1 Introduction

Codes over finite rings gained researchers interest after Hammons et al. developed binary images under a Gray map of linear cyclic codes over \({\mathbb {Z}}_{4}\) in [23]. For instance, the class of finite rings of the form \({\mathbb {F}}_{p^m} + u {\mathbb {F}}_{p^m}\) has been widely used as alphabets of certain constacyclic codes. In 2010, Dinh [10] determined the algebraic structures of constacyclic codes of length \(p^s\) over \({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}\) and their dual codes. In 2012, Dinh et al.  [8] gave the algebraic structures of constacyclic codes of length \(2p^s\) over \({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}\) and their dual codes. In 2018, Dinh et al. [11] investigated the algebraic structures of negacyclic codes of length \(4p^s\) over \({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}\) and their dual codes. In addition, constacyclic codes of length \(4p^s\) over \(\mathbb F_{p^m}+u{\mathbb {F}}_{p^m}\) are investigated in [12] and [13]. Moreover, Dinh et al. [14] provided all constacyclic codes of length \(3p^s\) over \(\mathbb F_{p^m}+u{\mathbb {F}}_{p^m}\).

It is well known that the ideals of \(\frac{{\mathbb {Z}}_{q}[x]}{\langle x^n-1 \rangle }\) are same as the cyclic codes over \({\mathbb {Z}}_{q}\) (see, for example, [29]). The researchers in [21, 28] introduced the additive cyclic codes, which are a special case of generalized quasi-cyclic codes. Moreover, Borges et al. [6] investigated \({\mathbb {Z}}_{2}{\mathbb {Z}}_{4}\)-additive codes which were later extended by Abualrub et al. for additive cyclic codes in [1] and Gao et al. for double cyclic codes over \({\mathbb {Z}}_{4}\) [20]. These works were further extended to \({\mathbb {Z}}_{2} {\mathbb {Z}}_{2} {\mathbb {Z}}_{4}\) by Wu et al. [30] and \({\mathbb {Z}}_{2} {\mathbb {Z}}_{4} \mathbb Z_{8}\)-additive cyclic codes by Aydogdu and Gursoy [3].

From the application point of view, it is necessary to study the asymptotic properties of these cyclic codes, because the rate of cyclic codes is used to measure the proportion of the number of information coordinates of a family of cyclic codes to the total number of coordinates, and the relative minimum distance of cyclic codes is used to measure error-correcting capability. In particular, it would be interesting to find out whether cyclic codes are asymptotically good, i.e., whether the rate and the relative minimum distance of cyclic codes are both positively bounded from below when the length of the code goes to infinity. This has been an open problem for quite fifty-five years as can be seen in [2]. In 2006, Martínez-Pérez and Willems, discussed in [25] whether the class of cyclic codes is asymptotically good. In 2015, Fan et al. showed that there exist numerous asymptotically good quasi-abelian codes attaining the GV-bound in [17], and in [15], they proved that quasi-cyclic codes of index \(1\frac{1}{2}\) are asymptotically good. Moreover, in 2016, they also showed that the quasi-cyclic codes of index \(1\frac{1}{3}\) are asymptotically good in [16]. Further, in [27], Shi et al. proved that there are additive cyclic codes that are asymptotically good.

In 2019, Fan and Liu proved that \({\mathbb {Z}}_{2} \mathbb Z_{4}\)-additive cyclic codes are asymptotically good by using a Bernoulli random variable in [18]. Few other works such as [31, 32] generalised [18] for \({\mathbb {Z}}_{p} {\mathbb {Z}}_{p^s}\) and \({\mathbb {Z}}_{p^r} {\mathbb {Z}}_{p^s}\) where p is any prime number and \(1\le r <s .\) Recently, Gao et al. [19] investigated the \({\mathbb {Z}}_{4}\)-double cyclic codes and found them asymptotically good.

The above mentioned literature is concerned with doubly additive cyclic codes. In this paper, we work on \({\mathbb {Z}}_{4}\mathbb Z_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes; we show that these codes are asymptotically good.

The paper is organized as follows: In Sect. 2, we discuss the algebraic structure of \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\mathbb Z_{4}\)-additive codes over \({\mathbb {Z}}_{4}\)-module. Then, we identify \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes of length \(n= \alpha + \beta + \gamma\) with \(\mathbb Z_{4}[x]\)-submodules of \({\mathbb {R}}_{\alpha } \times {\mathbb {R}}_{\beta } \times {\mathbb {R}}_{\gamma },\) where \({\mathbb {R}}_{\alpha } = \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^{\alpha } - 1 \rangle }\), \({\mathbb {R}}_{\beta } = \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^{\beta } - 1 \rangle }\) and \({\mathbb {R}}_{\gamma } = \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^{\gamma } - 1 \rangle }\). In Sect. 3, we define a class of cyclic codes \(C_{abc}\) as \(\mathbb Z_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes in \(\mathbb Z_{4}^{km} \times {\mathbb {Z}}_{4}^{lm} \times {\mathbb {Z}}_{4}^{tm}\) as

$$\begin{aligned} C_{abc} = \{ (f(x)a(x), f(x) b(x), f(x) c(x)) \in {\mathbb {R}}_{km} \times {\mathbb {R}}_{lm} \times {\mathbb {R}}_{tm} ~|~ f(x) \in {\mathbb {R}}_{kltm} \}, \end{aligned}$$

which can be seen as \({\mathbb {Z}}_{4}[x]\)-submodules of \({\mathbb {R}}_{km} \times {\mathbb {R}}_{lm} \times {\mathbb {R}}_{tm}\), for \((a(x), b(x), c(x)) \in {\mathbb {R}}_{km} \times {\mathbb {R}}_{lm} \times {\mathbb {R}}_{tm}\). Then we proved that \(C_{abc}\) is an \({\mathbb {R}}_{kltm}\)-submodule of \({\mathbb {R}}_{km} \times {\mathbb {R}}_{lm} \times {\mathbb {R}}_{tm}\) generated by (a(x), b(x), c(x)). In Sect. 4, we study the asymptotic properties of this class of cyclic codes using a Bernoulli random variable  \(Y_{f}\), which implies that \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\mathbb Z_{4}\)-additive cyclic codes are asymptotically good. In Sect. 5, we conclude the paper with some open directions for future work.

2 Preliminary

Consider the quaternary ring \({\mathbb {Z}}_{4}\) and define a Gray map \(\psi : {\mathbb {Z}}_{4} \longrightarrow {\mathbb {Z}}_{2}^{2}\) given as \(\psi (0)=(0,0),~\psi (1)=(0,1),~\psi (2)=(1,1),~\psi (3)=(1,0)\). It can also be extended for \({\mathbb {Z}}_{4}^{n}\) to \({\mathbb {Z}}_{2}^{2n}\), where n is an odd positive integer, given by

$$\begin{aligned} (x_{0},x_{1},\dots ,x_{n-1}) \longmapsto (\psi (x_{0}),\psi (x_{1}),\dots ,\psi (x_{n-1})). \end{aligned}$$

\({\mathbb {Z}}_{4}\) is equipped the Lee weight and the Gray image is equipped the Hamming weight. The Hamming weight is the number of non zero coordinates of the Gray image. The relation between the Lee weight \(wt_{L}\) and the Hamming weight \(wt_{H}\) for each element \(x_{i} \in {\mathbb {Z}}_{4},~ i= 0, \dots ,3\) is given by

$$\begin{aligned}wt_{L}(x_{i}) = wt_{H} ( \psi (x_{i})).\end{aligned}$$

For example, \(wt_{L}(0)=0, ~ wt_{L}(1)=1, ~wt_{L}(2)=2,~ wt_{L}(3)=1\). Therefore, for \(x=(x_{0},x_{1},\dots ,x_{n-1}) \in {\mathbb {Z}}_{4}^n,\) the Lee weight \(wt_L(x)\) can be defined as

$$\begin{aligned} wt_{L}(x) = wt_{H}(\psi (x) ) = \sum _{j=0}^{n-1} wt_{L}(x_{j}).\end{aligned}$$

The Lee distance between any two elements \(x=(x_{0},x_{1},\dots ,x_{n-1})\) and \(y=(y_{0},y_{1},\dots ,y_{n-1})\) in \({\mathbb {Z}}_4^{n}\) is defined as

$$\begin{aligned} d_{L}(x,y) = \sum _{j=0}^{n-1} wt_{L}(x_{j}-y_{j}).\end{aligned}$$

Now, it can be seen that \(\psi\) is a distance preserving map from \(({\mathbb {Z}}_{4}^n, d_{L})\) to \(({\mathbb {Z}}_{2}^{2n}, d_{H})\). Let C be a nonzero code of length n in \({\mathbb {Z}}_{4}^n\) then the minimum Lee weight \(wt_{L}(C)\) is defined as

$$\begin{aligned}wt_{L}(C)= \min \{ wt_{L}({x})~|~ {x} \in C, {x}\ne 0 \}.\end{aligned}$$

The minimum Lee distance of the code C is defined as

$$\begin{aligned}d_{L}(C) = \min \{wt_{L}({x-y})~|~ x,y \in C, {x} \ne {y} \}. \end{aligned}$$

Define

$$\begin{aligned}&{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4} =\{(\mu ,\nu ,\rho ) ~|~ \mu ,\nu ,\rho \in {\mathbb {Z}}_{4} \}, \\&\quad {\mathbb {Z}}_{4}^{\alpha } \times {\mathbb {Z}}_{4}^{\beta } \times {\mathbb {Z}}_{4}^{\gamma } = \{ (a,b,c) \in {\mathbb {Z}}_{4}^{\alpha } \times {\mathbb {Z}}_{4}^{\beta } \times {\mathbb {Z}}_{4}^{\gamma } ~|~ a \in {\mathbb {Z}}_{4}^{\alpha },b \in {\mathbb {Z}}_{4}^{\beta }, c \in {\mathbb {Z}}_{4}^{\gamma } \}, \end{aligned}$$

where \(\alpha , \beta ~\text {and}~ \gamma\) are positive integers. Thus, the set \({\mathbb {Z}}_{4}^{\alpha } \times {\mathbb {Z}}_{4}^{\beta } \times {\mathbb {Z}}_{4}^{\gamma }\) is an abelian group. For \((a,b,c) \in {\mathbb {Z}}_{4}^{\alpha } \times \mathbb Z_{4}^{\beta } \times {\mathbb {Z}}_{4}^{\gamma }\) and \(d \in \mathbb Z_{4},\) we define a multiplication operation \(\cdot\) as

$$\begin{aligned} d\cdot (a,b,c) = (d a\pmod 4,~ d b \pmod 4,~ d c \pmod 4).\end{aligned}$$

So, the set \({\mathbb {Z}}_{4}^{\alpha } \times \mathbb Z_{4}^{\beta } \times {\mathbb {Z}}_{4}^{\gamma }\) is closed with respect to multiplication for any \(d \in {\mathbb {Z}}_{4}\). Hence the abelian group \({\mathbb {Z}}_{4}^{\alpha } \times {\mathbb {Z}}_{4}^{\beta } \times {\mathbb {Z}}_{4}^{\gamma }\) is a \({\mathbb {Z}}_{4}\)-module. We now present some definitions related to this module \({\mathbb {Z}}_{4}^{\alpha } \times {\mathbb {Z}}_{4}^{\beta } \times {\mathbb {Z}}_{4}^{\gamma }\).

Definition 2.1

A subset C of \({\mathbb {Z}}_{4}^{\alpha } \times {\mathbb {Z}}_{4}^{\beta } \times {\mathbb {Z}}_{4}^{\gamma }\) is called a \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive code of length \(n=\alpha +\beta +\gamma ,\) if C is a subgroup of \({\mathbb {Z}}_{4}^{\alpha } \times {\mathbb {Z}}_{4}^{\beta } \times {\mathbb {Z}}_{4}^{\gamma }\), where the first \(\alpha\) coordinates of C are entries from \({\mathbb {Z}}_{4}\), which is also true for the next \(\beta\) and the last \(\gamma\) coordinates.

Definition 2.2

Let \(C \subseteq {\mathbb {Z}}_{4}^{\alpha } \times \mathbb Z_{4}^{\beta } \times {\mathbb {Z}}_{4}^{\gamma }\) be a \(\mathbb Z_{4}\)-additive code then C is called a \({\mathbb {Z}}_{4}\mathbb Z_{4}{\mathbb {Z}}_{4}\)-additive cyclic code of block length \((\alpha ,\beta , \gamma )\), if whenever \((a_{0}, \dots ,a_{\alpha -1},b_{0}, \dots ,\) \(b_{\beta -1},c_{0}, \dots ,c_{\gamma -1} )\) is in C, then \((a_{\alpha -1}, a_{0},\dots ,a_{\alpha -2},b_{\beta -1}, b_{0}, \dots , b_{\beta -2}, c_{\gamma -1}, c_{0}, \dots ,c_{\gamma -2} )\) is also in C.

Let \({\mathbb {R}}_{\alpha } = \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^{\alpha } - 1 \rangle }\), \({\mathbb {R}}_{\beta } = \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^{\beta } - 1 \rangle }\), \({\mathbb {R}}_{\gamma } = \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^{\gamma } - 1 \rangle }\) and define a map

$$\begin{aligned} \phi : {\mathbb {Z}}_{4}^{\alpha } \times {\mathbb {Z}}_{4}^{\beta } \times {\mathbb {Z}}_{4}^{\gamma } \longrightarrow {\mathbb {R}}_{\alpha } \times {\mathbb {R}}_{\beta } \times {\mathbb {R}}_{\gamma } \end{aligned}$$

given by

$$\begin{aligned} (a, b, c) \longmapsto (a(x), b(x), c(x)) \end{aligned}$$

where \(a(x)= a_{0} + a_{1}x + \dots + a_{\alpha -1}x^{\alpha -1}, ~b(x)= b_{0} + b_{1}x + \dots + b_{\beta -1}x^{\beta -1}, ~c(x)= c_{0} + c_{1}x + \dots + c_{\gamma -1}x^{\gamma -1}\). Thus, using the map \(\phi\) it can be seen clearly that \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes are \({\mathbb {Z}}_{4}[x]\)-submodules of \({\mathbb {R}}_{\alpha } \times {\mathbb {R}}_{\beta } \times {\mathbb {R}}_{\gamma }\).

Note that if C is a \({\mathbb {Z}}_{4}\)-free, then there exists \({\mathbb {Z}}_{4}\)-free basis for C. If cardinality of a \(\mathbb Z_{4}\)-free basis set is r then the rank of C is r. The rate of C is defined as \(R(C)=\frac{\text {rank} (C)}{n}\) and the relative distance of C is defined as \(\Delta (C)=\frac{d_{L}(C)}{n}.\)

Definition 2.3

[25] If there exists a sequences of \({\mathbb {Z}}_{4}\)-free \(\mathbb Z_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes \(\{C_{i}\}_{i=0}^\infty\) of length \(n_{i}\), where \({n_{i}\rightarrow \infty }\) and if the relative distance and rate of \(C_{i}\) are positively bounded from below, then these class of \({\mathbb {Z}}_{4}\mathbb Z_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes are said to be asymptotically good.

3 A class of \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes

Let \({\mathbb {R}}_{km} = \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^{km} - 1 \rangle }\), \({\mathbb {R}}_{lm} = \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^{lm} - 1 \rangle }\), \({\mathbb {R}}_{tm} = \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^{tm} - 1 \rangle }\) and \({\mathbb {R}}_{kltm} = \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^{kltm} - 1 \rangle }\), where m, k, l, t are positive integers such that \(\gcd (m,4)=1\) and k, l, t, 4 are pairwise co-prime positive integers. It is easy to see that \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes in \({\mathbb {Z}}_{4}^{km} \times {\mathbb {Z}}_{4}^{lm} \times \mathbb Z_{4}^{tm}\) are \({\mathbb {Z}}_{4}[x]\)-submodules of \({\mathbb {R}}_{km} \times {\mathbb {R}}_{lm} \times {\mathbb {R}}_{tm}\), for \((a(x), b(x), c(x)) \in {\mathbb {R}}_{km} \times {\mathbb {R}}_{lm} \times {\mathbb {R}}_{tm}\).

For any \(f(x)\in {\mathbb {Z}}_{4}[x]\) and \((a(x), b(x), c(x))\in {\mathbb {R}}_{km}\times {\mathbb {R}}_{lm}\times {\mathbb {R}}_{tm}\), we define the scalar multiplication, denoted by \(\star\), as follows \(f(x)\star (a(x), b(x), c(x))=(f(x)a(x)\mod (x^{km}-1), f(x)b(x)\mod (x^{lm}-1), f(x)c(x)\mod (x^{tm}-1)).\) For convenience, we write it as

$$\begin{aligned} f(x)\star (a(x), b(x), c(x))=(f(x)a(x), f(x)b(x), f(x)c(x)). \end{aligned}$$

Clearly, \({\mathbb {R}}_{km}\times {\mathbb {R}}_{lm}\times {\mathbb {R}}_{tm}\) is closed under the usual addition and scalar multiplication \(\star\) of \({\mathbb {R}}_{kltm} ={{\mathbb {Z}}_{4}[x]}/{ \langle x^{kltm} - 1 \rangle }\). Let

$$\begin{aligned} C_{abc} = \{ (f(x)a(x), f(x) b(x), f(x) c(x)) \in {\mathbb {R}}_{km} \times {\mathbb {R}}_{lm} \times {\mathbb {R}}_{tm} ~|~ f(x) \in {\mathbb {R}}_{kltm} \}, \end{aligned}$$

then \(C_{abc}\) is an \({\mathbb {R}}_{kltm}\)-submodule of \({\mathbb {R}}_{km} \times {\mathbb {R}}_{lm} \times {\mathbb {R}}_{tm}\) generated by (a(x), b(x), c(x)),  i.e., \(C_{abc}\) is a \(\mathbb Z_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic code generated by (a(x), b(x), c(x)).

We have the following lemma.

Lemma 3.1

Let \(C_{abc}\) be a \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\mathbb Z_{4}\)-additive cyclic code with the generator polynomial \(F(x)= (a(x), b(x), c(x)) \in {\mathbb {R}}_{km} \times {\mathbb {R}}_{lm} \times {\mathbb {R}}_{tm},\) where a(x), b(x) and  c(x) are \(\mathbb Z_{4}[x]\)-monic polynomials. Let

$$\begin{aligned}h(x)={{\,\mathrm{lcm}\,}}\left\{ \frac{x^{km}-1}{g_{1}(x)}, \frac{x^{lm}-1}{g_{2}(x)}, \frac{x^{tm}-1}{g_{3}(x)}\right\} \end{aligned}$$

be a monic parity-check polynomial of \(C_{abc}\) with degree \(h_{0},\) where \(g_{1}(x) = \gcd (a(x), x^{km}-1),~g_{2}(x) = \gcd (a(x), x^{lm}-1),~ \text {and}~ g_{3}(x) = \gcd (a(x), x^{tm}-1),\) then \(C_{abc}\) can be generated by the set \(\{ F(x), xF(x), \dots , x^{h_{0}-1}F(x)\}.\)

Proof

Let \(f(x) \in C_{abc}\), i.e., \(f(x)=v(x)F(x)\), where \(v(x) \in {\mathbb {Z}}_{4}[x]\). Since h(x) is monic, there exist polynomials \(p(x), r(x) \in {\mathbb {Z}}_{4}[x]\) such that

$$\begin{aligned}v(x) = p(x)h(x) +r(x) \end{aligned},$$

where \(\deg r(x) < \deg h(x)\) or \(r(x)=0\). Therefore,

$$\begin{aligned}f(x)=v(x)F(x)=( p(x)h(x) +r(x))F(x) = p(x)h(x)F(x) +r(x)F(x).\end{aligned}$$

Now since,

$$\begin{aligned}h(x)= {{\,\mathrm{lcm}\,}}\left\{ \frac{x^{km}-1}{g_{1}(x)}, \frac{x^{lm}-1}{g_{2}(x)}, \frac{x^{tm}-1}{g_{3}(x)} \right\} ,\end{aligned}$$

then there exist three polynomials \(d_{1}(x),~ d_{2}(x)\) and \(d_{3}(x)\) such that

$$\begin{aligned}h(x)=d_{1}(x)\left( \frac{x^{km}-1}{g_{1}(x)}\right) \text { or } h(x)=d_{2}(x)\left( \frac{x^{lm}-1}{g_{2}(x)}\right) \text { or }h(x)=d_{3}(x)\left( \frac{x^{tm}-1}{g_{3}(x)}\right) .\end{aligned}$$

It is also given that \(g_{1}(x) = \text {gcd}(a(x), x^{km}-1),~g_{2}(x) = \text {gcd}(a(x), x^{lm}-1) \text { and }g_{3}(x) = \text {gcd}(a(x), x^{tm}-1),\) there exist three polynomials \(e_{1}(x),~ e_{2}(x)\) and \(e_{3}(x)\) such that \(a(x)=e_{1}(x)g_{1}(x),~ b(x)=e_{2}(x)g_{2}(x) \text { and } c(x)=e_{3}(x)g_{3}(x).\) Therefore, \(h(x)F(x)=0\) in \({\mathbb {R}}_{km} \times {\mathbb {R}}_{lm} \times {\mathbb {R}}_{tm}.\) Consequently, \(f(x)=r(x)F(x).\) Let

$$\begin{aligned} f(x)&= (r_{0} +r_{1}x + \dots + r_{h_{0}-1} x^{h_{0}-1})F(x)\\&=r_{0}F(x) +r_{1}xF(x) + \dots + r_{h_{0}-1} x^{h_{0}-1}F(x), \end{aligned}$$

which implies that f(x) can be expressed as a \(\mathbb Z_{4}\)-linear combination of the elements \(F(x),~ xF(x), \dots , x^{h_{0}-1}F(x)\). This proves the lemma. \(\square\)

Lemma 3.2

[5] Let \(C= \langle f(x) \rangle\) be a \({\mathbb {Z}}_{4}\)-cyclic code of length m. Then C is \({\mathbb {Z}}_{4}\)-free if and only if there exists a polynomial \(q(x) \in {\mathbb {Z}}_{4}[x]\) such that \(f(x)=q(x)g(x)\) and \(C= \langle g(x) \rangle\), where \(g(x)|(x^{m}-1)\) and \(\gcd \left( q(x),~ \frac{x^m-1}{g(x)}\right) =1\).

Now by Lemmas 3.1 and 3.2, we get the following result.

Proposition 3.3

Let \(C_{abc}\) be a \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\mathbb Z_{4}\)-additive cyclic code with the generator polynomial \(F(x)= (a(x), b(x), c(x)) \in {\mathbb {R}}_{km} \times {\mathbb {R}}_{lm} \times {\mathbb {R}}_{tm},\) where a(x), b(x), c(x) are \({\mathbb {Z}}_{4}[x]\) monic polynomials. Let \(C_{1}= \langle a(x) \rangle , C_{2}= \langle b(x) \rangle\) and \(C_{3}= \langle c(x) \rangle\) be \(\mathbb Z_{4}\)-free cyclic codes and

$$\begin{aligned}g_{1}(x) = \gcd (a(x), x^{km}-1), g_{2}(x) = \gcd (b(x), x^{lm}-1), g_{3}(x) = \gcd (c (x), x^{tm}-1). \end{aligned}$$

If \(h(x)= {{\,\mathrm{lcm}\,}}\left\{ \frac{x^{km}-1}{g_{1}(x)}, \frac{x^{lm}-1}{g_{2}(x)} , \frac{x^{tm}-1}{g_{3}(x)}\right\}\) is a monic parity-check polynomial of \(C_{abc}\) with degree \(h_{0},\) then \(C_{abc}\) is a \({\mathbb {Z}}_{4}\)-free module of rank \(h_{0}\). Moreover, the set \(\{F(x), xF(x), \dots , x^{h_{0}-1}F(x)\}\) is a basis of \(C_{abc}.\)

Proof

By Lemma 3.1, we can see that \(C_{abc}\) can be generated by the set \(\{ F(x), xF(x), \dots ,x^{h_{0}-1}F(x)\}.\) Therefore, it is sufficient to show that \(\{ F(x), xF(x), \dots , x^{h_{0}-1}F(x)\}\) is linearly independent over \({\mathbb {Z}}_{4}\) . Now, suppose that there exist \(k_{0}, k_{1}, \dots , k_{h_{0}-1} \in {\mathbb {Z}}_{4}\) such that

$$\begin{aligned}k_{0}F(x)+ xF(x)+ \dots + x^{h_{0}-1}F(x) = \sum _{i=0}^{h_{0}-1}k_{i}x^{i}F(x)=0.\end{aligned}$$

Let \(k(x)=\sum _{i=0}^{h_{0}-1}k_{i}x^{i},\) then \(k(x)F(x)=0\) if and only if \(k(x)a(x)=0\), \(k(x)b(x)=0\) and \(k(x)c(x)=0\) in \(R_{kltm}\). In other words, we can say that \((x^{km}-1)|k(x)a(x)\),  \((x^{lm}-1)|k(x)b(x)\) and \((x^{tm}-1)|k(x)c(x)\), also that \(g_{1}|(x^{km}-1)\), \(g_{2}|(x^{lm}-1)\) and \(g_{3}|(x^{tm}-1)\). Now using Lemma 3.2, there exist \(q_{1}(x), ~q_{2}(x),~ q_{3}(x) \in {\mathbb {Z}}_{4}[x]\) such that

  1. 1.

    \(a(x)=q_{1}(x)g_{1}(x) \text {~and~}\gcd \left( q_{1}(x), \frac{x^{km}-1}{g_{1}(x)}\right) =1,\)

  2. 2.

    \(b(x)=q_{2}(x)g_{2}(x) \text {~and~} \gcd \left( q_{2}(x), \frac{x^{lm}-1}{g_{2}(x)}\right) =1,\)

  3. 3.

    \(c(x)=q_{3}(x)g_{3}(x) \text {~and~} \gcd \left( q_{3}(x), \frac{x^{tm}-1}{g_{3}(x)}\right) =1.\)

Since \((x^{km}-1)|k(x)a(x)\), \((x^{lm}-1)|k(x)b(x)\) and \((x^{tm}-1)|k(x)c(x).\) Therefore, \((x^{km}-1)|k(x)q_{1}(x)g_{1}(x)\), \((x^{lm}-1)|k(x)q_{2}(x)g_{2}(x)\) and \((x^{tm}-1)|k(x)q_{3}(x)g_{3}(x)\) which implies

$$\begin{aligned}\left( \frac{x^{km}-1}{g_{1}(x)}\right) |k(x)q_{1}(x),~ \left( \frac{x^{lm}-1}{g_{2}(x)}\right) |k(x)q_{2}(x) \text { and } \left( \frac{x^{tm}-1}{g_{3}(x)}\right) |k(x)q_{3}(x).\end{aligned}$$

Also, since

$$\begin{aligned}\gcd \left( q_{1}(x),~ \frac{x^{km}-1}{g_{1}(x)}\right) =1,~ \gcd \left( q_{2}(x),~ \frac{x^{lm}-1}{g_{2}(x)}\right) =1 \text { and } \gcd \left( q_{3}(x),~ \frac{x^{tm}-1}{g_{3}(x)}\right) =1.\end{aligned}$$

So \(\left( \frac{x^{km}-1}{g_{1}(x)}\right) |k(x)\), \(\left( \frac{x^{lm}-1}{g_{2}(x)}\right) |k(x)\) and \(\left( \frac{x^{tm}-1}{g_{3}(x)}\right) |k(x)\).

Therefore, \({{\,\mathrm{lcm}\,}}\left\{ \frac{x^{km}-1}{g_{1}(x)}, \frac{x^{lm}-1}{g_{2}(x)}, \frac{x^{tm}-1}{g_{3}(x)} \right\} |k(x)\), i.e., h(x)|k(x) and the monic polynomial h(x) has \(\deg\) \(h_{0}\). Now, if \(\deg ( h(x)) \le (h_{0}-1),\) then \(k(x)=0.\) This implies that \(F(x), xF(x), \dots ,x^{h_{0}-1}F(x)\) are linearly independent over \({\mathbb {Z}}_{4}.\) So the set \(\{F(x), xF(x), \dots ,x^{h_{0}-1}F(x)\}\) is a basis of \(C_{abc}.\) \(\square\)

Note that \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes are \({\mathbb {Z}}_{4}\)-free. Now, using the results of a Proposition 3.3, we shown a method to determine a generator matrix of the code \(C_{abc}.\) For the polynomials \(a(x) = a_{0} + a_{1}x + \dots + a_{km-1} x^{km-1}\), \(b(x) = b_{0} + b_{1}x + \dots + b_{lm-1} x^{lm-1}\) and \(c(x) = c_{0} + c_{1}x + \dots + c_{tm-1} x^{tm-1}\), the circulant matrices A, B and C are defined as follows:

$$\begin{aligned} A= & {} \begin{pmatrix} a_{0} &{} a_{1} &{} \dots &{} a_{km-1} \\ a_{km-1} &{} a_{0} &{} \dots &{} a_{km-2} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ a_{1} &{} a_{2} &{} \dots &{} a_{0} \\ \end{pmatrix}, \\ B= & {} \begin{pmatrix} b_{0} &{} b_{1} &{} \dots &{} b_{lm-1} \\ b_{lm-1} &{} b_{0} &{} \dots &{} b_{lm-2} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ c_{1} &{} c_{2} &{} \dots &{} c_{0} \\ \end{pmatrix} , \\ C= & {} \begin{pmatrix} c_{0} &{} c_{1} &{} \dots &{} c_{tm-1} \\ c_{tm-1} &{} c_{0} &{} \dots &{} c_{tm-2} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ c_{1} &{} c_{2} &{} \dots &{} c_{0} \\ \end{pmatrix}, \end{aligned}$$

thus the circulant matrix for \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\) over \({\mathbb {Z}}_{4}\) can be constructed as

$$\begin{aligned} M = \begin{pmatrix} A &{} B &{} C \\ A &{} B &{} C \\ \vdots &{} \vdots &{} \vdots \\ A &{} B &{} C \\ \end{pmatrix}_{kltm \times (k+l+t)m}. \end{aligned}$$
(1)

Thus,

$$\begin{aligned} C_{abc}=\{(x_{0},x_{1},\dots , x_{kltm-1})M \in \mathbb Z_{4}^{km} \times {\mathbb {Z}}_{4}^{lm} \times {\mathbb {Z}}_{4}^{tm}~ | ~(x_{0},x_{1},\dots , x_{kltm-1}) \in {\mathbb {Z}}_{4}^{kltm} \}. \end{aligned}$$

If the parity-check polynomial of \(C_{abc},\) \(h(x)= {{\,\mathrm{lcm}\,}}\left\{ \frac{x^{km}-1}{g_{1}(x)}, \frac{x^{lm}-1}{g_{2}(x)}, \frac{x^{tm}-1}{g_{3}(x)}\right\}\) has deg \(h_{0},\) then \(\text {rank}(C_{abc})=h_{0}.\) Therefore, the first \(h_{0}\) rows of M form a generator matrix of \(C_{abc}\). Now, we present an example to illustrate the method discussed above.

Example 3.4

Let \(m=9,~ k=l=t=1\),  \(a(x)= x^2 + x + 1,~ b(x) = x^6 + x^3 + 1,~ c(x) = x^2 + x + 1\), we find \(\text {rank}(C_{abc}).\)

At first, we find that \(g_{1}(x) = \gcd (a(x), x^{9}-1) = x^2 + x + 1\), \(g_{2}(x) = \gcd (b(x), x^{9}-1)=x^6 + x^3 + 1\), \(g_{3}(x) = \gcd (c (x), x^{9}-1) =x^2 + x + 1\). Therefore, \(h(x)= \text {lcm} \{ \frac{x^{9}-1}{g_{1}(x)},~ \frac{x^{9}-1}{g_{2}(x)},~ \frac{x^{9}-1}{g_{3}(x)} \} = (x-1)(x^2 + x + 1)(x^6 + x^3 + 1)\).

The circulant matrices corresponding to the polynomials \(a(x),b(x)\text { and } c(x)\) are

$$\begin{aligned} A= & {} C = \begin{pmatrix} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ \end{pmatrix}, \\ B= & {} \begin{pmatrix} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \\ \end{pmatrix}. \end{aligned}$$

Therefore, from (1), we have

$$\begin{aligned} M = \begin{pmatrix} A &{} B &{} C \\ \end{pmatrix}_{9\times 27.} \end{aligned}$$

Hence, the first 9 rows of M form a generator matrix for \(C_{abc}.\) So, by Proposition 3.3, we have \(\text {rank}(C_{abc}) = \deg (h(x)) = 9\).

4 Asymptotically good \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes

There is a long standing question whether the class of cyclic codes is asymptotically good. This has been an open problem for more than half a century as can be seen in [2]. Important research has been done related to this question by many researchers (see [15,16,17,18, 24] etc). To consider this question, entropy function has an important role (see [9]). Define a forth order entropy function \(h_{4}(x)\) as follows,

$$\begin{aligned}h_{4}(x)= x \log _{4}3 -x\log _{4}x-(1-x)\log _{4}(1-x),\end{aligned}$$

where, \(0 \le x \le 1.\) Further, let \(\delta\) be a real number such that \(0< \delta < 1\) and \(h_{4}(\frac{\delta }{2}) < \frac{1}{4}\).

We can see that \(x^m-1=(x-1)(x^{m-1}+x^{m-2}+ \dots +1),\) and using the Chinese Remainder Theorem (CRT), we have

$$\begin{aligned} \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^m - 1 \rangle } = \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^{m-1} + x^{m-2} + \dots + 1 \rangle } \oplus \frac{{\mathbb {Z}}_{4}[x]}{ \langle x - 1 \rangle }. \end{aligned}$$

The cyclic code generated by \(x^{m-1} + x^{m-2} + \dots + 1\) is just the code consisting of multiple of the all-one vector, and then we only consider the cyclic codes generated by \(x-1\) which are defined as,

$$\begin{aligned}&{\mathbb {J}}_{m} = \left\langle x-1 \right\rangle _{{\mathbb {R}}_{m}},~~ {\mathbb {J}}_{kltm}= \left\langle \frac{x^{kltm}-1}{x^{m}-1}(x-1) \right\rangle _{{\mathbb {R}}_{kltm}},\\&{\mathbb {J}}_{km}= \left\langle \frac{x^{km}-1}{x^{m}-1}(x-1) \right\rangle _{{\mathbb {R}}_{km}},~{\mathbb {J}}_{lm}= \left\langle \frac{x^{lm}-1}{x^{m}-1}(x-1) \right\rangle _{{\mathbb {R}}_{lm}},~ {\mathbb {J}}_{tm}= \left\langle \frac{x^{tm}-1}{x^{m}-1}(x-1) \right\rangle _{{\mathbb {R}}_{tm}}. \end{aligned}$$

Now, for \((a(x), b(x), c(x)) \in {\mathbb {J}}_{km} \times {\mathbb {J}}_{lm} \times {\mathbb {J}}_{tm}\), let

$$\begin{aligned} C_{abc} = \{ (f(x)a(x), f(x)b(x), f(x)c(x)) \in {\mathbb {R}}_{km} \times {\mathbb {R}}_{lm} \times {\mathbb {R}}_{tm} ~|~ f(x) \in {\mathbb {J}}_{kltm} \} . \end{aligned}$$

Then reformulating \(C_{abc}\) as a \({\mathbb {Z}}_{4}\mathbb Z_{4}{\mathbb {Z}}_{4}\)-additive cyclic code, we want to discuss the asymptotic properties of the rate \(R(C_{abc})\) and the relative distance \(\Delta (C_{abc})\) of \(C_{abc}.\) First, we will have discussion on the asymptotic properties of

$$\begin{aligned} C_{a'b'c'} = \{ (f(x)(a'(x), f(x) b'(x), f(x) c'(x)) \in {\mathbb {R}}_{m} \times {\mathbb {R}}_{m} \times {\mathbb {R}}_{m} ~|~ f(x) \in {\mathbb {J}}_{m} \}, \end{aligned}$$

where \((a'(x), b'(x), c'(x)) \in {\mathbb {J}}_{m} \times {\mathbb {J}}_{m} \times {\mathbb {J}}_{m}\) .

Thus \({\mathbb {J}}_{m} \times {\mathbb {J}}_{m} \times {\mathbb {J}}_{m}\) and \({\mathbb {J}}_{km} \times {\mathbb {J}}_{lm} \times {\mathbb {J}}_{tm}\) can be viewed as probability spaces of \({\mathbb {R}}_{m} \times {\mathbb {R}}_{m} \times {\mathbb {R}}_{m}\) and \({\mathbb {R}}_{km} \times {\mathbb {R}}_{lm} \times {\mathbb {R}}_{tm}\), respectively. Moreover, let \(C_{abc}\) be a random code of the probability space \({\mathbb {J}}_{km} \times {\mathbb {J}}_{lm} \times {\mathbb {J}}_{tm}\) with random variable \(R(C_{abc})\) and \(\Delta (C_{abc}).\) Also, let \(C_{a'b'c'}\) be a random code of the probability space \({\mathbb {J}}_{m} \times {\mathbb {J}}_{m} \times {\mathbb {J}}_{m}\) with random variable \(R(C_{a'b'c'})\) and \(\Delta (C_{a'b'c'}).\) Clearly, if we are using \(R(C_{abc})\) and \(\Delta (C_{abc})\) as random variables on the probability space \({\mathbb {J}}_{km} \times {\mathbb {J}}_{lm} \times {\mathbb {J}}_{tm}\), then by the definition of asymptotically good codes, the problem has been transformed into studying of probabilities of \(\mathbb P_r(\Delta (C_{abc}) \ge \delta )\) and \({\mathbb {P}}_r(\text {rank} (C_{abc}) = m-1 )\), where \(\delta\) is a real number such that \(0< \delta < 1\) and \({\mathbb {P}}_{r}\) denotes the probabilities of random variables \(R(C_{abc})\) and \(\Delta (C_{abc}).\)

To see the relation between \(R(C_{abc})\) and \(R(C_{a'b'c'}),\) we define a map \(\psi ^{\prime}\) as

$$\begin{aligned}&\psi ^{\prime}: {\mathbb {J}}_{m} \times {\mathbb {J}}_{m} \times {\mathbb {J}}_{m} \longrightarrow {\mathbb {J}}_{km} \times {\mathbb {J}}_{lm} \times {\mathbb {J}}_{tm} \\&\quad (a'(x), b'(x), c'(x)) \longmapsto (a(x), b(x), c(x)) \end{aligned}$$

where \((a(x), b(x), c(x))= \left( a'(x)\frac{x^{{km}}-1}{x^{m}-1}, b'(x)\frac{x^{{lm}}-1}{x^{m}-1}, c'(x)\frac{x^{{tm}}-1}{x^{m}-1}\right)\). Clearly, \(\psi '\) is a \({\mathbb {R}}_{kltm}\)-isomorphism and

$$\begin{aligned}(a(x), b(x), c(x))= \psi '(a'(x), b'(x), c'(x)),~ C_{abc}=\psi '(C_{a'b'c'}).\end{aligned}$$

Moreover, this also implies

$$\begin{aligned} wt_{L}(a(x), b(x), c(x))&= wt_{L}(a(x))+ wt_{L}(b(x)) + wt_{L}(c(x)) \\&= kwt_{L}(a'(x))+ lwt_{L}(b'(x)) + twt_{L}(c'(x))\\&\ge wt_{L}(a'(x), b'(x), c'(x)). \end{aligned}$$

By using the definition of relative distance, define,

$$\begin{aligned}\Delta (C_{abc})= \frac{d_{L}(C_{abc})}{(k+l+t)m} = \frac{wt_{L}(C_{abc})}{(k+l+t)m}\end{aligned}$$

and

$$\begin{aligned}\Delta (C_{a'b'c'})= \frac{d_{L}(C_{a'b'c'})}{3m} = \frac{wt_{L}(C_{a'b'c'})}{3m}.\end{aligned}$$

Now, if \(\Delta (C_{abc})\ge \Delta (C_{a'b'c'})\) then

$$\begin{aligned} (k+l+t)m\Delta (C_{abc})\ge 3m\Delta (C_{a'b'c'}), \text { i.e., }\Delta (C_{abc})\ge \frac{3}{k+l+t} \Delta (C_{a'b'c'}).\end{aligned}$$

Lemma 4.1

\({\mathbb {P}}_r(\Delta (C_{abc}) \ge \delta ) \ge \mathbb P_r(\Delta (C_{a'b'c'}) \ge \frac{k+l+t}{3}\delta )\).

Proof

Let \(\Delta (C_{a'b'c'}) \ge \frac{k+l+t}{3}\delta\) and \(\Delta (C_{abc})\ge \frac{3}{k+l+t} \Delta (C_{a'b'c'})\) then \(\Delta (C_{abc})\ge \delta\). Thus,

$$\begin{aligned} |\Delta (C_{abc})\ge \delta | \ge \left| \Delta (C_{a'b'c'}) \ge \frac{k+l+t}{3}\delta \right| . \end{aligned}$$

Since \(\psi ^{\prime}\) is an isomorphism, we have \(|{\mathbb {J}}_{m} \times {\mathbb {J}}_{m} \times {\mathbb {J}}_{m}| = |{\mathbb {J}}_{km} \times {\mathbb {J}}_{lm} \times {\mathbb {J}}_{tm}|.\) So, we get

$$\begin{aligned} {\mathbb {P}}_r(\Delta (C_{abc}) \ge \delta )&= \frac{|(\Delta (C_{abc}) \ge \delta )|}{|{\mathbb {J}}_{km} \times {\mathbb {J}}_{lm} \times {\mathbb {J}}_{tm}|} \ge \frac{|\Delta (C_{a'b'c'}) \ge \frac{k+l+t}{3}\delta |}{|{\mathbb {J}}_{m} \times {\mathbb {J}}_{m} \times {\mathbb {J}}_{m}|} \\&= {\mathbb {P}}_r\left( \Delta (C_{a'b'c'}) \ge \frac{k+l+t}{3}\delta \right) . \end{aligned}$$

\(\square\)

Now, in order to study the asymptotic properties of \(\mathbb P_r(\Delta (C_{abc}) \ge \delta )\) using Lemma 4.1, we need to study the asymptotic properties of \(\mathbb P_r(\Delta (C_{a'b'c'}) \ge \frac{k+l+t}{3}\delta )\). For that we need the following definition. For any \(f(x) \in {\mathbb {J}}_{m}\) and \((a'(x), b'(x), c'(x)) \in {\mathbb {J}}_{m} \times {\mathbb {J}}_{m} \times {\mathbb {J}}_{m}\) over the probability space \({\mathbb {J}}_{m} \times {\mathbb {J}}_{m} \times {\mathbb {J}}_{m}\). We have

Definition 4.2

The Bernoulli random variable \(Y_{f}\) is defined as

$$\begin{aligned} Y_{f} = {\left\{ \begin{array}{ll} 1 ~~~ 1 \le wt_{L}(a'(x), b'(x), c'(x)) \le 3m\delta \ \\ 0 ~~~ otherwise. \end{array}\right. } \end{aligned}$$

Given that \(f(x) \in {\mathbb {J}}_{m}\), consider the set \(\{f(x)a'(x) \in {\mathbb {R}}_{m}~|~ a'(x) \in {\mathbb {J}}_{m} \}\). It can be inferred that this set is an ideal of \({\mathbb {R}}_{m}\) generated by f(x). Let \({\mathbb {I}}_{f} = \langle f(x) \rangle \subseteq {\mathbb {J}}_{m}\) and \(|{\mathbb {I}}_{f}|=2^{d_{f}}.\)

We have the following:

Lemma 4.3

If \({\mathbb {I}}_{f} \times {\mathbb {I}}_{f} \times {\mathbb {I}}_{f} \subseteq {\mathbb {R}}_{m} \times {\mathbb {R}}_{m} \times {\mathbb {R}}_{m},\) and

$$\begin{aligned}({\mathbb {I}}_{f} \times {\mathbb {I}}_{f} \times {\mathbb {I}}_{f})^{\le 3m\delta } = \{ (f_{1}(x),f_{2}(x), f_{3}(x)) \in {\mathbb {I}}_{f} \times {\mathbb {I}}_{f} \times {\mathbb {I}}_{f}~|~ wt_{L}(f_{1}(x),f_{2}(x), f_{3}(x)) \le 3m\delta \},\end{aligned}$$

then

$$\begin{aligned}|({\mathbb {I}}_{f} \times {\mathbb {I}}_{f} \times {\mathbb {I}}_{f})^{\le 3m\delta }| \le 4^{3d_{f}h_{4}(\frac{\delta }{2})} = 2^{6d_{f}h_{4}(\frac{\delta }{2})}.\end{aligned}$$

Proof

Since \(|{\mathbb {R}}_{m} \times {\mathbb {R}}_{m} \times {\mathbb {R}}_{m}|= 4^{3m}=2^{6m}\) and \(|{\mathbb {I}}_{f} \times {\mathbb {I}}_{f} \times {\mathbb {I}}_{f}|= 2^{3d_{f}}\) then the fraction of \(3m\delta\) over the length 6m is \(\frac{3m\delta }{6m}=\frac{\delta }{2}.\) Additionally \(0< \delta < 1,\) so, \(0< \frac{\delta }{2}< \frac{1}{2} < \frac{3}{4}.\) Therefore, by extending the results in [ [17], Corollary  3.5, Remark  3.2] for \(\mathbb Z_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4},\) we have

$$\begin{aligned}|({\mathbb {I}}_{f} \times {\mathbb {I}}_{f} \times {\mathbb {I}}_{f})^{\le 3m\delta }| \le 4^{3d_{f}h_{4}(\frac{\delta }{2})} = 2^{6d_{f}h_{4}(\frac{\delta }{2})}.\end{aligned}$$

\(\square\)

Now, by Lemma 4.3 we have the following:

Lemma 4.4

\({\mathbb {E}}(Y_{f}) \le 4^{3d_{f}h_{4}(\frac{\delta }{2})-\frac{3d_{f}}{2}}\), where \({\mathbb {E}}\) denotes the expectation of a random variable.

Proof

From Lemma 4.3, \(|({\mathbb {I}}_{f} \times {\mathbb {I}}_{f} \times {\mathbb {I}}_{f})^{\le 3m\delta }| \le 4^{3d_{f}h_{4}(\frac{\delta }{2})}\). So

$$\begin{aligned} {\mathbb {E}}(Y_{f}) = {\mathbb {P}}_r(Y_{f}=1)&= \frac{|({\mathbb {I}}_{f} \times {\mathbb {I}}_{f} \times {\mathbb {I}}_{f})^{\le 3m\delta }|-1}{|{\mathbb {I}}_{f} \times {\mathbb {I}}_{f} \times {\mathbb {I}}_{f}|} \\&\le \frac{4^{3d_{f}h_{4}(\frac{\delta }{2})}}{2^{3d_{f}}={4^{\frac{3d_{f}}{2}}}} \\&= 4^{3d_{f}h_{4}(\frac{\delta }{2})}4^{-\frac{3d_{f}}{2}} \\&= 4^{3d_{f}h_{4}(\frac{\delta }{2})-\frac{3d_{f}}{2}}. \end{aligned}$$

\(\square\)

By CRT, we have

$$\begin{aligned} {\mathbb {J}}_{m} =\langle x-1 \rangle _{R_{m}}&\cong \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^{m-1} + x^{m-2} + \dots + 1\rangle } \\&= \frac{{\mathbb {Z}}_{4}[x]}{ \langle q_{1}(x)\rangle } \times \frac{{\mathbb {Z}}_{4}[x]}{ \langle q_{2}(x)\rangle } \times \dots \times \frac{{\mathbb {Z}}_{4}[x]}{ \langle q_{r}(x)\rangle }, \end{aligned}$$

where \(q_{1}(x), q_{2}(x), \dots , q_{r}(x)\) are monic basic irreducible factors of \(x^{m-1} + x^{m-2} + \dots + 1 \in \mathbb Z_{4}[x]\). Let \(q_{k}(x),\) for \(1\le k \le r,\) be a polynomial lowest degree among \(q_{1}(x), q_{2}(x), \dots , q_{r}(x)\). Then the minimal Galois ring among them is \(\frac{{\mathbb {Z}}_{4}[x]}{ \langle q_{k}(x)\rangle }\) and it contains a non-zero ring of least size \(2^{k_{m}}.\) By CRT, the ideals in \({\mathbb {J}}_{m}\) correspond to the ideals in \(\frac{{\mathbb {Z}}_{4}[x]}{ \langle q_{1}(x)\rangle } \times \frac{{\mathbb {Z}}_{4}[x]}{ \langle q_{2}(x)\rangle } \times \dots \times \frac{{\mathbb {Z}}_{4}[x]}{ \langle q_{r}(x)\rangle }\) (see [19,  Lemma 9] and [7]). So, the minimal size of the non-zero ideal contained in \({\mathbb {J}}_{m}\) is equal to \(2^{k_{m}}\).

Lemma 4.5

[19] The number of non-zero ideals of size \(2^{d}\) contained in \(\mathbb J_{m}\) is at most \((2m)^{\frac{d}{k_{m}}}\), where \(k_{m} \le d \le 2(m-1)\).

Now, we will show that \(\lim \nolimits _{i \rightarrow \infty } {\mathbb {P}}_{r} (\Delta (C_{a'b'c'}^{i}) \ge \delta )=1\). For that, by Lemmas 4.4 and 4.5, we prove an useful lemma:

Lemma 4.6

Let \(0< \delta < 1\) be a real number and \(h_{4}(\frac{\delta }{2}) < \frac{1}{4},\) then

$$\begin{aligned} {\mathbb {P}}_{r} (\Delta (C_{a'b'c'}) \le \delta ) \le \sum \limits _{i=k_{m}}^{2(m-1)} 4^{-3j (\frac{1}{3} - h_{4}(\delta _{2}) - \frac{\log _{4}2m }{3k_{m}})}.\end{aligned}$$

Proof

Let \(Y_{f}\) for \(f(x) \in J_{m}\) be a Bernoulli variable with a value 0 or 1. Let \(Y =\sum \nolimits _{f(x) \in J_{m}} Y_{f},\) then Y is a non-negative integer random variable over the probability space \({\mathbb {J}}_{m} \times {\mathbb {J}}_{m} \times {\mathbb {J}}_{m}\). Y stands for the cardinality of \(f(x) \in {\mathbb {J}}_{m}\) such that the weight of the codewords is at most \(3m\delta\) and \(\Delta (C_{a'b'c'}) = \frac{wt_{L}(C_{a'b'c'})}{3m}\), we get \({\mathbb {P}}_{r} (\Delta (C_{a'b'c'}) \le \delta ) = {\mathbb {P}}_{r} (Y > 0)\). By Markov’s inequality [26,  Theorem 3.1], \({\mathbb {P}}_{r} (Y > 0) \le {\mathbb {E}}(Y)\). So, we only need to find the value of \({\mathbb {E}}(Y)\). From [22], we have

$$\begin{aligned}{\mathbb {E}}(\alpha Y_{1} + Y_{2}) =\alpha {\mathbb {E}}(Y_{1}) + {\mathbb {E}}(Y_{2}).\end{aligned}$$

So, \({\mathbb {E}}(Y) = {\mathbb {E}}(\sum \nolimits _{f(x) \in {\mathbb {J}}_{m}} Y_{f})\), for any ideal \({\mathbb {I}}\) of \({\mathbb {J}}_{m}\), denoted as \(({\mathbb {I}} \le {\mathbb {J}}_{m})\). Let \({\mathbb {I}}^{*} = \{ f(x) \in {\mathbb {I}} ~|~ {\mathbb {I}}_{f} = {\mathbb {I}}~\}\), where \({\mathbb {I}}_{f} = \langle f(x) \rangle _{{\mathbb {R}}_{m}} \subseteq {\mathbb {J}}_{m}\). Since \(d_{f} = \text {rank}({\mathbb {I}}_{f})\) then \({\mathbb {I}}^{*} = \{ f(x) \in {\mathbb {I}} ~|~ d_{f} = \text {rank} ({\mathbb {I}})~\}\). Therefore,

$$\begin{aligned} {\mathbb {J}}_{m} = \bigcup \limits _{{\mathbb {I}} \subseteq {\mathbb {J}}_{m}}{\mathbb {I}}^{*} \end{aligned}$$

and \(0 \ne {\mathbb {I}} \le {\mathbb {J}}_{m}\) then \(k_{m}\le \text {rank}({\mathbb {I}}) = d \le 2(m-1)\). So,

$$\begin{aligned} {\mathbb {E}}(Y) = \sum \limits _{{\mathbb {I}} \le {\mathbb {J}}_{m}} \sum \limits _{f(x) \in {\mathbb {I}}^{*}} {\mathbb {E}}(Y_{f}) = \sum \limits _{i = k_{m}}^{2(m-1)} \sum \limits _{\begin{array}{c} {\mathbb {I}} \le {\mathbb {J}}_{m} \\ \text {rank}{\mathbb {I}} = j \end{array}} \sum \limits _{f(x) \in {\mathbb {I}}^{*}} {\mathbb {E}}(Y_{f}). \end{aligned}$$

For \({\mathbb {I}} \le {\mathbb {J}}_{m}\) with \(\text {rank}({\mathbb {I}}) = j ~ \& ~ |{\mathbb {I}}^*| \le |{\mathbb {I}}| = 2^{i}\). Using Lemma 4.4, we have

$$\begin{aligned} \sum \limits _{f \in {\mathbb {I}}^{*}} {\mathbb {E}}(Y_{f})&\le \sum \limits _{f \in {\mathbb {I}}^{*}} 4^{3 d_{f} h_{4}(\frac{\delta }{2})-\frac{3 d_{f}}{2}} \\&= \sum \limits _{f \in {\mathbb {I}}^{*}} 4^{3j h_{4} (\frac{\delta }{2}) - \frac{3j}{2}}. \end{aligned}$$

By Lemma 4.5, for \({\mathbb {I}} \le {\mathbb {J}}_{m}\) with rank\({\mathbb {I}} = j\) which is less than \((2m)^{\frac{j}{k_{m}}}\) and we know that \(\log _{4} 2m \le \frac{j\log _{4}2m}{k_{m}} ~ \text {as}~ k_{m} \le j\), so

$$\begin{aligned} {\mathbb {E}}(Y)&\le \sum \limits _{j= k_{m}}^{2(m-1)}(2m)^{\frac{j}{k_{m}}} 4^{-j + 3j h_{4}(\frac{\delta }{2})} \\&= \sum \limits _{j= k_{m}}^{2(m-1)} 4^{ \frac{3j}{3k_{m}} \log _{4}2m} 4^{\frac{-3j}{3} + 3j h_{4}(\frac{\delta }{2})} \end{aligned}$$
$$\begin{aligned} {\mathbb {E}}(Y) \le \sum \limits _{j=k_{m}}^{2(m-1)} 4^{-3j (\frac{1}{3} - h_{4}(\delta _{2}) - \frac{\log _{4}2m }{3k_{m}})}. \end{aligned}$$

Thus, we have

$$\begin{aligned} {\mathbb {P}}_{r} (\Delta (C_{a'b'c'}) \le \delta ) \le \sum \limits _{i=k_{m}}^{2(m-1)} 4^{-3j (\frac{1}{3} - h_{4}(\delta _{2}) - \frac{\log _{4}2m }{3k_{m}})}.\end{aligned}$$

\(\square\)

Remark 4.7

By [4,  Lemma 2.6] there exist positive integers \(m_{1}, m_{2}, \dots\) such that \(\gcd (m_{i}, 4) = 1, m_{i} \rightarrow \infty , \lim \nolimits _{i \rightarrow \infty } \frac{\log _{4} m_{i}}{k_{m_{i}}} = 0\) where \(k_{m_{i}}\) are defined as in Lemma 4.5.

Let

$$\begin{aligned} C_{a'b'c'}^{i} = \{ f(x)a'(x), f(x)b'(x), f(x)c'(x) \in {\mathbb {R}}_{m_{i}} \times {\mathbb {R}}_{m_{i}} \times {\mathbb {R}}_{m_{i}} | f(x) \in {\mathbb {J}}_{m_{i}}\} \end{aligned}$$

be a random \({\mathbb {Z}}_{4} {\mathbb {Z}}_{4} {\mathbb {Z}}_{4}\)-cyclic code of length \(3m_{i}\), where \((a'(x), b'(x), c'(x)) \in {\mathbb {J}}_{m_{i}} \times {\mathbb {J}}_{m_{i}} \times {\mathbb {J}}_{m_{i}}\).

Now, by using Lemma 4.6, we have one of the main results of the paper in the following proposition.

Proposition 4.8

Let \(0< \delta < 1\) be a real number and \(h_{4}(\frac{\delta }{2}) < \frac{1}{4}\) then

$$\begin{aligned}\lim _{i \rightarrow \infty } {\mathbb {P}}_{r} (\Delta (C_{a'b'c'}^{i}) \ge \delta )=1.\end{aligned}$$

Proof

From the assumptions on \(\delta\) and \(h_{4},\) we have \(h_{4}(\frac{\delta }{2})< \frac{1}{4} < \frac{1}{3}\) which implies that \(\frac{1}{3}-h_{4}(\frac{\delta }{2})>0\). Since \(\lim \nolimits _{i \rightarrow \infty } \frac{\log _{4} m_{i}}{k_{m_{i}}} = 0 ,\) then \(\lim \nolimits _{i \rightarrow \infty } \frac{\log _{4} 2m_{i}}{k_{m_{i}}} = 0\). Therefore, for a given \(\epsilon >0\) there exists a non-negative integer N such that for \(i>N\), we have \(\frac{1}{3} - h_{4}(\frac{\delta }{2}) - \frac{\log _{4}2m_i }{3k_{m_i}} \ge \epsilon >0.\) From Lemma 4.6, we have

$$\begin{aligned} \lim _{i \rightarrow \infty }{\mathbb {P}}_{r} (\Delta (C_{a'b'c'}^{i}) \le \delta )&\le \lim _{i \rightarrow \infty }\sum \limits _{j=k_{m_i}}^{2(m-1)} 4^{-3j (\frac{1}{3} - h_{4}(\frac{\delta }{2}) - \frac{\log _{4}2m_i }{3k_{m_i}})}\\&\le \lim _{i \rightarrow \infty }\sum \limits _{j=k_{m_i}}^{2(m-1)} 4^{-3j\epsilon } \\&\le \lim _{i \rightarrow \infty }\sum \limits _{j=k_{m_i}}^{2(m-1)} 4^{-3k_{m_{i}}\epsilon } \\&\le \lim _{i \rightarrow \infty } 2m_{i} 4^{-3k_{m_{i}}\epsilon } \\&= \lim _{i \rightarrow \infty } 4^{-3k_{m_{i}}(\epsilon -\frac{\log _{4} 2m_{i}}{3k_{m_{i}}})}. \end{aligned}$$

Also, since \(\lim \nolimits _{i \rightarrow \infty } \frac{\log _{4} m_{i}}{k_{m_{i}}} = 0,\) then \(\lim \nolimits _{i \rightarrow \infty } \frac{\log _{4} 2m_{i}}{3k_{m_{i}}} = 0\) which yields \(\lim \nolimits _{i \rightarrow \infty }~ 3m_{i} \rightarrow \infty .\) Therefore, \(\lim \nolimits _{i \rightarrow \infty } 4^{-3k_{m_{i}}(\epsilon -\frac{\log _{4} 2m_{i}}{3k_{m_{i}}})}=0\), i.e., \(\lim \nolimits _{i \rightarrow \infty }{\mathbb {P}}_{r} (\Delta (C_{a'b'c'}^{i}) \le \delta )=0\) which implies that

$$\begin{aligned} \lim \limits _{i \rightarrow \infty } {\mathbb {P}}_{r} (\Delta (C_{a'b'c'}^{i}) \ge \delta )=1. \end{aligned}$$

\(\square\)

From Proposition 4.8, \(0< \delta < 1\) and \(h_{4}(\frac{\delta }{2}) < \frac{1}{4}\) it can be seen that,

$$\begin{aligned}\lim \limits _{i \rightarrow \infty } {\mathbb {P}}_{r} (\Delta (C_{a'b'c'}^{i}) \ge \delta )=1.\end{aligned}$$

In other words, we can say that if \(0< \delta < 1\) and \(h_{4}(\frac{1}{2}\frac{k+l+t}{3}\delta ) < \frac{1}{4}\), i.e., \(h_{4}(\frac{k+l+t}{6}\delta ) < \frac{1}{4}\), then we have

$$\begin{aligned}\lim \limits _{i \rightarrow \infty } {\mathbb {P}}_{r} (\Delta (C_{a'b'c'}^{i}) \ge \frac{k+l+t}{3}\delta )=1.\end{aligned}$$

Now, by Proposition 4.8 and Lemma 4.1, we have one of the main results of the paper in the following proposition.

Proposition 4.9

If \(h_{4}(\frac{k+l+t}{6}\delta ) < \frac{1}{4}\) then \(\lim \nolimits _{i \rightarrow \infty } {\mathbb {P}}_{r} (\Delta (C_{abc}^{i}) \ge \delta )=1\).

Proof

By Proposition 4.8, \(0< \delta < 1\) and \(h_{4}(\frac{k+l+t}{6}\delta ) < \frac{1}{4}\), we have

$$\begin{aligned}\lim \limits _{i \rightarrow \infty } {\mathbb {P}}_{r} (\Delta (C_{a'b'c'}^{i}) \ge \frac{k+l+t}{3}\delta )=1.\end{aligned}$$

From Lemma 4.1, we have

$$\begin{aligned}\lim \limits _{i \rightarrow \infty } {\mathbb {P}}_{r} (\Delta (C_{abc}^{i}) \ge \delta )\ge \lim \limits _{i \rightarrow \infty } {\mathbb {P}}_{r} (\Delta (C_{a'b'c'}^{i}) \ge \frac{k+l+t}{3}\delta )=1.\end{aligned}$$

So \(\lim \nolimits _{i \rightarrow \infty } {\mathbb {P}}_{r} (\Delta (C_{abc}^{i}) \ge \delta )=1\). \(\square\)

Now, we will prove that \(\lim \nolimits _{i \rightarrow \infty } {\mathbb {P}}_{r} (\text {rank} (C_{abc}^{i}) = m_{i}-1)=1\). For that, we need the following lemma:

Lemma 4.10

Let

$$\begin{aligned} C_{a'b'c'} = \{ (f(x)a'(x), f(x) b'(x), f(x) c'(x)) \in {\mathbb {R}}_{m} \times {\mathbb {R}}_{m} \times {\mathbb {R}}_{m}~ |~ f(x) \in {\mathbb {J}}_{m} \},\end{aligned}$$

where \((a'(x), b'(x), c'(x)) \in {\mathbb {J}}_{m} \times {\mathbb {J}}_{m} \times {\mathbb {J}}_{m}\). Then \(\mathrm{rank} (C_{a'b'c'}) \le m-1\). Note that \(\mathrm{rank}(C_{a'b'c'})=m-1\) if and only if there is no basic irreducible factor q(x) of \(\frac{x^{m}-1}{x-1}\) in \({\mathbb {Z}}_{4}[x]\) such that

$$\begin{aligned}q(x)|a'(x), q(x)|b'(x)\text { and }q(x)|c'(x).\end{aligned}$$

Proof

Suppose \(g_{a'b'c'}(x) = \text {gcd}(a'(x), b'(x), c'(x), x^{m}-1)\) and consider

$$\begin{aligned}(a'(x), b'(x), c'(x)) \in {\mathbb {J}}_{m} \times {\mathbb {J}}_{m} \times {\mathbb {J}}_{m}.\end{aligned}$$

We have \((x-1)|g_{a'b'c'}(x)\), i.e., \(\langle g_{a'b'c'}(x) \rangle \subseteq \langle x-1 \rangle = {\mathbb {J}}_{m}\), which implies that

$$\begin{aligned}\text {rank}(C_{a'b'c'}) = \text {deg}(\frac{x^{m}-1}{g_{a'b'c'}(x)}) \le m-1.\end{aligned}$$

Clearly, \(\text {rank}(C_{a'b'c'}) < m-1 \text { if and only if } \text {deg}({g_{a'b'c'}(x)}> 1)\) if and only if there is a basic irreducible factor q(x) of \(\frac{x^{m}-1}{x-1}\) in \(\mathbb Z_{4}[x]\) such that

$$\begin{aligned}q(x)|a'(x), q(x)|b'(x) \text { and } q(x)|c'(x).\end{aligned}$$

Therefore, it is easy to see that \(\text {rank}(C_{a'b'c'})=m-1\) if and only if \(g_{a'b'c'}(x)=x-1\) if and only if there is no basic irreducible factor q(x) of \(\frac{x^{m}-1}{x-1}\) in \({\mathbb {Z}}_{4}[x]\) such that

$$\begin{aligned}q(x)|a'(x), q(x)|b'(x)\text { and }q(x)|c'(x).\end{aligned}$$

\(\square\)

Proposition 4.11

Let \(m_{1}, m_{2}, \dots\) be positive integers such that \(\gcd (m_{i}, 4) = 1\) and \(\lim \nolimits _{i \rightarrow \infty } \frac{\log _{4} m_{i}}{k_{m_{i}}} = 0,\) for \(m_{i} \rightarrow \infty\) where \(k_{m_{i}}\) are as defined in Lemma 4.5. Let

$$\begin{aligned} C_{a'b'c'}^{i} = \{ (f(x)a'(x), f(x)b'(x), f(x)c'(x)) \in {\mathbb {R}}_{m_{i}} \times {\mathbb {R}}_{m_{i}} \times {\mathbb {R}}_{m_{i}} | f(x) \in {\mathbb {J}}_{m_{i}}\} \end{aligned}$$

then \(\lim \nolimits _{i \rightarrow \infty } {\mathbb {P}}_{r} (\)rank\((C_{a'b'c'}^{i}) = m_{i}-1)=1\).

Proof

For any i, suppose that

$$\begin{aligned} x^{m_{i}}-1&= (x-1)(x^{m_{i}-1} + x^{m_{i}-2} + \dots + 1) \\&= (x-1)q_{1}(x), q_{2}(x), \dots , q_{r_{i}}(x). \end{aligned}$$

where \(q_{1}(x), q_{2}(x), \dots , q_{r_i}(x)\) are monic basic irreducible factors of \(x^{m-1} + x^{m-2} + \dots + 1 \in \mathbb Z_{4}[x]\). Using CRT, we have

$$\begin{aligned} {\mathbb {J}}_{m_{i}} =\langle x-1 \rangle _{R_{m_{i}}}&\cong \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^{m_{i}-1} + x^{m_{i}-2} + \dots + 1\rangle } \\&= \frac{{\mathbb {Z}}_{4}[x]}{ \langle q_{1}(x)\rangle } \times \frac{{\mathbb {Z}}_{4}[x]}{ \langle q_{2}(x)\rangle } \times \dots \times \frac{{\mathbb {Z}}_{4}[x]}{ \langle q_{r_{i}}(x)\rangle }, \end{aligned}$$

define a function

$$\begin{aligned} (a'(x)) \longmapsto (a'_{1}(x), a'_{2}(x), \dots , a'_{r_{i}}(x)) \end{aligned}$$

where \(a'_{j}(x)= a'(x)(\mod q_{j}),~ j= 1,2, \dots , r_{i}\) for \((a'(x), b'(x), c'(x)) \in {\mathbb {J}}_{m_{i}} \times {\mathbb {J}}_{m_{i}} \times {\mathbb {J}}_{m_{i}}.\) By Lemma 4.10, we have \(\text {rank} (C_{a'b'c'}^{i}) \le m_{i}-1\) and \(\text {rank} (C_{a'b'c'}^{i}) < m_{i}-1\) if and only if there is basic irreducible factor \(q_{j}(x), j= 1, 2, \dots , r_{i}\) of \(\frac{x^{m_{i}}-1}{x-1}\) in \({\mathbb {Z}}_{4}[x]\) such that \(q_{j}(x)|a'(x),~ q_{j}(x)|b'(x)\) and \(q_{j}(x)|c'(x)\) which can only defined when \(a'_{j}(x)=b'_{j}(x)=c'_{j}(x)=0\). In other words, \(\text {rank} (C_{a'b'c'}^{i}) = m_{i}-1\) if and only if \((a'_{j}(x), b'_{j}(x), c'_{j}(x)) \ne (0, 0, 0)\). Let \(k_{j}=\deg q_{j}(x)\) then \(|\frac{{\mathbb {Z}}_{4}[x]}{\langle q_{j}(x) \rangle }|= 4^{k_{j}}.\) Since there is a surjective homomorphism

$$\begin{aligned} \mathbb J_{m_{i}} \longrightarrow \frac{{\mathbb {Z}}_{4}[x]}{\langle q_{j}(x) \rangle }, \end{aligned}$$

so there are \(4^{3k_{j}}-1\) polynomial triples \((a'_{j}(x), b'_{j}(x), c'_{j}(x)) \ne (0, 0, 0)\). i.e., \({\mathbb {P}}_r((a'_{j}(x), b'_{j}(x), c'_{j}(x)) \ne (0, 0, 0))= \frac{4^{3k_{j}}-1}{4^{3k_{j}}}= 1- 4^{-3k_{j}}\) which yields,

$$\begin{aligned} {\mathbb {P}}_{r} (\text {rank} (C_{a'b'c'}^{i}) = m_{i}-1) = \prod \limits _{j=1}^{r_{i}} (1- 4^{-3k_{j}}). \end{aligned}$$

Since \(k_{m_{i}} \le k_{j}\) then \(r_{i}\le \frac{m_{i}-1}{k_{m_{i}}} \le \frac{m_{i}}{k_{m_{i}}}\) (Lemma 4.5).

Therefore,

$$\begin{aligned} {\mathbb {P}}_{r} (\text {rank} (C_{a'b'c'}^{i}) = m_{i}-1)&\ge (1-4^{-3k_{m_{i}}})^{\frac{m_{i}}{k_{m_{i}}}} \\&= (1-4^{-3k_{m_{i}}})^{4^{3k_{m_{i}}}\frac{m_{i}}{k_{m_{i}}4^{3k_{m_{i}}}}}. \end{aligned}$$

Since \(\lim \nolimits _{i \rightarrow \infty }\frac{m_{i}}{k_{m_{i}}4^{3k_{m_{i}}}} = 0\) and \(\lim \nolimits _{i \rightarrow \infty }(1-4^{-3k_{m_{i}}})^{4^{3k_{m_{i}}}}= \frac{1}{e},\) therefore

$$\begin{aligned}\lim _{i \rightarrow \infty }(1-4^{-3k_{m_{i}}})^{4^{3k_{m_{i}}}\frac{m_{i}}{k_{m_{i}}4^{3k_{m_{i}}}}}=(\frac{1}{e})^{0}=1. \end{aligned}$$

Thus, \(\lim \nolimits _{i \rightarrow \infty }{\mathbb {P}}_{r} (\text {rank} (C_{a'b'c'}^{i}) = m_{i}-1) \ge 1\), i.e., \(\lim \nolimits _{i \rightarrow \infty }{\mathbb {P}}_{r} (\text {rank} (C_{a'b'c'}^{i}) = m_{i}-1)=1\). \(\square\)

By the isomorphism \(\psi '\), it gives us \(C_{abc}^{i}=\psi '(C_{a'b'c'}^{i})\) and using Proposition 4.11, we have one of the main results of the paper in the following proposition.

Proposition 4.12

\(\lim \nolimits _{i \rightarrow \infty }{\mathbb {P}}_{r} (\)rank\((C_{abc}^{i}) = m_{i}-1)=1\).

Proof

From isomorphism \(\psi '\), \(C_{abc}^{i}=\psi '(C_{a'b'c'}^{i})\) and \(\text {rank}(C_{abc}^{i})= \text {rank} (\psi '(C_{a'b'c'}^{i})) = \text {rank}(C_{a'b'c'}^{i})\) and using Proposition 4.11 we have \(\lim \nolimits _{i \rightarrow \infty }{\mathbb {P}}_{r} (\text {rank} (C_{abc}^{i}) = m_{i}-1)=1\). \(\square\)

Now, by using Propositions 4.9 and 4.12 we get the asymptotic properties of \({\mathbb {P}}_{r} (\Delta (C_{abc}^{i}) \ge \delta )\) and \({\mathbb {P}}_{r} (\text {rank} (C_{abc}^{i}) = m_{i}-1)\) as follows.

Corollary 4.13

Let \(C_{abc}^{i} = \{ (f(x)a(x), f(x)b(x), f(x)c(x)) \in {\mathbb {R}}_{km_{i}} \times {\mathbb {R}}_{lm_{i}} \times {\mathbb {R}}_{tm_{i}} | f(x) \in {\mathbb {J}}_{kltm_{i}}\}\) and \(m_{1}, m_{2}, \dots\) such that \(\gcd (m_{i}, 4) = 1\) and \(\lim \nolimits _{i \rightarrow \infty } \frac{\log _{4} m_{i}}{k_{m_{i}}} = 0\) for \(m_{i} \rightarrow \infty .\)

  • If \(h_{4}(\frac{k+l+t}{6} \delta ) < \frac{1}{4},\) then \(\lim \nolimits _{i \rightarrow \infty } {\mathbb {P}}_{r}(\Delta C_{abc}^{i} \ge \delta ) = 1\).

  • \(\lim \nolimits _{i \rightarrow \infty } {\mathbb {P}}_{r}(\)rank\((C_{abc}^{i}) = m_{i}-1) =1\).

Considering all the results mentioned above, a main result of this paper can be stated in the following theorem.

Theorem 4.14

Let \(0< \delta < 1\) be a real number and \(h_{4}(\frac{k+l+t}{6} \delta ) < \frac{1}{4}\) then there exists a sequence of \(\mathbb Z_{4}\)-free \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes \(\{C_{i}\}_{i=0}^\infty\) of block length \((km_{i}, lm_{i}, tm_{i})\), when \(m_{i} \rightarrow \infty\), such that

  • \(\lim \nolimits _{i \rightarrow \infty } R(C_{i}) = \frac{1}{k+l+t}\)

  • \(\Delta (C_{i}) \ge \delta\)

Consequently, \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes are asymptotically good.

Proof

By Corollary 4.13, if \(h_{4}(\frac{k+l+t}{6} \delta ) < \frac{1}{4}\) then \(\lim \nolimits _{i \rightarrow \infty } {\mathbb {P}}_{r}(\Delta C_{i} \ge \delta ) = 1\) and \(\lim \nolimits _{i \rightarrow \infty } {\mathbb {P}}_{r} (\text {rank}(C_{i})= m_{i}-1) =1.\) It implies that, there exists an integer \(N >0\) such that for \(i>N\), we have \(\text {rank}(C_{i})= m_{i}-1\) and \(\Delta (C_{i}) \ge \delta .\) Thus, if we delete the first N codes and then for the remaining codes we have \(\text {rank}(C_{i})= m_{i}-1\) and \(\Delta (C_{i}) \ge \delta .\) The asymptotic rate of \(C_{i}\) is

$$\begin{aligned}\lim \limits _{i \rightarrow \infty } R(C_{i}) = \lim \limits _{i \rightarrow \infty } \frac{\text {rank}(C_{i})}{km_{i}+lm_{i}+tm_{i}}= \lim \limits _{i \rightarrow \infty } \frac{m_{i}-1}{(k+l+t)m_{i}}= \frac{1}{k+l+t}\end{aligned}$$

and the asymptotic relative distance of \(C_{i}\) is \(\Delta (C_{i}) \ge \delta\). Now, it can be seen that the relative distance and the rate of \(C_{i}\) are positively bounded from below. So, by definition, \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes are asymptotically good. \(\square\)

Example 4.15

We find a sequence of codes \(\{C_{i}\}_{i=0}^\infty\) of \(\mathbb Z_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes and their rate converges to \(\frac{1}{3}\) and relative distance greater than or equal to \(\frac{1}{8}\), and to show they are asymptotically good.

Assume that \(k=l=t=1\), let \(\delta = \frac{1}{8}\) and \(h_{4}(\frac{1}{16})=.21817511< .25\). So, \({\mathbb {R}}_{km}={\mathbb {R}}_{m} = \frac{{\mathbb {Z}}_{4}[x]}{ \langle x^{m} - 1 \rangle } = {\mathbb {R}}_{lm}={\mathbb {R}}_{tm}={\mathbb {R}}_{kltlm}\), where \(m,k,l~\text {and}~t\) are positive integers such that \(\gcd (m,4)=1\) and \(k, l, t~\text {and}~ 4\) are pairwise co-prime. Therefore, it is easy to see that \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes in \({\mathbb {Z}}_{4}^{m} \times {\mathbb {Z}}_{4}^{m} \times {\mathbb {Z}}_{4}^{m}\) are \({\mathbb {Z}}_{4}[x]\)-submodules of \({\mathbb {R}}_{m} \times {\mathbb {R}}_{m} \times {\mathbb {R}}_{m}\), for \((a(x), b(x), c(x)) \in {\mathbb {R}}_{m} \times {\mathbb {R}}_{m} \times {\mathbb {R}}_{m}\). Hence, consider a sequence of codes \(\{C_{i}\}_{i=0}^\infty\) of \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\mathbb Z_{4}\)-additive cyclic codes as follows.

Let \(C_{abc}^{i} = \{ (f(x)a(x), f(x)b(x), f(x)c(x)) \in {\mathbb {R}}_{m_{i}} \times {\mathbb {R}}_{m_{i}} \times {\mathbb {R}}_{m_{i}} | f(x) \in {\mathbb {J}}_{m_{i}}\}\) and \(m_{i}\) be the positive integers such that \(\gcd (m_{i}, 4) = 1\). Further, \(\lim \nolimits _{i \rightarrow \infty } \frac{\log _{4} m_{i}}{k_{m_{i}}} = 0\) for \(m_{i} \rightarrow \infty\), where \(k_{m_{i}}\) is as defined in Lemma 4.5. Now by Corollary 4.13, we get \(\lim \nolimits _{i \rightarrow \infty } {\mathbb {P}}_{r}(\Delta C_{abc}^{i} \ge \frac{1}{8} ) = 1\) and \(\lim \nolimits _{i \rightarrow \infty } {\mathbb {P}}_{r} (\text {rank}(C_{abc}^{i})= m_{i}-1) =1\). Therefor, by Theorem 4.14

  • \(\lim \nolimits _{i \rightarrow \infty } R(C_{i}) = \frac{1}{3}\)

  • \(\Delta (C_{i}) \ge \frac{1}{8}\)

Now, it can be seen that the relative distance and the rate of \(C_{i}\) are positively bounded from below. Hence, the sequence of codes \(\{C_{i}\}_{i=0}^\infty\) of \({\mathbb {Z}}_{4}\mathbb Z_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes is asymptotically good.

5 Conclusion

In this paper, we have discussed \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\mathbb Z_{4}\)-additive cyclic codes of different component lengths and constructed a class of \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\mathbb Z_{4}\)-additive cyclic codes \(C_{abc}\). Moreover, we have found a basis set for \(C_{abc}\) and presented a method to determine a generator matrix for the code \(C_{abc}\). By using a probabilistic method, we have constructed a random sequence of codes \(C_{abc}^{i}\) of \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes. Moreover, we have studied the asymptotic properties of these classes of \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes and then we proved \(\lim \nolimits _{i \rightarrow \infty } {\mathbb {P}}_{r}(\Delta C_{abc}^{i} \ge \delta ) = 1\) and \(\lim \nolimits _{i \rightarrow \infty } {\mathbb {P}}_{r} (\text {rank}(C_{abc}^{i})= m_{i}-1) =1\). Additionally, we have determined the asymptotic rates and relative distances of these classes of codes using probabilistic methods and found that they are asymptotically good. Also, we have presented a supporting example for these classes of codes.

In the future, it would be interesting to study the asymptotic properties of other families of codes, such as other additive cyclic codes generated by 3-tuples of polynomials of different code lengths.