EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 2, 2019, 668-679
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Counting Z2Z4Z8-additive codes

Basri Çalışkan1,∗, Kemal Balıkçı2

1 Department of Mathematics, Faculty of Arts and Science,
Osmaniye Korkut Ata University, Osmaniye, Turkey
2 Department of Electrical and Electronic, Engineering Faculty,
Osmaniye Korkut Ata University, Osmaniye, Turkey

Abstract. In Algebraic Coding Theory, all linear codes can be described by generator matrices.
Any linear code has different generator matrices. It is important to find the number of the generator
matrices for the construction of these codes. In this paper, we study Z2Z4Z8-additive codes, which
are an extension of Z2Z4-additive codes. We count the number of arbitrary Z2Z4Z8-additive codes.
Then we investigate connections to Z2Z4 and Z2Z8-additive codes with Z2Z4Z8-additive codes, and
give some illustrative examples.

2010 Mathematics Subject Classifications: 94B15, 94B60

Key Words and Phrases: Additive codes, Z2Z4Z8-additive codes, Gaussian numbers.

1. Introduction

Coding Theory is important in modern communication systems and has become
applicable to many areas such as data storage devices, mobile phones and the Internet.
One of the main problems in communication is ”How accurately can the symbols of com-
munication be transmitted?”. This problem is concerned with the accuracy of transference
from sender to receiver of sets of symbols; that is, it involves transmission of finite discrete
symbols [12]. For this purpose, linear codes are widely used in coding theory. A linear
code of length n over Fq is a subspace C of the vector space Fnq where Fq is a finite field
of size q which is prime or a power of a prime. Although the finite fields are widely used
in coding theory, the studies of the codes on different rings have attracted the interest of
the researchers [9].

Let Zm be the ring of integers modulo m. Znm is called the set of Cartesian product
of n copies of Zm. Any nonempty subset C of Znm is a code and a submodule of a Znm
is called a linear code of length n over Zm. Specially, for m=2 and m = 4 the codes
are called binary (Z2) and quaternary codes (Z4), respectively. It was shown that specific

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3427

Email addresses: bcaliskan@osmaniye.edu.tr (B. Çalışkan), kbalikci@osmaniye.edu.tr (K. Balıkçı)

http://www.ejpam.com 668 c© 2019 EJPAM All rights reserved.



B. Çalışkan, K. Balıkçı / Eur. J. Pure Appl. Math, 12 (2) (2019), 668-679 669

good non-linear binary codes which simplifies encoding and decoding can be seen as binary
images of linear codes over Z4 under the Gray map [9].

Additive codes were first defined by Delsarte [7] in terms of association schemes. He
defined additive codes as subgroups of the underlying abelian group in a translation asso-
ciation scheme. According to this, for the binary Hamming scheme, when the underlying
abelian group is of order 2n, the only structures for the abelian group are those of the
form Zα2 × Zβ4 where α and β are positive integers (n = α + 2β). Later, translation in-
variant propelinear codes were first introduced by Pujol et al.[11]. They showed that all

these binary codes are group-isomorphic to subgroups of Zα2 × Zβ4 × Qσ
8 , being Q8 the

non-abelian quaternion group with eight elements. Recently, in particular, Z2Z4-additive
codes and Z2Z4Z8-additive codes, which are a generalization of Z2Z4-additive codes, have
been extensively studied [1–6, 8].

In order to examine the Z2Z4Z8-additive codes of a given length and type, it is neces-
sary to construct the generator matrices of these codes and to find the number of these
generator matrices. In fact, the counting problem is the calculation of the number of
subspaces created by the rows of the generator matrices over the finite fields. Recently,
in this regard, there has been some researches of particular types over the different rings.
For example, while Dougherty et al. [8] counted the Z2Z4-additive codes, Aydogdu et al.
[3] performed the count of the Z2Z8-additive codes. In this paper, we aim to focus on the
number of matrices that generate different (not necessarily equivalent) Z2Z4Z8-additive
codes. First of all, we will introduce the structure of Z2Z4Z8-additive codes and then give
the basic theorem of our article and a simple example. In addition, we will demonstrate
the relationship of Z2Z4Z8-additive codes with Z2Z4 and Z2Z8-additive codes.

2. Preliminaries

In this section, we briefly introduce the backgrounds of our proceeding works.

2.1. Z2Z4Z8-Additive Codes

Definition 1. Let Z2, Z4 and Z8 be the rings of integers modulo 2, 4 and 8, respectively.
Then, C is called a Z2Z4Z8-additive code if it is a subgroup of Zα2 × Zβ4 × Zθ8 where α, β
and θ are positive integers [2].

According to this definition, the first α coordinates of C consist of entries from Z2, the
next β coordinates are elements from Z4 and remaining θ coordinates are the elements of
the ring Z8. From the Fundamental Theorem of Finite Abelian Groups, such an additive
code C which is a subgroup of Zα2 × Zβ4 × Zθ8 is group isomorphic to the abelian structure

Zk02 × Zk14 × Zk22 × Zk38 × Zk44 × Zk52 .

By this isomorphism, such a Z2Z4Z8-additive code is classified as of type
(α, β, θ; k0, k1, k2, k3, k4, k5) [2].



B. Çalışkan, K. Balıkçı / Eur. J. Pure Appl. Math, 12 (2) (2019), 668-679 670

2.2. Generator and Parity-check matrices of Z2Z4Z8-additive codes

Let C be a Z2Z4Z8-additive code of type (α, β, θ; k0, k1, k2, k3, k4, k5). Then, it is
shown that C is permutation equivalent to a Z2Z4Z8-additive code which has the following
generator matrix in standard form:

Ik0 A01 0 0 2T1 0 0 0 4T2
0 S1 Ik1 B01 B02 0 0 2T3 2T4
0 0 0 2Ik2 2B12 0 0 0 4T5
0 S2 0 S01 S02 Ik3 A01 A02 A03

0 S3 0 0 2S12 0 2Ik4 2A12 2A13

0 0 0 0 0 0 0 4Ik5 4A23

 (1)

where A01, S1, S2, S3 are matrices with all entries from Z2 and B02, B12, S02 and S12
are matrices over Z4. Also, T4, T5 and Ai3 are matrices over Z8 for 0 ≤ i ≤ 2. All the
entries in B01, S01 and T1 are in {0, 1} ⊆ Z4. Likewise, A01 and T2 are matrices over Z4

whose all entries are from {0, 1}. T3, A12 and A02 are matrices over Z8, but all values are
the elements of the set {0, 1, 2, 3}. Also, C has 2k022k12k223k322k42k5 codewords [2].

In (1), k0, k2 and k5 represent the number of order 2 generators that are contributed
through Z2, Z4 and Z8 parts, respectively. Also, k1 and k4 represent the number of
order 4 generators that are contributed through Z4 and Z8 parts, respectively. k3 repre-
sents the number of order 8 generators that are contributed through the Z8 part. Note
that, the order 2 elements from the Z4 and Z8 parts have the first α coordinates all zero [2].

The inner product for the elements u,v ∈ Zα2 × Zβ4 × Zθ8 as follows

〈u · v〉 = 4

(
α∑
i=1

uivi

)
+ 2

 α+β∑
j=α+1

ujvj

+

α+β+θ∑
k=α+β+1

ukvk.

Definition 2. The set of all vectors which are orthogonal to every vector in C is called
the dual code of C, and denoted by C⊥. The dual code C⊥ can be defined in the usual way
with respect to inner product

C⊥ =
{
v ∈ Zα2 × Zβ4 × Zθ8| 〈u · v〉 = 0 for all u ∈ C

}
.

A generator matrix for C⊥ is called a parity-check matrix of C [2].

Let C be a Z2Z4Z8-additive code with the generator matrix (1), then we have the
following parity-check matrix:

−At01 Iα−k0 −2St1 0 0
−T t1 0 Bt

02 −Bt
12B

t
01 Bt

12 Iβ−k1−k2
0 0 −2Bt

01 2Ik2 0 P
−T t2 0 −T t4 + T t3A

t
23 + T t5B

t
01 −T t5 0

0 0 −2T t3 0 0
0 0 0 0 0

 (2)



B. Çalışkan, K. Balıkçı / Eur. J. Pure Appl. Math, 12 (2) (2019), 668-679 671

where P is the matrix:

P =


4S2

t − 2S3
t
At01 −2S3

t
0 0

−2St01Bt12 − 2St02 + 2St12A
t
01 −2St12 0 0

−4St01 0 0 0
−At03 +At13A

t
01 +At23A

t
02 −At23At12At01 + 2St01T

t
5 −At13 +At23A

t
12 −At23 Iθ−k3−k4−k5

−2At02 + 2At12A
t
01 −2At12 2Ik5 0

−4At01 4Ik4 0 0

 · (3)

So, the dual code C⊥ is of type (α, β, θ;α− k0, β − k1 − k2, k2, θ − k3 − k4 − k5, k5, k4) .

3. The Number of Generator Matrices for Z2Z4Z8-Additive Codes

In this section, we shall count the number of arbitrary Z2Z4Z8-additive codes of type
(α, β, θ; k0, k1, k2, k3, k4, k5). For this purpose, we will give two lemmas and the definition
of Gaussian coefficient. We will also present our main theorem and give some examples.

Lemma 1. The number of ways of choosing elements (or vectors) to generate a Z2Z4Z8-
additive code of type (α, β, θ; k0, k1, k2, k3, k4, k5) is

∏6
j=1Nj, where

N1 =

k0−1∏
i=0

(
2α − 2i

)
2β+θ,

N2 =

k1−1∏
i=0

(
4β − 2β+i

)
2α+2θ,

N3 =

k2−1∏
i=0

(
2β+θ − 2θ+k1+i

)
,

N4 =

k3−1∏
i=0

(
8θ − 4θ2i

)
2α+2β,

N5 =

k4−1∏
i=0

(
4θ − 2θ+k3+i

)
2α+2β,

N6 =

k5−1∏
i=0

(
2θ − 2k3+k4+i

)
.

Proof. Recall the generating matrix given in (1), we will examine six major parts

for proof. In Zα2Z
β
4Zθ8, because of the number of all vectors of order 2 is 2α2β2θ, we can

choose first element of order 2 that contributes through Z2 part in (2α − 1) 2β2θ ways.
Next, (2α − 2) 2β2θ selections can be made for the second element. Similarly, the last
element can be selected by

(
2α − 2k0−1

)
2β2θ ways. Thus, the k0 elements of order 2 that

contribute only to the binary part are selected in N1 different ways.
Next, we will choose k1 vectors of order 4 that from the Z4 part. However, in those

vectors, there should not be vectors of order 4 that contribute to the Z8 part. There are



B. Çalışkan, K. Balıkçı / Eur. J. Pure Appl. Math, 12 (2) (2019), 668-679 672

2α4β4θ vectors of order 4 in all space. Since the number of elements of order 4 in the Z4

part is 2α
(
4β − 2β

)
4θ, the first element can be selected in 2α

(
4β − 2β · 1

)
4θ ways. The

number of choosing for the next element is 2α
(
4β − 2β · 2

)
4θ and 2α

(
4β − 2β2k1−1

)
4θ for

the last element. So, there are N2 different ways for k1 elements of order 4 that contribute
to the Z4 part.

Next, we need to choose k2 vectors of order 2 that are not in the space generated by
k1 vectors of order 4. This imposes that the first α entries of such elements to be all
zero. Thus, in order to pick such an element first we subtract elements of order 2 that
are obtained through already chosen k1 elements for the Z4 part. So, there are 2β2θ such
vectors in the Z4 part. Then, first element can be chosen in 2β2θ−2θ2k1 ways, next choice
comes from 2β2θ − 2 · 2θ2k1 and inductively we reach N3.

Now, there are 2α4β8θ vectors in total but the number of all vectors of order 8 are
2α4β8θ − 2α4β4θ. So, we choose the first one of order 8 in 2α4β

(
8θ − 4θ

)
different ways.

Then we choose the remaining k3−1 elements of order 8 in a similar way as before. Hence,
the last element can be chosen in 2α4β

(
8θ − 4θ2k3−1

)
ways. So, we obtain N4.

Next, to choose vectors of order 4 that are contributed through the Z8 part. There are
2α4β

(
4θ − 2θ2k3

)
selections for the first vector. Note that, the elements of order 4 that are

formed from the k3 elements of order 8 by taking their 2 multiples need to be considered.
Next, 2α4β

(
4θ − 2 · 2θ2k3

)
selections can be made for the second element. Similarly, the

last element can be selected by 2α4β
(
4θ − 2k4−1 · 2θ2k3

)
ways. Thus, the k4 elements of

order 4 that contribute only to the Z8 part are selected in N5 different ways.
Finally, we need to choose k5 vectors of order 2 that are not generated by k3 and k4

vectors of order 2. Note that, the first entries of such elements to be all zero. So, there are
2θ vectors of order 2. In order to pick such an element first we subtract order 2 elements
that are obtained through already chosen k3 and k4 elements for the Z8 part. So, we have
2θ − 2k3+k4 choices. The second element comes from 2θ − 2 · 2k3+k4 and inductively we
reach N6.

Lemma 2. The number of distinct generator matrices of a Z2Z4Z8-additive code of type
(α, β, θ; k0, k1, k2, k3, k4, k5) is

∏6
j=1Dj, where

D1 = 2k0(k1+k2+k3+k4+k5)
k0−1∏
i=0

(
2k0 − 2i

)
,

D2 = 2k1(k0+k1+k2+2k3+2k4+k5)
k1−1∏
i=0

(
2k1 − 2i

)
,

D3 = 2k2(k1+3k3+2k4+k5)
k2−1∏
i=0

(
2k2 − 2i

)
,

D4 = 2k3(k0+2k1+k2+2k3+2k4+k5)
k3−1∏
i=0

(
2k3 − 2i

)
,



B. Çalışkan, K. Balıkçı / Eur. J. Pure Appl. Math, 12 (2) (2019), 668-679 673

D5 = 2k4(k0+2k1+k2+2k3+k4+k5)
k4−1∏
i=0

(
2k4 − 2i

)
,

D6 = 2k5(k3+k4)
k5−1∏
i=0

(
2k5 − 2i

)
.

Proof. As we did in Lemma 1, we will apply the same procedure in a group of the
type (k0, k1, k2, k3, k4, k5). In order to choose first element of order 2 that contributes
through the Z2 part, we subtract all elements of order 2 in the group from the ones that
are not coming through the Z2 part, i.e. 2k0+k1+k2+k3+k4+k5 − 2k1+k2+k3+k4+k5 . Next, the
second element can be chosen in 2k0+k1+k2+k3+k4+k5 − 2 · 2k1+k2+k3+k4+k5 different ways
and inductively we reach at D1.

Next, to choose an element of order 4 in the group, we have
(
4k1 − 2k1

)
2k0+k2+2k3+2k4+k5

choices. Then, there are
(
4k1 − 2 · 2k1

)
2k0+k2+2k3+2k4+k5 choices for second element of or-

der 4. So, inductively, we have D2.
Now, to choose an element of order 2 in the group that only contributes through the

Z4 part. There are
(
2k1+k2 − 2k1

)
23k3+2k4+k5 choices for the first element. Next, the

second element can be chosen
(
2k1+k2 − 2 · 2k1

)
23k3+2k4+k5 different ways and inductively

we reach at D3.
After the Z2 and Z4 parts, we now calculate the selections for the Z8 part. First, in

order to choose an element of order 8 in the group, we have
(
8k3 − 4k3

)
2k0+2k1+k2+2k4+k5

choices. Next, we can choose second element in
(
8k3 − 2 · 4k3

)
2k0+2k1+k2+2k4+k5 different

ways. Similarly, if we continue, we reach at D4.
Next, to choose an element of order 4 within the group that only contributes through

the Z8 part. We have
(
4k4 − 2k4

)
2k0+2k1+k2+2k3+k5 choices. Therefore, there are(

4k4 − 2 · 2k4
)

2k0+2k1+k2+2k3+k5 different choices for the second element. So, inductively,
the last element can be chosen in

(
4k4 − 2k4−12k4

)
2k0+2k1+k2+2k3+k5 different ways. As a

result, we reach at D5.
Finally, to choose an element of order 2 within the group that solely contributes through

the Z8 part. For this, there are 2k3+k4+k5 − 2k3+k4 choices. The second element can be
chosen in 2k3+k4+k5 − 2 · 2k3+k4 different ways. So, the last element can be chosen in
2k3+k4+k5 − 2k5−12k3+k4 different ways. In this way, D6 is obtained and the proof is
completed.

Definition 3. [10] Let n and q be two positive integers

[n]q = 1 + q + q2 + . . .+ qn−1 =
qn − 1

q − 1

and the q-factorial is defined as

[n]q! = [n]q · [n− 1]q · · · [2]q · [1]q



B. Çalışkan, K. Balıkçı / Eur. J. Pure Appl. Math, 12 (2) (2019), 668-679 674

and

a−1∏
j=0

(
2b − 2j

)
=

(
2b − 1

)(
2b − 2

)(
2b − 22

)
· · ·
(

2b − 2a−2
)(

2b − 2(a−1)
)

= 21+2+...+(a−1)
(

2b − 1
)(

2b−1 − 1
)
· · ·
(

2b−(a−1) − 1
)

= 2

(a
2
)

[b]2!

[b− a]2!
·

Definition 4. [10] Let n, k and q be non-negative integers such that k ≤ n. Then,[
n
k

]
q

=
[n]q!

[k]q![n− k]q!
·

Here, [0]q! = 1 and if k = 0 then the q binomial coefficient is equal to 1. Also we have an
algebraic expression of Gaussian coefficients as follows;[

n
k1, k2, . . . , km

]
2

=

[
n
k1

]
2

·
[
n− k1
k2

]
2

· · ·
[
n−

∑m−1
i=1

km

]
2

·

Theorem 1. The number of distinct Z2Z4Z8-additive codes of type (α, β, θ; k0, k1, k2, k3, k4, k5)
is

N2×4×8 = 2δ
[
α
k0

]
2

[
β

k1, k2

]
2

[
θ

k3, k4, k5

]
2

where,

δ = k0 [(β − s) + (θ − t)]
+k1 [(α− k0) + (β − s) + 2 (θ − t) + k5]

+k2 [(θ − t)− 2k3 − k4]
+k3 [(α− k0) + 2 (β − s) + 2 (θ − t) + k2 + k5]

+k4 [(α− k0) + 2 (β − s) + (θ − t) + k2]

s = k1 + k2 and t = k3 + k4 + k5.

Proof. In order to prove this theorem we count ordered generators for the code of the
type (α, β, θ; k0, k1, k2, k3, k4, k5). Firstly, we count the ordered generators by choosing

them from the all space Zα2 ×Zβ4 ×Zθ8 which gives say A. Let the number of codes of type
(k0, k1, k2, k3, k4, k5) be N2×4×8. Secondly, we choose the ordered generators from these
codes which gives say B. So, we have the following equation

N2×4×8 =
A

B
·



B. Çalışkan, K. Balıkçı / Eur. J. Pure Appl. Math, 12 (2) (2019), 668-679 675

Note that, Lemma 1 gives the numerator A and Lemma 2 gives the denominator B. From
Definition 3, we have,

N1 = 2
k0(β+θ)+

(
k0
2

)
[α]2!

[α− k0]2!
,

N2 = 2
k1(α+β+2θ)+

(
k1
2

)
[β]2!

[β − k1]2!
,

N3 = 2
k2(θ+k1)+

(
k2
2

)
[β − k1]2!

[β − k1 − k2]2!
,

N4 = 2
k3(α+2β+2θ)+

(
k3
2

)
[θ]2!

[θ − k3]2!
,

N5 = 2
k4(α+2β+θ+k3)+

(
k4
2

)
[θ − k3]2!

[θ − k3 − k4]2!
,

N6 = 2
k5(k3+k4)+

(
k5
2

)
[θ − k3 − k4]2!

[θ − k3 − k4 − k5]2!
.

Also, from Definition 3, we have,

D1 = 2
k0(k1+k2+k3+k4+k5)+

(
k0
2

)
[k0]2!,

D2 = 2
k1(k0+k1+k2+2k3+2k4+k5)+

(
k1
2

)
[k1]2!,

D3 = 2
k1+k2(3k3+2k4+k5)+

(
k2
2

)
[k2]2!,

D4 = 2
k3(k0+2k1+k2+2k3+2k4+k5)+

(
k3
2

)
[k3]2!,

D5 = 2
k4(k0+2k1+k2+2k3+k4+k5)+

(
k4
2

)
[k4]2!,

D6 = 2
k5(k3+k4)+

(
k5
2

)
[k5]2! ·

Then,

N2×4×8 =
A

B
=

∏6
j=1Nj∏6
j=1Dj

·

So, we have

N2×4×8 = 2δ
[
α
k0

]
2

[
β
k1

]
2

[
β − k1
k2

]
2

[
θ
k3

]
2

[
θ − k3
k4

]
2

[
θ − k3 − k4

k5

]
2

.



B. Çalışkan, K. Balıkçı / Eur. J. Pure Appl. Math, 12 (2) (2019), 668-679 676

Hence the number of distinct Z2Z4Z8-additive codes of type (α, β, θ; k0, k1, k2, k3, k4, k5)
is

N2×4×8 = 2δ
[
α
k0

]
2

[
β

k1, k2

]
2

[
θ

k3, k4, k5

]
2

where, δ, s and t are as defined above.

Example 1. Let C be a Z2Z4Z8-additive code of type (2, 1, 1, 1, 1, 0, 1, 0, 0), then all
possible generator matrices are 12 matrices that generate different codes. Here α = 2,
β = θ = 1, k0 = k1 = k3 = 1 and k2 = k4 = k5 = 0.

Here, N1, N2 and N4 are calculated as 12, 32 and 64, respectively. Also, D1, D2 and
D4 are 4, 16 and 32, respectively. Because of k2 = k4 = k5 = 0, N3, N5, N6, D3, D5 and
D6 do not exist. So, we skip these terms, hence

N2×4×8 =
N1N2N4

D1D2D4
=

24576

2048
= 12.

These 12 generator matrices can be obtained as follows:

•

 1 x 0 0
0 y 1 0
0 z 0 1

 here, we have 8 possible matrices,

•

 0 1 0 0
y 0 1 0
z 0 0 1

 here, we have 4 possible matrices,

where x, y and z are either 0 or 1.

From Theorem 1, we have the following corollary.

Corollary 1. The dual of a Z2Z4Z8-additive code of type (α, β, θ; k0, k1, k2, k3, k4, k5) is
of type (α, β, θ;α− k0, β− k1− k2, k2, θ− k3− k4− k5, k5, k4) and the number of generator
matrices of this dual code is

N2×4×8 = 2δ
[

α
α− k0

]
2

[
β

β − s, k2

]
2

[
θ

θ − t, k5, k4

]
2

where

δ = (α− k0) (k1 + k3)

+ (β − s) (k0 + k1 + 2k3 + k5)

+k2 (k3 − 2 (θ − t)− k5)
+ (θ − t) (k0 + 2k1 + k2 + 2k3 + k4)

+k5 (k0 + 2k1 + k2 + k3) .

where s = k1 + k2 and t = k3 + k4 + k5.



B. Çalışkan, K. Balıkçı / Eur. J. Pure Appl. Math, 12 (2) (2019), 668-679 677

Example 2. Let us show that for the code in the Example 1, the number of generator
matrices of the dual code is 12. Since α − k0 = 2 − 1, β − k1 − k2 = 1 − 1 − 0 = 0,
k2 = 0, θ − k3 − k4 − k5 = 1− 1− 0− 0 = 0 and k5 = k4 = 0; the type of the dual code is
(2, 1, 1, 1, 0, 0, 0, 0, 0) and δ = (2− 1)(1 + 1) + 0 + 0 + 0 + 0 = 2. So,

N2×4×8(2, 1, 1, 1, 0, 0, 0, 0, 0) = 22
[

2
2

]
2

[
1

0, 0

]
2

[
1

0, 0, 0

]
2

= 12.

4. Connections to Z2Z4 and Z2Z8-Additive Codes

Here, we relate the number of the Z2Z4Z8-additive codes with the number of additive
codes over Z2Z4 and Z2Z8, respectively. Hence, we also obtain formulas for counting the
number of matrices that generate all these codes.

Corollary 2. Let C be a Z2Z4Z8-additive code of type (α, β, θ; k0, k1, k2, k3, k4, k5). If
θ = k3 = k4 = k5 = 0, then we get a Z2Z4-additive code of type (α, β; k0, k1, k2). The
number of these distinct codes is as follows

N2×4×8(α, β, 0; k0, k1, k2, 0, 0, 0) = 2k0(β−k1−k2)+k1(α+β−k0−k1−k2)
[
α
k0

]
2

[
β

k1, k2

]
2

= N2×4(α, β; k0, k1, k2)

Proof. It is known [5] that the generator matrix for a Z2Z4-additive code C of type
(α, β; γ, δ, κ) has the following standard form Iκ Tb 2T2 0 0

0 0 2T1 2Iγ−κ 0
0 Sb Sq R Iδ

 (4)

where Tb, Sb are matrices over Z2; T1, T2, R are matrices over Z4 with all entries in
{0, 1} ∈ Z4 and Sq is a matrix over Z4. Also, from [13], we know that the generator
matrix of C of type (α, β; k0, k1, k2) is in the form of Ik0 A01 0 0 2T02

0 S1 Ik1 A01 A02

0 0 0 2Ik2 2A12

 (5)

which is permutation equivalent to a matrix of the form (4). So, from the generator
matrices (4) and (5) we have the equations such that κ = k0, δ = k1 and γ − κ = k2.

In Theorem 1, it is clear that if θ = k3 = k4 = k5 = 0, then we have a Z2Z4Z8-additive
code of type (α, β, 0; k0, k1, k2, 0, 0, 0). Also, the number of these codes is

N2×4×8(α, β, 0; k0, k1, k2, 0, 0, 0) = 2k0(β−k1−k2)+k1(α+β−k0−k1−k2)
[
α
k0

]
2

[
β

k1, k2

]
2

= N2×4(α, β; k0, k1, k2) (6)



B. Çalışkan, K. Balıkçı / Eur. J. Pure Appl. Math, 12 (2) (2019), 668-679 678

In (6), if we replace κ = k0, δ = k1 and γ − κ = k2, then it is obtained the number of
Z2Z4-additive codes C of type (α, β; γ, δ, κ) as in [8], (Theorem 3.3).

Corollary 3. Let C be a Z2Z4Z8-additive code of type (α, β, θ; k0, k1, k2, k3, k4, k5). If
β = k1 = k2 = 0, then we get a Z2Z8-additive code of type (α, θ; k0, k3, k4, k5). The
number of these distinct codes is as follows

N2×4×8(α, 0, θ; k0, 0, 0, k3, k4, k5) = 2k0(θ−k3−k4−k5)+k3[(α−k0)+2(θ−k3−k4−k5)+k5]
[
α
k0

]
2

[
θ

k3, k4, k5

]
2

= N2×8(α, θ; k0, k3, k4, k5).

Proof. In Theorem 1, it is clear that If β = k1 = k2 = 0, then we have a Z2Z4Z8-
additive code of type (α, 0, θ; k0, 0, 0, k3, k4, k5). Also, the number of these codes is

N2×4×8(α, 0, θ; k0, 0, 0, k3, k4, k5) = 2k0(θ−k3−k4−k5)+k3[(α−k0)+2(θ−k3−k4−k5)+k5]
[
α
k0

]
2

[
θ

k3, k4, k5

]
2

= N2×8(α, θ; k0, k3, k4, k5). (7)

In (7), if we replace both k3 = k1, k4 = k2, k5 = k3 and k1 + k2 + k3 = l, then it is
obtained the number of Z2Z8-additive codes of type (α, θ; k0, k3, k4, k5) as in [3], (Theorem
2.1).

Example 3.

N2×4(3, 4; 2, 1, 2) = N2×4×8(3, 4, 0; 2, 1, 2, 0, 0, 0) = 11760

N2×8(3, 4; 2, 0, 1, 2) = N2×4×8(3, 0, 4; 2, 0, 0, 0, 1, 2) = 11760

5. Conclusions

In this work we established a formula that gives the number of distinct Z2Z4Z8-additive
codes and their dual codes. Also, we showed the relationships among the numbers of
Z2Z4, Z2Z8 and Z2Z4Z8-additive codes. Also, we gave some examples that show the
relationships.

Moreover, it is clear that in Theorem 1, if the parameters of Z2Z4Z8-additive codes of
type (α, β, θ; k0, k1, k2, k3, k4, k5) are appropriated, the numbers of Z2, Z4 and Z8 linear
codes can be obtained.

Acknowledgements

We would like to thank the referees for their valuable remarks.



REFERENCES 679

References

[1] T Abualrub, I Siap, and N Aydin. Z2Z4-additive cyclic codes. IEEE Trans. Inform.
Theory, 60(3):15081514, 2014.

[2] I Aydogdu and F Gursoy. Z2Z4Z8-Cyclic Codes. J. Appl. Math. Comput., 60(1-
2):327341, 2019.

[3] I Aydogdu and I Siap. Counting The Generator Matrices of Z2Z8-Codes. Mathemat-
ical Sciences And Applications E-Notes, 1(2):143–149, 2013.

[4] I Aydogdu and I Siap. On ZprZps-additive codes. Linear Multilinear Algebra,
63(10):20892102, 2015.

[5] J Borges, C Fernndez-Crdoba, J Pujol, J Rif, and M Villanueva. Z2Z4-linear codes:
generator matrices and duality. Designs, Codes Cryptogr., 54(2):167179, 2010.

[6] J Borges, C Fernndez-Crdoba, and R Ten-Valls. Z2Z4-additive cyclic codes, generator
polynomials and dual codes. IEEE Trans. Inform. Theory, 62(11):63486354, 2016.

[7] P Delsarte. An algebraic approach to the association schemes of coding theory.
Philips Research Rep.Suppl., 10, 1973.

[8] S.T Dougherty and E Salturk. Counting Z2Z4-Additive Codes. Noncommutative
Rings and Their Applications, Contemporary Mathematics, 634:137–147, 2015.

[9] A.R Hammons, P.V Kumar, A.R Calderbank, N.J.A Sloane, and P Sol. The Z4-
linearity of kerdock, preparata, goethals and related codes. IEEE Trans. Inform.
Theory, 40:301319, 1994.

[10] F. J MacWilliams and N.J.A Sloane. The Theory of Error-Correcting Codes. North-
Holland Pub. Co., New York, NY, 1977.

[11] J Pujol and J Rif. Translation invariant propelinear codes. IEEE Trans. Inform.
Theory, 43:590598, 1997.

[12] C.E Shannon and W Weaver. The Mathematical Theory of Communication. The
University of Illinois, Urbana, 1964.

[13] I Siap and I Aydogdu. The Structure of Z2Z2s-Additive Codes: Bounds on the
Minimum Distance. Appl. Math. Inf. Sci., 7(6):2271–2278, 2013.