of these codes. A comprehensive study of these codes can be found in [1] and in [3].

Definition 1

A non-empty subset ${\mathcal {C}}$ of ${{\mathbb {Z}}_{2}^{r}\times {{\mathbb {Z}} _{4}^{s}}}$ is called a ${{\mathbb {Z}}_2{{\mathbb {Z}}_4}}$-additive code if ${ \mathcal {C}}$ is a subgroup of ${{\mathbb {Z}}_{2}^{r}\times {{\mathbb {Z}}_{4}^{s} }}.$

If ${\mathcal {C}}$ is a ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive code, then it is isomorphic to an abelian group ${{\mathbb {Z}}}_{2}^{\gamma }\times {{\mathbb {Z}}}_{4}^{\delta }$ with the order of ${\mathcal {C}}$ given by $|{ \mathcal {C}}|=2^{\gamma }4^{\delta }.$ Also, the number of order two codewords in ${\mathcal {C}}$ is $2^{\gamma +\delta }.$ Let $\kappa$ be the dimension of the binary linear code obtained by taking the subcode of ${ \mathcal {C}}$ containing all order-two codewords. In this case, the code ${\mathcal {C}}$ will be referred to as of type $\left( r,s;\gamma ,\delta ;\kappa \right) .$

Let $\varphi :{{\mathbb {Z}}}_{4}\rightarrow {{\mathbb {Z}}}_{2}^{2}$ be the usual Gray map defined by $\varphi (0)=00$, $\varphi (1)=01$, $\varphi (2)=11$ and $\varphi (3)=10.$ $\varphi$ can be extended to a map $\Phi$ defined by

$$\begin{aligned} \Phi :{{\mathbb {Z}}_{2}^{r}\times {{\mathbb {Z}}_{4}^{s}}}\rightarrow & {} {\mathbb { Z}}_{2}^{n} \\ \left( u_{0},u_{1},\ldots u_{r-1}|v_{0},v_{1},\ldots v_{s-1}\right)\rightarrow & {} \left( u_{0},u_{1},\ldots u_{r-1}|\varphi (v_{0}),\varphi (v_{1}),\ldots \varphi (v_{s-1})\right) \end{aligned}$$

where $n=r+2s$, $\left( u_{0},u_{1},\ldots u_{r-1}|v_{0},v_{1},\ldots v_{s-1}\right) \in {{\mathbb {Z}}_{2}^{r}\times {{\mathbb {Z}}_{4}^{s}}}.$ The Gray image $\Phi ({\mathcal {C}})$ is a binary code (not necessary linear since $\Phi$ is not linear).

Example 2

Let ${\mathcal {C}}$ be a ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive code generated by

$$\begin{aligned} \left( \begin{array}{cc|cccc} 1 &{} 0 &{} 0 &{} 0 &{} 2 &{} 2 \\ 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 2 \end{array} \right) . \end{aligned}$$

Hence, ${\mathcal {C}} =\{00|0000,10|0022,11|1102,01|1120,00|2200,10|2222,11|3302,01|3320\}$.

• The order of ${\mathcal {C}}$ is $2^{1}4^{1}$, so $\gamma =1$ and $\delta =1.$

• $r=2,s=4$ and $\kappa =1$.

• Therefore, ${\mathcal {C}}$ is of type (2, 4; 1, 1; 1).

Definition 3

Let ${\mathcal {C}}$ be a ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive code of length $n=r+s.$ ${\mathcal {C}}$ is called a ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic code if $c=\left( u_{0},u_{1},\ldots u_{r-1}|v_{0},v_{1},\ldots v_{s-1}\right)$ is a codeword in ${\mathcal {C}},$ then

$$\begin{aligned} \sigma \left( c\right) =\left( u_{r-1},u_{0},\ldots u_{r-2}|v_{s-1},v_{0},\ldots v_{s-2}\right) \end{aligned}$$

is also in ${\mathcal {C}}.$

Let $\mathcal {R}_{r,s}={{\mathbb {Z}}}_{2}[x]/\langle x^{r}-1\rangle \times { {\mathbb {Z}}}_{4}[x]/\langle x^{s}-1\rangle .$ Then any element $c=\left( u_{0},u_{1},\ldots u_{r-1}|v_{0},v_{1},\ldots v_{s-1}\right)$ $\in {\mathbb { Z}_{2}^{r}\times {{\mathbb {Z}}_{4}^{s}}}$ can be identified with an element in $\mathcal {R}_{r,s}$ as follows:

$$\begin{aligned} c(x)= & {} \left( u_{0}+u_{1}x+\cdots +u_{r-1}x^{r-1},v_{0}+v_{1}x+\cdots +v_{s-1}x^{s-1}\right) \\= & {} \left( u(x),v(x)\right) \end{aligned}$$

This is one-one correspondence between the elements of ${{\mathbb {Z}} _{2}^{r}\times {{\mathbb {Z}}_{4}^{s}}}$ and the elements of $\mathcal {R}_{r,s}.$ Therefore, we can identify ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes with polynomials of $\mathcal {R}_{r,s}$. The following theorem gives the generator polynomials of ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes when s is an odd integer. Throughout this paper, we will use the notation f instead of the polynomial $f\left( x\right) .$

Theorem 4

([1]) Let ${\mathcal {C}}$ be a ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$ -additive cyclic code in $\mathcal {R}_{r,s}$ with odd integer s. Then ${ \mathcal {C}}$ can be identified as

$$\begin{aligned} {\mathcal {C}}=\langle (f,0),(l,g+2a)\rangle , \end{aligned}$$

where $f|(x^{r}-1)\,mod\,2$, $a|g|(x^{s}-1)\,mod\,4,$ l is a binary polynomial satisfying $\deg (l)<\deg (f),$ and $f|\dfrac{x^{s}-1}{a}l.$

Lemma 5

Let ${\mathcal {C}}=\left\langle (f,0),(l,g+2a)\right\rangle$ be a ${\mathbb { Z}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic code in $\mathcal {R}_{r,s}$ with odd integer s,  where the generators satisfy the conditions in Theorem 4. Then the generators $f,\,l,\,g$ and a are unique.

Proof

The proof is similar to the proof of Theorem 3 in [2]. $\square$

Example 6

Let ${\mathcal {C}}$ be a ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic code in ${{\mathbb {Z}}}_{2}[x]/\langle x^{7}-1\rangle \times {{\mathbb {Z}}} _{4}[x]/\langle x^{7}-1\rangle$ generated by $\langle (f,0),(l,g+2a)\rangle ,$ where

$$\begin{aligned} f= & {} x^{7}-1,\,l=1+x^{2}+x^{3}, \\ a= & {} 3+2x+3x^{2}+x^{3},\,g=1+x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}. \end{aligned}$$

The code ${\mathcal {C}}$ has the following generator matrix

$$\begin{aligned} G=\left( \begin{array}{cccccccccccccc} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 3 &{} 1 &{} 3 &{} 3 &{} 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 2 &{} 2 &{} 2 &{} 0 &{} 2 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 2 &{} 2 &{} 2 &{} 0 &{} 2 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 2 &{} 2 &{} 2 &{} 0 &{} 2 \end{array} \right) . \end{aligned}$$

Furthermore, the binary image of ${\mathcal {C}}$ under the Gray map that we defined above is an optimal binary linear code with parameters [21, 5, 10].

Definition 7

Let ${\mathcal {C}}$ be a ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive code. ${ \mathcal {C}}$ is called separable if ${\mathcal {C}}={\mathcal {C}}_{X}\times { \mathcal {C}}_{Y}$, where

$$\begin{aligned} {\mathcal {C}}_{X}\times {\mathcal {C}}_{Y}=\{(a,b)\,|\,\text {there are codewords }(a,c_{2}),(c_{1},b)\in {\mathcal {C}}\}. \end{aligned}$$

Corollary 8

([4]) Let ${\mathcal {C}}=\langle (f,0),(l,g+2a)\rangle$ be a ${ {\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic code. Then, ${\mathcal {C}}$ is separable if and only if $l=0.$

3 The number of ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes

Let ${\mathcal {C}}$ be a ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic code in $\mathcal {R}_{r,s},$ where s is an odd integer. Then ${\mathcal {C}}$ can be uniquely identified as

$$\begin{aligned} {\mathcal {C}}=\langle (f,0),(l,g+2a)\rangle , \end{aligned}$$

(1)

where $f|(x^{r}-1)\,mod\,2$, $a|g|(x^{s}-1)\,mod\,4,$ l is a binary polynomial satisfying $\deg (l)<\deg (f)$ and $f|\dfrac{\left( x^{s}-1\right) }{a}l.$ In this section, we are interested to determine a formula for the number of distinct ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes of length $n=r+s.$ Before starting our main work, we will give a few remarks which are related to our work.

Remark

• 1. The generator polynomials in Eq. 1 are unique.

• 2. The only restrictions on the polynomial l are $\deg (l)<\deg (f)$ and $f|\dfrac{\left( x^{s}-1\right) }{a}l.$ This makes the number of ${ {\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic code in $\mathcal {R}_{r,s}$ to be huge compared to the number of cyclic codes over ${{\mathbb {Z}}_{2}}$ or over ${{{\mathbb {Z}}_{4}.}}$ Moreover, the existence of the polynomial l as a part of the generators will make the problem of finding a general formula for the number of ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic code a challenging problem.

• 3. If r is odd then, $\left( x^{r}-1\right) =\widetilde{f_{1}} \widetilde{f_{2}}\ldots \widetilde{f_{t}}\mod 2,$ is factored as a product of the irreducible factors $\widetilde{f_{1},}\widetilde{f_{2},}\ldots , \widetilde{f_{t}}$. Any factor (not equal 1) of $\left( x^{r}-1\right)$ will be labeled as $f_{i}$ where $i\in \left\{ 1,2,\ldots ,2^{t}-1\right\} .$ The same is applied for $\left( x^{s}-1\right) \mod 4.$

The number of ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes of length $n=r+s,$ where r is any integer and s is an odd integer will be given in Corollary 14. But first we will find the number of these codes when r and s are odd positive integers. For the results from Lemma 9 until Theorem 13, we will always assume that r and s are any odd positive integers.

Lemma 9

Let ${\mathcal {C}}=\langle \left( f,0\right) ,\left( l,g+2a\right) \rangle$ be a cyclic code in ${\mathbb {\ Z}}_{2}[x]/\langle x^{r}-1\rangle \times {{\mathbb {Z}}}_{4}[x]/\langle x^{s}-1\rangle$, where $f|(x^{r}-1)\,mod\,2$, $a|g|(x^{s}-1)\,mod\,4,\,l$ is a binary polynomial satisfying $\deg (l)<\deg (f)$ and $f|\dfrac{\left( x^{s}-1\right) }{a}l.$ If $\gcd \left( f,\dfrac{\left( x^{s}-1\right) }{a}\right) =1,$ then ${ \mathcal {C}}$ is a separable code.

Proof

By Corollary 12 in [1], the polynomial $l=0.$ Hence, ${\mathcal {C}}$ is separable. $\square$

Lemma 10

The number of separable ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes in $\mathcal {R}_{r,s}$ is $2^{w_{1}}3^{w_{2}}$ where $w_{1}$ is the number of irreducible factors of $\left( x^{r}-1\right) \text { mod }2$ and $w_{2}$ is the number of irreducible factors of $\left( x^{s}-1\right) \text {mod }4.$

Proof

Since ${\mathcal {C}}$ is separable then ${\mathcal {C}}=\langle \left( f,0\right) ,\left( 0,g+2a\right) \rangle ={\mathcal {C}}_{1}\times {\mathcal {C }}_{2},$ where ${\mathcal {C}}_{1}=\left\langle f\right\rangle$ is a binary cyclic code of length r and ${\mathcal {C}}_{2}=\left\langle g+2a\right\rangle$ is a quaternary cyclic code over ${\mathbb {Z}}_{4}$ of length s. The result follows from the fact that there are $2^{w_{1}}$ binary cyclic codes of length r and $3^{w_{2}}$ quaternary cyclic codes over ${\mathbb {Z}}_{4}$ of length s. $\square$

In order to count the number of non-separable cyclic codes in ${\mathbb {\ Z}} _{2}[x]/\langle x^{r}-1\rangle \times {{\mathbb {Z}}}_{4}[x]/\langle x^{s}-1\rangle$, by Lemma 9 we must always have $\gcd \left( f, \dfrac{\left( x^{s}-1\right) }{a}\right) >1.$ Hence, when we consider non-separable ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes, we will always assume that $\gcd \left( f,\dfrac{\left( x^{s}-1\right) }{a} \right) >1.$

Lemma 11

Suppose that ${\mathcal {C}}=\left\langle \left( f,0\right) ,\left( l,g+2a\right) \right\rangle$ is a non-separable $\mathbb { Z}_{2}{\mathbb {Z}}_{4}$-additive cyclic code in ${\mathbb {\ Z}}_{2}[x]/\langle x^{r}-1\rangle \times {{\mathbb {Z}}}_{4}[x]/\langle x^{s}-1\rangle$ with $\gcd \left( r,s\right) =1.$ Then

$$\begin{aligned} {\mathcal {C}}=\langle \left( \left( x-1\right) Q_{1},0\right) ,\left( Q_{1},g+2a\right) \rangle , \end{aligned}$$

where $Q_{1}|(x^{r}-1)\,mod\,2,\,a|g|(x^{s}-1)\,mod\,4$ and $\left( x-1\right)$ is not a factor of a.

Proof

Let ${\mathcal {C}}=\langle \left( f,0\right) ,\left( l,g+2a\right) \rangle$ be a non-separable cyclic code in ${\mathbb {\ Z}}_{2}[x]/\langle x^{r}-1\rangle \times {{\mathbb {Z}}}_{4}[x]/\langle x^{s}-1\rangle ,$ with $\gcd (r,s)=1.$ Since $\gcd \left( r,s\right) =1,$ then the only common factors of $\left( x^{r}-1\right)$ and $\left( x^{s}-1\right)$ $\mod 2$ are 1 and $\left( x-1\right) .$ Suppose that $a=\left( x-1\right) J$ for some binary polynomial J. Since $f|\dfrac{\left( x^{s}-1\right) }{a}l$ and $\gcd \left( f,\dfrac{\left( x^{s}-1\right) }{a}\right) =1$, we get $f|l,$ which is a contradiction unless $l=0$, and hence the code is separable. Now, suppose that $\left( x-1\right)$ is not a factor of f. Then, $\gcd \left( f,\frac{x^{s}-1}{a}\right) =1$ and again l must be zero giving that ${ \mathcal {C}}$ is a separable code. Hence, in order for ${\mathcal {C}}$ to be a non-separable code, we must have $\gcd \left( f,\frac{x^{s}-1}{a}\right) =x-1.$ This implies that $f=\left( x-1\right) Q_{1}$ and $\dfrac{x^{s}-1}{a} =\left( x-1\right) Q_{2},$ with $\gcd \left( Q_{1},Q_{2}\right) =1.$ Since $f|\dfrac{\left( x^{s}-1\right) }{a}l,$ then $Q_{1}|Q_{2}l$ which implies that $Q_{1}|l$ and $l=Q_{1}V.$ Since $\deg l<\deg f$ and $f=\left( x-1\right) Q_{1}$, then $l=Q_{1}.$ Thus, ${\mathcal {C}}=\langle \left( \left( x-1\right) Q_{1},0\right) ,\left( Q_{1},g+2a\right) \rangle ,$ where $\left( x-1\right)$ is not a factor of a. $\square$

Theorem 12

Let ${\mathcal {C}}=\langle \left( f,0\right) ,\left( l,g+2a\right) \rangle$ be a non-separable cyclic code in ${\mathbb {\ Z}} _{2}[x]/\langle x^{r}-1\rangle \times {{\mathbb {Z}}}_{4}[x]/\langle x^{s}-1\rangle$ and let $x^{r}-1=\widetilde{f_{1}}\widetilde{f_{2}}\ldots \widetilde{f_{t}}\mod 2$ and $x^{s}-1=\widetilde{g_{1}}\widetilde{g_{2}} \ldots \widetilde{g_{w}}\mod 4$ be the factorizations of $x^{r}-1$ and $x^{s}-1$ into irreducible polynomials in ${{\mathbb {Z}}}_{2}[x]$ and ${\mathbb { Z}}_{4}[x]$, respectively, with $\gcd (r,s)=1.$ Then, the number of non-separable ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes is given by

$$\begin{aligned} 2^{t}3^{w-1}. \end{aligned}$$

Proof

By Lemma 11, we know that ${\mathcal {C}}=\langle \left( \left( x-1\right) Q_{1},0\right) ,\left( Q_{1},g+2a\right) \rangle ,$ where $Q_{1}|(x^{r}-1)\,mod\,2,\,a|g|(x^{s}-1)\,mod\,4$ and $\left( x-1\right)$ is not a factor of a. Since $x^{r}-1=\widetilde{f_{1}}\widetilde{f_{2}}\ldots \widetilde{f_{t}}\mod 2,$ then $\left( x^{r}-1\right)$ has $2^{t}$ different factors and $Q_{1}$ has $2^{t-1}$ choices (because $\left( x-1\right)$ cannot be a factor of $Q_{1}$). For the polynomials a and g , we must have $a\,|\,g\,|\left( x^{s}-1\right)$ and $\left( x-1\right)$ is not a factor of a. Hence, the number of choices for a and g is

$$\begin{aligned}&\left( \begin{array}{c} w \\ 0 \end{array} \right) 2^{w}+\left( \begin{array}{c} w-1 \\ 1 \end{array} \right) 2^{w-1}+\left( \begin{array}{c} w-2 \\ 2 \end{array} \right) 2^{w-2}+\ldots +\left( \begin{array}{c} w-1 \\ w-1 \end{array} \right) 2^{1} \\&\quad =2^{w}+2\left[ \left( \begin{array}{c} w-1 \\ 1 \end{array} \right) 2^{w-2}+\left( \begin{array}{c} w-2 \\ 2 \end{array} \right) 2^{w-3}+\ldots +\left( \begin{array}{c} w-1 \\ w-2 \end{array} \right) 2^{1}+\left( \begin{array}{c} w-1 \\ w-1 \end{array} \right) 2^{0}\right] \\&\quad =2^{w}+2\left[ \begin{array}{c} \left( \begin{array}{c} w-1 \\ 0 \end{array} \right) 2^{w-1}+\left( \begin{array}{c} w-1 \\ 1 \end{array} \right) 2^{w-2}+\left( \begin{array}{c} w-2 \\ 2 \end{array} \right) 2^{w-3}+\ldots +\left( \begin{array}{c} w-1 \\ w-2 \end{array} \right) 2^{1} \\ +\left( \begin{array}{c} w-1 \\ w-1 \end{array} \right) 2^{0}-\left( \begin{array}{c} w-1 \\ 0 \end{array} \right) 2^{w-1} \end{array} \right] \\&\qquad 2^{w}+2\left[ 3^{w-1}-2^{w-1}\right] \\&\quad =2\times 3^{w-1}. \end{aligned}$$

Therefore, if $\gcd (r,s)=1,$ then the number of non-separable cyclic codes is $2^{t-1}\times 2\times 3^{w-1}=2^{t}3^{w-1}.$ $\square$

Our next theorem gives the number of non-separable cyclic codes for any odd integers r and s.

Theorem 13

Let ${\mathcal {C}}=\langle \left( f,0\right) ,\left( l,g+2a\right) \rangle$ be a non-separable cyclic code in ${\mathbb {\ Z}} _{2}[x]/\langle x^{r}-1\rangle \times {{\mathbb {Z}}}_{4}[x]/\langle x^{s}-1\rangle$. Assume that $x^{r}-1=\widetilde{f_{1}}\widetilde{f_{2}} \ldots \widetilde{f_{t}}$ and $x^{s}-1=\widetilde{g_{1}}\widetilde{g_{2}} \ldots \widetilde{g_{w}}$ are the factorizations of $x^{r}-1$ and $x^{s}-1$ into irreducible polynomials in ${{\mathbb {Z}}}_{2}[x]$ and ${{\mathbb {Z}}} _{4}[x]$, respectively. The number of non-separable ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes is given by

$$\begin{aligned} \left[ \sum _{i=1}^{2^{t}-1}\left( \sum _{j=0}^{w-1}2^{w-j}\sum _{k}^{{}}\left( 2^{\deg (m_{ijk})}-1\right) \right) \right] , \end{aligned}$$

(2)

where $m_{ijk}=\gcd \left( f_{i},\frac{x^{s}-1}{a_{ijk}}\right) >1$ and $a=a_{ijk}$ is the collection of all polynomials that satisfy the following conditions:

• 1. $f_{i}|\left( \frac{x^{s}-1}{a_{ijk}}l\right) mod\,2$.

• 2. $f_{i}$ is not a factor of $a_{ijk}$ $mod\,2.$

• 3. $a_{ijk}$ has exactly j factors of $x^{s}-1.$

• 4. The sum k runs over all the choices for a satisfying the above conditions.

Proof

Suppose that ${\mathcal {C}}$ is a non-separable ${{\mathbb {Z}}_{2}{{\mathbb {Z}} _{4}}}$-additive cyclic code in $\mathcal {R}_{r,s}$ of the form ${\mathcal {C} }=$ $\langle (f,0),(l,g+2a)\rangle$ where

$$\begin{aligned} l\ne 0,\,f\,|\,\left( x^{r}-1\right) mod\,2,\,a\,|\,g\,|\,\left( x^{s}-1\right) mod\,4 \, \text {and} \,f\,|\,\left( \frac{x^{s}-1}{a}l\right) mod\,2\,\text {with} \,\deg (l)<\deg (f). \end{aligned}$$

We use the following diagram in order to give a clear picture of the proof. In the above theorem, we get the first sum by considering the condition $f\,|x^{r}-1$ for a ${{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}$-additive cyclic code ${ \mathcal {C}}$ and we have the other sums in a similar approach.

If $f=1,$ then l must be 0 and hence the code is separable. Thus f is a polynomial of degree at least 1 satisfying the condition $f\,|\,\left( x^{r}-1\right) .$ This will give $\left( {\begin{array}{c}t\\ 1\end{array}}\right) +\left( {\begin{array}{c}t\\ 2\end{array}}\right) +\cdots +\left( {\begin{array}{c}t \\ t-1\end{array}}\right) +\left( {\begin{array}{c}t\\ t\end{array}}\right) =2^{t}-1$ different choices for f. So f runs over all the factors of $x^{r}-1$ except for 1. That is, $f=f_{i},$ $i\in \{1,2,\ldots ,2^{t}-1\}$. This explains the first sum in Eq. 2. Now we will consider the polynomials g and a. We choose these polynomials among the ones that satisfy $a|\,g|\left( x^{s}-1\right) mod\,4.$

Case 1:

$a=1$. Since $f_{i}\,|\left( \frac{x^{s}-1}{a} l\right) ,$ then $f_{i}\,|\left( x^{s}-1\right) l.$ This will produce $\left( {\begin{array}{c} w\\ 0\end{array}}\right) +\left( {\begin{array}{c}w\\ 1\end{array}}\right) +\left( {\begin{array}{c}w\\ 2\end{array}}\right) +\cdots +\left( {\begin{array}{c}w\\ w\end{array}}\right) =2^{w}$ different choices for g with $a_{{}}\,|\,g_{{}}\,|\,x^{s}-1$.

Case 2:

$a=\widetilde{g_{i_1}}$, $i\in \{1,2,...,w\}$, i.e., a has only one factor of $x^s-1$. Again, since we know that $a_{}\,|\,g_{}\,|\,x^{s}-1,$ then, we have $\left( {\begin{array}{c}w-1\\ 0\end{array}}\right) +\left( {\begin{array}{c}w-1\\ 1\end{array}}\right) +\left( {\begin{array}{c} w-1\\ 2\end{array}}\right) +\cdots +\left( {\begin{array}{c}w-1\\ w-1\end{array}}\right) =2^{w-1}$ different choices for $g_{}.$

Case 3:

$a=\widetilde{g_{i_{1}}}\widetilde{g_{i_{2}}}\ldots \widetilde{g_{i_{j}}},$ i.e., a has exactly j irreducible factors of $x^{s}-1$, $\,2\le j\le w-1$. Similar to the above cases we have $2^{w-j}$ different choices for $g_{{}}.$ It is important to emphasize that a cannot be equal to $x^{s}-1$ since we must have $f_{i}\,|\,\frac{x^{s}-1}{a}l$ with $\deg (l)<\deg (f_{i}).$ So, we take $j<w.$

Note that the polynomial l satisfies the condition (1) in the theorem above. Suppose that $f_{i}$ is a factor of $a_{ijk}$ $mod\,2.$ Then $a_{ijk}=f_{i}T\,mod\,2.$ If $f_{i}\,|\,\left( \frac{x^{s}-1}{f_{i}T}l\right)$ $\text {mod }2$ and since s is odd, then $f_{i}|\,l$ which contradicts the fact that $\deg l<\deg f_{i}.$ Thus, $f_{i}$ is not a factor of $a_{ijk}$ $mod\,2.$ This implies that the polynomial a must satisfy the conditions in the theorem to be one of the generators.

Finally, we will consider the polynomial l. Let $m_{ijk}=\gcd \left( f_{i}, \dfrac{x^{s}-1}{a_{ijk}}\right) .$ Then, $f_{i}=q_{1}m_{ijk}$ and $\dfrac{ x^{s}-1}{a_{ijk}}=q_{2}m_{ijk}$ with $\gcd \left( q_{1},q_{2}\right) =1.$ Since $f_{i}\,|\,\left( \dfrac{x^{s}-1}{a_{ijk}}l\right) ,$

$$\begin{aligned} \frac{x^{s}-1}{a_{ijk}}l= & {} f_{i}M \\ q_{2}m_{ijk}l= & {} q_{1}m_{ijk}M \\ q_{2}l= & {} q_{1}M. \end{aligned}$$

Hence, $q_{1}|\,q_{2}l.$ Since $\gcd \left( q_{1},q_{2}\right) =1,$ $q_{1}|\,l,$ and $l=q_{1}q_{3}=\dfrac{f_{i}}{m_{ijk}}q_{3}.$ Since $\deg l<\deg f_{i}$, $q_{3}$ may be any polynomial of degree less than the degree of $m_{ijk}.$ Hence, there are $2^{\deg \left( m_{ijk}\right) }$ different choices for l. However, if $l=0$ then we get a separable code. Thus, there are $2^{\deg \left( m_{ijk}\right) }-1$ choices for l which produces non-separable codes. Consequently, the number of non-separable ${\mathbb {Z}} _{2}{\mathbb {Z}}_{4}$-additive cyclic codes is

$$\begin{aligned} \left[ \sum _{i=1}^{2^{t}-1}\left( \sum _{j=0}^{w-1}2^{w-j}\sum _{k}^{{}}\left( 2^{\deg (m_{ijk}(x))}-1\right) \right) \right] . \end{aligned}$$

$\square$

Our next result gives the number of ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes for any integer r and any odd integer s. Let $r=2^{v}N$ where N is an odd integer. Then, we know that $\left( x^{r}-1\right) =\left( x^{N}-1\right) ^{2^{v}}=\widetilde{f_{1}}^{2^{v}} \widetilde{f_{2}}^{2^{v}}\ldots \widetilde{f_{t}}^{2^{v}}$ is the factorization of $\left( x^{r}-1\right)$ into powers of irreducible polynomials. The number of binary cyclic codes of length r is $\left( 2^{v}+1\right) ^{t}.$ Based on this fact, our previous results can be applied for any integer r.

Corollary 14

Suppose that $\left( x^{r}-1\right) =\left( x^{N}-1\right) ^{2^{v}}=\widetilde{f_{1}}^{2^{v}}\widetilde{f_{2}} ^{2^{v}}\ldots \widetilde{f_{t}}^{2^{v}}$ is the factorization of $\left( x^{r}-1\right)$ into powers of irreducible polynomials in ${{\mathbb {Z}}} _{2}[x]$ and $x^{s}-1=\widetilde{g_{1}}\widetilde{g_{2}}...\widetilde{g_{w}}$ be the factorization $x^{s}-1$ into irreducible polynomials in ${{\mathbb {Z}}} _{4}[x].$

• 1. The number of separable ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$ -additive cyclic codes is $\left( 2^{v}+1\right) ^{t}3^{w}.$

• 2. If $\left( r,s\right) =1$, then the number of non-separable ${\mathbb {Z }_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes is $\left( 2^{v}+1\right) ^{t}3^{w-1}.$

• 3. If $\left( r,s\right) \ne 1$, then the number of non-separable ${ {\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes is

  $$\begin{aligned} \left[ \sum _{i=1}^{\left( 2^{v}+1\right) ^{t}-1} \left( \sum _{j=0}^{w-1}2^{w-j}\sum _{k}^{{}}\left( 2^{\deg (m_{ijk}(x))}-1\right) \right) \right] . \end{aligned}$$

Proof

The proof follows from Lemma 10, Theorems 12 and 13$\square$

4 Examples

Example 15

Let $r=9$ and $s=7$. Then,

$$\begin{aligned} x^{9}-1= & {} x^{9}-1=(1+x)(1+x+x^{2})(1+x^{3}+x^{6})\text { in}\,{{\mathbb {Z}}} _{2}[x]\text { and} \\ x^{7}-1= & {} (x+3)(x^{3}+2x^{2}+x+3)(x^{3}+3x^{2}+2x+3)\,\text {in}\,{{\mathbb {Z}}} _{4}[x]. \end{aligned}$$

The number of separable ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes is $2^{3}3^{3}=216$. Since $\gcd \left( r,s\right) =1$, the number of non-separable ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes is $2^{3}3^{2}=72$ by Theorem 12. Hence, the total number of ${\mathbb { Z}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes of length $n=r+s=16$ is $216+72=288$.

Example 16

Let $r=7=s$. Then

$$\begin{aligned} x^{7}-1= \;& {} (x-1)(x^{3}+x+1)(x^{3}+x^{2}+1)\,\text{in}\,{{\mathbb {Z}}}_{2}[x]\,\text {and} \\ x^{7}-1= \;& {} (x+3)(x^{3}+2x^{2}+x+3)(x^{3}+3x^{2}+2x+3)\,\text{in}\,{{\mathbb {Z}}}_{4}[x]. \end{aligned}$$

Label the factors of $\left( x^{7}-1\right) \mod 2$ as: $f_{1}=(1+x),\,f_{2}=(1+x+x^{3}),\,f_{3}=(1+x^{2}+x^{3}),\,f_{4}=(1+x)(1+x+x^{3}),\,f_{5}=(1+x)(1+x^{2}+x^{3}),\,f_{6}=(1+x+x^{3})(1+x^{2}+x^{3}),\,f_{7}=x^{7}-1.$ Label the factors of $\left( x^{7}-1\right)$ in ${{\mathbb {Z}}}_{4}[x]$ as

$$\begin{aligned} g_{1}= & {} (3+x),g_{2}=(3+x+2x^{2}+x^{3}),g_{3}=(3+2x+3x^{2}+x^{3}), \\ g_{4}= & {} (3+x)(3+x+2x^{2}+x^{3}),g_{5}=(3+x)(3+2x+3x^{2}+x^{3}), \\ g_{6}= & {} (3+x+2x^{2}+x^{3})(3+2x+3x^{2}+x^{3}),g_{7}=x^{7}-1. \end{aligned}$$

First, let ${\mathcal {C}}$ be a separable ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic code with ${\mathcal {C}}=\langle \left( f,0\right) ,\left( 0,g+2a\right) \rangle$. By Lemma 10, there are $2^{3}3^{3}=216$ separable ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes.

Now, we will find the number of non-separable ${{\mathbb {Z}}_{2}{\ {\mathbb {Z}} _{4}}}$-additive cyclic codes. According to Theorem 13, the number of non-separable ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes with $r=s=7$ is

$$\begin{aligned} \sum _{i=1}^{7}\left( \sum _{j=0}^{2}2^{3-j}\sum _{k}^{{}}\left( 2^{\deg (m_{ijk})}-1\right) \right) , \end{aligned}$$

where the number of choices for the polynomial f is 7. Let $f=(1+x)=f_{1}$. Based on Theorem 13, we have the number of codes for this choice of f to be

$$\begin{aligned} \sum _{j=0}^{2}2^{3-j}\sum _{k}^{{}}\left( 2^{\deg (m_{1jk})}-1\right) . \end{aligned}$$

If $j=0,$ then $a_{1,0,k}$ is the collection of all polynomials that do not contain $f_{1}\mod 2$ and have 0 factors of $x^{7}-1.$ Hence, there is only one choice for $a=1$ and in this case $k=1$ with

$$\begin{aligned} m_{1,0,1}(x)=\gcd \left( 1+x,x^{7}-1\right) =(1+x)=f_{1}(x). \end{aligned}$$

If $j=1,$ then $a_{1,1,k}$ is the collection of all polynomials that do not contain $f_{1}\mod 2$ and have 1 factor of $\left( x^{7}-1\right) \mod 2.$ Hence, there are two choices as $g_{2},\,g_{3}$ and in this case $k=1,2$ with

$$\begin{aligned} m_{1,1,1}= & {} \gcd \left( 1+x,\frac{x^{7}-1}{g_{2}}\right) =(1+x)=f_{1},\text { and} \\ m_{1,1,2}= & {} \gcd \left( 1+x,\frac{x^{7}-1}{g_{3}}\right) =(1+x)=f_{1}. \end{aligned}$$

If $j=2,$ then $a_{1,2,k}$ is the collection of all polynomials that do not contain $f_{1} \mod 2$ and have 2 factors of $x^{7}-1.$ Hence there is only 1 choice as $g_{6}$ and in this case $k=1$ with

$$\begin{aligned} m_{1,2,1}=\gcd \left( 1+x,\frac{x^{7}-1}{g_{6}}\right) =(1+x)=f_{1}. \end{aligned}$$

Thus the number of codes when $f=f_{1}$ is

$$\begin{aligned} 8\left( 2^{1}-1\right) +4\left[ \left( 2^{1}-1\right) +\left( 2^{1}-1\right) \right] +2\left( 2^{1}-1\right) =18. \end{aligned}$$

If $f=f_{2}=(1+x+x^{3}),$ then a similar approach as above will give

$$\begin{aligned} 8\left( 2^{3}-1\right) +4\left[ \left( 2^{3}-1\right) +\left( 2^{3}-1\right) \right] +2\left( 2^{3}-1\right) =126\text { codes.} \end{aligned}$$

If $f=f_{3}=(1+x^{2}+x^{3}),$ then a similar approach as above will give

$$\begin{aligned} 8\left( 2^{3}-1\right) +4\left[ \left( 2^{3}-1\right) +\left( 2^{3}-1\right) \right] +2\left( 2^{3}-1\right) =126\text { codes.} \end{aligned}$$

If $f=(1+x)(1+x+x^{3})=f_{4}$ then a similar approach as above will give

$$\begin{aligned} 8\left[ 2^{4}-1\right] +4\left[ (2^{3}-1)+(2^{1}-1)+(2^{4}-1)\right] +2\left[ (2^{3}-1)+(2^{1}-1)\right] =228\text { codes.} \end{aligned}$$

If $f=(1+x)(1+x^{2}+x^{3})=f_{5}$ then we get the same number of codes as in the case $f=f_{4}$ above. Hence, there are 228 codes with $f=f_{5}.$

If $f=f_{6}=f=(1+x+x^{3})(1+x^{2}+x^{3}),$ then we have

$j=0$. In this case there is only one choice for $a=1$ and $k=1$ with

$$\begin{aligned} m_{6,0,1}=\gcd \left( f_{6},x^{7}-1\right) =f_{6}. \end{aligned}$$

If $j=1$, then there are 3 choices for a and $k=1,2,3$ with

$$\begin{aligned} m_{6,1,1}= & {} \gcd \left( f_{6},\frac{x^{7}-1}{g_{1}}\right) =f_{6} \\ m_{6,1,2}= & {} \gcd \left( f_{6},\frac{x^{7}-1}{g_{2}}\right) =\left( 1+x^{2}+x^{3}\right) \\ m_{6,1,3}= & {} \gcd \left( f_{6},\frac{x^{7}-1}{g_{3}}\right) =\left( 1+x+x^{3}\right) . \end{aligned}$$

If $j=2,$ then there are 2 choices for a and $k=1,2$ with

$$\begin{aligned} m_{6,2,1}= & {} \gcd \left( f_{6},\frac{x^{7}-1}{g_{4}}\right) =\left( 1+x^{2}+x^{3}\right) \\ m_{6,2,2}= & {} \gcd \left( f_{6},\frac{x^{7}-1}{g_{5}}\right) =\left( 1+x+x^{3}\right) . \end{aligned}$$

Hence in this case, the number of codes is

$$\begin{aligned} \sum _{j=0}^{2}2^{3-j}\sum _{k}^{{}}\left( 2^{\deg (m_{i,j,k})}-1\right)= & {} 8 \left[ 2^{6}-1\right] +4\left[ (2^{6}-1)+(2^{3}-1)+(2^{3}-1)\right] \\&+\,2\left[ (2^{3}-1)+(2^{3}-1)\right] \\= & {} 504+308+28=840. \end{aligned}$$

If $f=f_{7}=x^{7}-1,$ then we get

$$\begin{aligned} \sum _{j=0}^{2}2^{3-j}\sum _{k}^{{}}\left( 2^{\deg (m_{i,j,k})}-1\right)= & {} 2^{3}\left[ 2^{7}-1\right] +2^{2}\left[ \left( 2^{6}-1\right) +\left( 2^{4}-1\right) +\left( 2^{4}-1\right) \right] \\&+\,2\left[ \left( 2^{3}-1\right) +\left( 2^{3}-1\right) +\left( 2^{1}-1\right) \right] \\= & {} 1016+372+30=1418\text { codes.} \end{aligned}$$

Therefore, the total number of non-separable ${{\mathbb {Z}}_{2}{{\mathbb {Z}} _{4} }}$-additive cyclic codes when $r=s=7$ is

$$\begin{aligned} 18+126+126+228+228+840+1418=2984. \end{aligned}$$

Example 17

Let $r=9$ and $s=15$. Then,

$$\begin{aligned} x^{9}-1= & {} (1+x)(1+x+x^{2})(1+x^{3}+x^{6})\text { in}\,{{\mathbb {Z}}}_{2}[x]\, \text {and} \\ x^{15}-1= & {} (3+x)(1+x+x^{2})(1+3x+2x^{2}+x^{4})(1+2x^{2}+3x^{3}+x^{4}) \\&(1+x+x^{2}+x^{3}+x^{4})\text { in}\,{{\mathbb {Z}}}_{4}[x]. \end{aligned}$$

Hence, by Lemma 10, the number of separable ${{\mathbb {Z}}_{2}{ {\mathbb {Z}}_{4}}}$-additive cyclic codes for $r=9$ and $s=7$ is $2^{3}3^{5}=1944$. By Theorem 13, the number of non-separable ${ {\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes is

$$\begin{aligned} \sum _{i=1}^{2^{3}-1}\left( \sum _{j=0}^{4}2^{5-j}\sum _{k}^{{}}\left( 2^{\deg (m_{i,j,k})}-1\right) \right) . \end{aligned}$$

Let us label the factors of $x^{9}-1$ as

$$\begin{aligned} f_{1}= & {} (1+x),f_{2}=(1+x+x^{2}),f_{3}=(1+x^{3}+x^{6}), \\ f_{4}= & {} (1+x)(1+x+x^{2}),f_{5}=(1+x)(1+x^{3}+x^{6}), \\ f_{6}= & {} (1+x+x^{2})(1+x^{3}+x^{6}),f_{7}=x^{9}-1, \end{aligned}$$

and label the factors of $x^{15}-1$ as

$$\begin{aligned} \begin{array}{rcl} g_{1} &{}=&{}(3+x),g_{2}=(1+x+x^{2}),g_{3}=(1+3x+2x^{2}+x^{4}), \\ g_{4} &{}=&{}(1+2x^{2}+3x^{3}+x^{4}),g_{5}=(1+x+x^{2}+x^{3}+x^{4}), \\ g_{6} &{}=&{}(3+x)(1+x+x^{2}),g_{7}=(3+x)(1+3x+2x^{2}+x^{4}), \\ \vdots &{}\vdots &{}\vdots \\ g_{30} &{}=&{}(3+x)(1+x+x^{2})(1+3x+2x^{2}+x^{4})(1+2x^{2}+3x^{3}+x^{4}), \\ g_{31} &{}=&{}x^{15}-1. \end{array} \end{aligned}$$

Note that since $\gcd \left( f_{3},x^{15}-1\right) =1$, f cannot be chosen as to be $f_{3}.$ We start calculating the cyclic codes which correspond to $f=f_{1}=1+x.$

If $j=0,$ then $k=1,\,a=1,$ and

$$\begin{aligned} m_{1,0,1}=\gcd \left( 1+x,x^{15}-1\right) =1+x. \end{aligned}$$

If $j=1,$ then, $k\in \{1,2,3,4\}$ and

$$\begin{aligned} m_{1,1,1}=m_{1,1,2}=m_{1,1,3}=m_{1,1,4}=1+x \end{aligned}$$

If $j=2,$ then, $k\in \{1,2,3,4,5,6\}$ and

$$\begin{aligned} m_{1,2,1}=m_{1,2,2}=\cdots =m_{1,2,6}=1+x \end{aligned}$$

For $j=3,$ then, $k\in \{1,2,3,4\}$ and

$$\begin{aligned} m_{1,3,1}=m_{1,3,2}=m_{1,3,3}=m_{1,3,4}=1+x \end{aligned}$$

Finally, for $j=4,$

$$\begin{aligned} m_{1,4,1}= & {} \gcd \left( f_{1},\frac{x^{15}-1}{a_{1,4,1}(x)}\right) =1+x, \text { where} \\ a_{1,4,1}= & {} (1+x+x^{2})(1+3x+2x^{2}+x^{4})(1+2x^{2}+3x^{3}+x^{4})(1+x+x^{2}+x^{3}+x^{4}). \end{aligned}$$

Consequently, we have

$$\begin{aligned}&32\cdot (2^{1}-1)+16\cdot [\underbrace{(2^{1}-1)+\cdots +(2^{1}-1)} _{\text {4 times}}\ ]+8\cdot [\underbrace{(2^{1}-1)+\cdots +(2^{1}-1)} _{\text {6 times}}\ ] \\&\quad +4\cdot [\underbrace{(2^{1}-1)+\cdots +(2^{1}-1)}_{\text {4 times}}\ ]+2\cdot (2^{1}-1)=32+64+48+16+2=162 \end{aligned}$$

${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes for $f=f_{1}=1+x.$ If we take $f=f_{2}$, then we have

$$\begin{aligned}&32\cdot (2^{2}-1)+16\cdot [\underbrace{(2^{2}-1)+\cdots +(2^{2}-1)} _{\text {4 times}}\ ]+8\cdot [\underbrace{(2^{2}-1)+\cdots +(2^{2}-1)} _{\text {6 times}}\ ] \\&\quad +4\cdot [\underbrace{(2^{2}-1)+\cdots +(2^{2}-1)}_{\text {4 times}}\ ]+2\cdot (2^{2}-1)=96+192+144+48+6=486\text { codes.} \end{aligned}$$

For $f=f_{4},$ by applying Theorem 13, we get

$$\begin{aligned}&32\cdot (2^{3}-1)+16\cdot [(2^{2}-1)+(2^{1}-1)+\underbrace{ (2^{3}-1)+\cdots +(2^{3}-1)}_{\text {3 times}}\ ] \\&\quad +8\cdot [\underbrace{(2^{2}-1)+\cdots +(2^{2}-1)}_{\text {3 times}}+ \underbrace{(2^{1}-1)+\cdots +(2^{1}-1)}_{\text {3 times}}+\underbrace{ (2^{3}-1)+\cdots +(2^{3}-1)}_{\text {3 times}}\ ] \\&\quad +4\cdot [\underbrace{(2^{2}-1)+\cdots +(2^{2}-1)}_{\text {3 times}}+ \underbrace{(2^{1}-1)+\cdots +(2^{1}-1)}_{\text {3 times}}+(2^{3}-1)] \\&\quad +2\cdot [(2^{2}-1)+(2^{1}-1)]=32\cdot 7+16\cdot 25+8\cdot 33+4\cdot 19+2\cdot 4=972 \end{aligned}$$

${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes. Furthermore, we calculate the number of ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes as

$$\begin{aligned} \text {for }f_{5}\longrightarrow & {} 162, \\ \text {for }f_{6}\longrightarrow & {} 486, \\ \text {for }f_{7}\longrightarrow & {} 972. \end{aligned}$$

Finally the total number of non-separable additive cyclic code ${\mathcal {C}} \subseteq {{\mathbb {Z}}}_{2}[x]/\langle x^{9}-1\rangle \times {{\mathbb {Z}}} _{4}[x]/\langle x^{15}-1\rangle$ is

$$\begin{aligned} 162+486+972+162+486+972=3240, \end{aligned}$$

and the total number of ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes is $1944+3240=5184$.

5 The number of ${{\mathbb {Z}}_{p}{{\mathbb {Z}}_{p^{2}}}}$-additive cyclic codes

Let p be any prime number, r is any positive integer and s is any positive integer relatively prime with p. In this case, the ring ${ {\mathbb {Z}}}_{p^{2}}[x]/\langle x^{s}-1\rangle$ will be a principal ideal ring. In this section, we are interested to generalize our previous results and find formulas for the number of separable and non-separable ${{\mathbb {Z}} _{p}{{\mathbb {Z}}_{p^{2}}}}$-additive cyclic codes of length $n=r+s.$ In [5], Borges et. al. studied the structure of ${{\mathbb {Z}}_{p^{r}}{ {\mathbb {Z}}_{p^{s}}}}$-additive cyclic codes. Hence, based on this work if ${ \mathcal {C}}$ is an additive cyclic code over ${{\mathbb {Z}}_{p}{{\mathbb {Z}} _{p^{2}}}}$ of length $n=r+s$, then ${\mathcal {C}}$ is generated by

$$\begin{aligned} {\mathcal {C}}=\langle \left( f,0\right) ,\left( l,g+pa\right) \rangle \end{aligned}$$

where $f|(x^{r}-1)\,mod\,p$, $a|g|(x^{s}-1)\,mod\,p^{2},$ l is a polynomial over ${\mathbb {Z}}_{p}[x]$ satisfying $\deg (l)<\deg (f),$ and $f|\dfrac{ x^{s}-1}{a}l.$ As in the case of ${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes, the above generators are unique. Moreover, the code ${\mathcal {C}}$ is separable if and only if the polynomial $l=0.$

Lemma 18

The number of separable ${{\mathbb {Z}}_{p}{{\mathbb {Z}}_{p^{2}} }}$-additive cyclic codes of length $n=r+s$ is ${(p^v+1)}^{w_{1}}3^{w_{2}}$ where $\left( x^{r}-1\right)=\left( x^{N}-1\right)^{p^v}, \text{ }w_{1}$ is the number of irreducible factors of $\left( x^{r}-1\right) \text { mod }p$ and $w_{2}$ is the number of irreducible factors of $\left( x^{s}-1\right) \text {mod }p^{2}.$

Proof

The proof is similar to the proof of Lemma 10. $\square$

In fact, as we have showed in the proof of Theorem 13, the number of ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$-additive cyclic codes are determined only by the generator polynomials of the code ${\mathcal {C}}$. Hence, the same proof can easily be applied to give the exact number of ${{\mathbb {Z}}_{p}{{\mathbb {Z}} _{p^{2}}}}$-additive cyclic codes of length $n=r+s.$

Corollary 19

Let ${\mathcal {C}}=\langle \left( f,0\right) ,\left( l,g+pa\right) \rangle$ be a non-separable cyclic code in ${{\mathbb {Z}}} _{p}[x]/\langle x^{r}-1\rangle \times {{\mathbb {Z}}}_{p^{2}}[x]/\langle x^{s}-1\rangle \text{ with } (r,s)\neq 1$. Assume that $x^{r}-1=(\widetilde{f_{1}}\widetilde{f_{2}} \ldots \widetilde{f_{t}})^{p^v}$ and $x^{s}-1=\widetilde{g_{1}}\widetilde{g_{2}} \ldots \widetilde{g_{w}}$ are the factorizations of $x^{r}-1$ and $x^{s}-1$ into irreducible polynomials in ${{\mathbb {Z}}}_{p}[x]$ and ${{\mathbb {Z}}} _{p^{2}}[x]$, respectively. The number of ${{\mathbb {Z}}_{p}{{\mathbb {Z}}_{p^{2}} }}$-additive cyclic codes is given by

$$\begin{aligned} \left[ \sum _{i=1}^{({p^v+1})^{t}-1}\left( \sum _{j=0}^{w-1}2^{w-j}\sum _{k}^{{}}\left( p^{\deg (m_{ijk})}-1\right) \right) \right] , \end{aligned}$$

where $m_{ijk}=\gcd \left( f_{i},\frac{x^{s}-1}{a_{ijk}}\right) >1$ and $a=a_{ijk}$ is the collection of all polynomials that satisfy the following conditions:

• 1. $f_{i}|\left( \frac{x^{s}-1}{a_{ijk}}l\right) mod\,p$.

• 2. $f_{i}$ is not a factor of $a_{ijk}$ $mod\,p.$

• 3. $a_{ijk}$ has exactly j factors of $x^{s}-1.$

• 4. The sum k runs over all the choices for a satisfying the above conditions.

Proof

The proof of this corollary is very similar to the proof of Theorem 13. So we skip it. $\square$

Example 20

Let ${\mathcal {C}}$ be a ${{\mathbb {Z}}}_{3}{{\mathbb {Z}}}_{9}$-additive cyclic code in ${\mathbb {\ Z}}_{3}[x]/\langle x^{7}-1\rangle \times {{\mathbb {Z}}} _{9}[x]/\langle x^{7}-1\rangle$. Hence, $p=3,\,r=7=s.$ Therefore,

$$\begin{aligned} x^{7}-1= & {} (2+x)(1+x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6})\,in\,{{\mathbb {Z}}}_{3}[x]\, \text {and} \\ x^{7}-1= & {} (8+x)(1+x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6})\,in\,{{\mathbb {Z}}}_{9}[x]. \end{aligned}$$

Label the factors of $\left( x^{7}-1\right) \mod 3$ as: $f_{1}=(2+x),\,f_{2}=(1+x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}),$ and $f_{3}=\left( x^{7}-1\right) .$ Label the factors of $\left( x^{7}-1\right)$ in ${\mathbb { Z}}_{9}[x]$ as: $g_{1}=(8+x),\,g_{2}=(1+x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}),$ and $g_{3}=\left( x^{7}-1\right)$.

First, let ${\mathcal {C}}$ be a separable ${{\mathbb {Z}}_{3}{{\mathbb {Z}}_{9}}}$-additive cyclic code with ${\mathcal {C}}=\langle \left( f,0\right) ,\left( 0,g+3a\right) \rangle$. By Lemma 18, there are $2^{3}3^{3}=216$ separable ${{\mathbb {Z}}_{3}{{\mathbb {Z}}_{9}}}$-additive cyclic codes.

Based on Corollary 19, the number of non-separable ${ {\mathbb {Z}}_{3}{{\mathbb {Z}}_{9}}}$-additive cyclic codes with $r=s=7$ is

$$\begin{aligned} \sum _{i=1}^{3}\left( \sum _{j=0}^{1}2^{2-j}\sum _{k}^{{}}\left( 3^{\deg (m_{ijk})}-1\right) \right) , \end{aligned}$$

where the number of choices for the polynomial f is 3. First, take $f=(2+x)=f_{1}$. Hence, the number of codes for this choice of f is

$$\begin{aligned} \sum _{j=0}^{1}2^{2-j}\sum _{k}^{{}}\left( 3^{\deg (m_{1jk})}-1\right) . \end{aligned}$$

If $j=0,$ then $a_{1,0,k}$ is the collection of all polynomials that do not contain $f_{1}\mod 3$ and have 0 factors of $x^{7}-1.$ Hence, there is only one choice for $a=1$ and in this case $k=1$ with

$$\begin{aligned} m_{1,0,1}(x)=\gcd \left( 2+x,x^{7}-1\right) =(2+x)=f_{1}(x). \end{aligned}$$

If $j=1,$ then $a_{1,1,k}$ is the collection of all polynomials that do not contain $f_{1}\mod 3$ and have 1 factor of $\left( x^{7}-1\right) \mod 3.$ Hence, there is only one choice which is $g_{2}$ and in this case $k=1$ with

$$\begin{aligned} m_{1,1,1}=\gcd \left( 2+x,\dfrac{x^{7}-1}{g_{2}}\right) =(2+x)=f_{1}. \end{aligned}$$

Thus the number of codes when $f=f_{1}$ is

$$\begin{aligned} 4\left( 3^{1}-1\right) +2\left( 3^{1}-1\right) =12. \end{aligned}$$

Now, if $f=f_{2}=(1+x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6})$ then similarly we have

$$\begin{aligned} 4\left( 3^{6}-1\right) +2\left( 3^{6}-1\right) =4368\text { codes.} \end{aligned}$$

If $f=f_{3}=x^{7}-1,$ then we get

$$\begin{aligned} \sum _{j=0}^{1}2^{2-j}\sum _{k}^{{}}\left( 3^{\deg (m_{i,j,k})}-1\right)= & {} 2^{2}\left[ 3^{7}-1\right] +2\left[ \left( 3^{6}-1\right) +\left( 3^{1}-1\right) \right] \\= & {} 8744+1460=10204\text { codes.} \end{aligned}$$

Therefore, the total number of non-separable ${{\mathbb {Z}}_{3}{{\mathbb {Z}}_{9}} }$-additive cyclic codes when $r=s=7$ is

$$\begin{aligned} 12+4368+10204=14584. \end{aligned}$$

Note that the number of non-separable ${{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}$-additive cyclic codes for $r=s=7$ is 2984.

6 Conclusion

${{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}$-additive cyclic codes were studied recently by many researchers [1, 3, 4]. In this paper, we focused on counting the exact number of ${{\mathbb {Z}}_{2}{ {\mathbb {Z}}_{4}}}$-additive cyclic codes of length $n=r+s,$ for any positive integer r and any positive odd integer s. Moreover, we provided formulas which give the exact number of separable and non-separable ${{\mathbb {Z}}_{2}{ {\mathbb {Z}}_{4}}}$-additive cyclic codes. We then generalized our results to find the number of separable and non-separable ${{\mathbb {Z}}_{p}{{\mathbb {Z}} _{p^{2}}}}$-additive cyclic codes of length $n=r+s,$ for any prime p,  any positive integer r and any positive integer s where $\gcd \left( p,s\right) =1$.