ISSN 2148-838Xhttps://doi.org/10.13069/jacodesmath.1056624

J. Algebra Comb. Discrete Appl.
9(1) • 57–70

Received: 11 July 2021
Accepted: 8 October 2021

Journal of Algebra Combinatorics Discrete Structures and Applications

1-generator two-dimensional quasi-cyclic codes over
Z4[u]/〈u2 −1〉

Research Article

Arazgol Ghajari, Kazem Khashyarmanesh, Zohreh Rajabi

Abstract: In this paper, we obtain generating set of polynomials of two-dimensional cyclic codes over the ring
R = Z4[u]/〈u2−1〉, where u2 = 1. Moreover, we find generator polynomials for two-dimensional quasi-
cyclic codes and two-dimensional generalized quasi-cyclic codes over R and specify a lower bound
on minimum distance of free 1-generator two-dimensional quasi-cyclic codes and two-dimensional
generalized quasi-cyclic codes over R.

2010 MSC: 12E20, 94B05, 94B15, 94B60

Keywords: Two-dimensional cyclic codes, Two-dimensional quasi-cyclic codes, Two-dimensional generalized quasi-cyclic
codes

1. Introduction

There are many generalizations of cyclic codes. One of them is two-dimensional cyclic codes. A lot of
works on two-dimensional cyclic codes has done. Ikai et al. first introduced the concept of common zeros
for characterizing two-dimensional codes [6], and showed the existence of two-dimensional codes that can
be characterized by the common zeros. After that the researchers have studied with different concepts
in these codes. The reader can find some of such studies in the papers [11–13]. Moreover, Lalason et al.
[7] construct a basis of an s-dimensional cyclic code over a finite field. On the other hand, quasi-cyclic
codes are another natural generalizations of cyclic codes. The study of quasi-cyclic codes over finite rings
has provided useful information in coding theory. We shall use the phrase ‘QC code’ as an abbreviation
for ‘quasi-cyclic code’ and ‘GQC code’ for ‘generalized quasi-cyclic codes’. QC codes form an important
class of linear codes which also include cyclic codes (when we consider the case ` = 1). Ling and Solé
studied the algebraic structure of QC codes over finite fields and provided a new algebraic approach
to QC codes (see also [8]). There have been a lot of investigations of QC codes and GQC codes over

Arazgol Ghajari, Kazem Khashyarmanesh (Corresponding Author), Zohreh Rajabi; Department of Pure Math-
ematics, Ferdowsi University of Mashhad, Mashhad, Iran (email: Ghajari3051@gmail.com, khashyar@ipm.ir,
rajabi261@yahoo.com).

57

https://orcid.org/0000-0002-3675-2832
https://orcid.org/0000-0003-3314-7298
https://orcid.org/0000-0001-7857-2672


A. Ghajari et. al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 57–70

the rings, for example [1–3, 5, 9, 14]. In [10], the authors studied the constacyclic codes over the finite
non-chain ring Z4 + uZ4 with u2 = 1 and obtained some new Z4-linear codes. Finally, Gao et al. [4] have
generalized QC codes and GQC codes over the finite non-chain ring R = Z4[u]/〈u2 − 1〉 with u2 = 1.
They have determined the structure of the generators and the minimal generating sets of 1-generator QC
codes and GQC codes. They also have given a lower bound on minimum distance of free 1-generator
QC codes and GQC codes over R. Furthermore, in [4], some new Z4-linear codes were constructed by
1-generated QC codes and GQC codes over R. Hence, there are many examples of cyclic codes and QC
codes over R. There exist many researches of two-dimensional cyclic codes over finite fields. However,
the research of two-dimensional cyclic codes over R has not been considered by any coding scientist.
Moreover, quasi-cyclic codes perform very well on the codes have great lengths. Therefore, these codes
are the important and most intensively studies classes of linear codes. The ring R = Z4[u]/〈u2 − 1〉 with
u2 = 1 is a Frobenius non-chain ring with 16 elements. There are some examples of cyclic codes over R
whose Z4 Gray images have better parameters than previous best-known Z4-linear codes were presented
(see for example [4] and [10]). The main purpose of this paper is to obtain sets of generator polynomials
of two-dimensional cyclic codes over R. We also determine the structure of the generators and the
minimal generating sets of 1-generator two-dimensional QC codes and two-dimensional GQC codes. This
method probably helps to decode two-dimensional cyclic codes and two-dimensional QC codes as it has
done for cyclic codes and QC codes. This paper is organized as follows: at first, we find the generator
polynomials corresponding to two-dimensional cyclic codes over R. Then, by using these polynomials, we
obtain generator polynomials for two-dimensional QC codes over R. Moreover, we study the structure
of generators two-dimensional QC codes. The last part of the paper is devoted to obtain 1-generator
polynomial two-dimensional GQC codes and determine a lower bound for the minimum distance of free
1-generator GQC codes.

2. Generator polynomials

As was mentioned in the Introduction, the purpose of this section is to obtain a generating set of
polynomials for two-dimensional QC codes over the ring R = Z4[u]/ < u2 − 1 > with u2 = 1. Assume
that S := Z4[x]/ < xm − 1 >, R′ := R[x,y]/ < xm − 1,y3 − 1 >, where y3 = 1, xm = 1 and m is an
odd positive integer. Suppose that n = 3m` and R := R

′`. As before, Fq denotes a finite field with q
elements. Recall that a linear code C′ of length ms over a finite field F is a two-dimensional cyclic code,
if it is closed under row shift and column shift of codewords, whose codewords are viewed as ms arrays.
This means that for every codeword c of the form

c =




c0,0 c0,1 · · · c0,s−1
c1,0 c1,1 · · · c1,s−1
...

...
...

cm−1,0 cm−1,1 · · · cm−1,s−1




in C′, the codewords 

cm−1,0 cm−1,1 · · · cm−1,s−1

c0,0 c0,1 · · · c0,s−1
...

...
...

cm−2,0 cm−2,1 · · · cm−2,s−1




and 


c0,s−1 c0,0 · · · c0,s−2
c1,s−1 c1,0 · · · c1,s−2

...
...

...
cm−1,s−1 cm−1,01 · · · cm−1,s−2




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A. Ghajari et. al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 57–70

also belong to C′.

It is well known that these codes are the ideals of the quotient ring F [x,y]/ < xm − 1,ys − 1 >.
Similarly, we consider the above definition for a two-dimensional cyclic code C′ of length ms over the
ring R. So we define a two-dimensional QC codes over R as follows.

Definition 2.1. Let C be a linear code of length n. If there exists a least positive integer ` such that C is
closed the `th composition under the row shift and the column shift, then we call C is a two-dimensional
QC code over R.

Clearly, ` is a divisor of n. If ` = 1, then C is a two-dimensional cyclic code over R. An r-generator
two-dimensional QC code is an ideal of C with r generators. In the rest of this section, we shall focus on
1-generator two-dimensional QC code over R. According to Gao et al.[3], 1-generator two-dimensional
QC code C over R can be generated by element (b1(x,y), . . . ,b`(x,y)) ∈R, and so

C = {f(x,y)(b1(x,y), . . . ,b`(x,y))|f(x,y) ∈ R[x,y]}
= {(f(x,y)b1(x,y), . . . ,f(x,y)b`(x,y))|f(x,y) ∈ R[x,y]}.

Özen et al.[10] have studied cyclic codes over R. In fact, they determined a generators of the cyclic codes
over R. In [10], it was proved that if m is odd, then S is a principal ideal ring. Now, by using a method
similar that used for two-dimensional cyclic codes over a field in [13], we obtain a generator polynomials for
two-dimensional cyclic codes over R. Our generating set has an important role in determining generator
polynomials two-dimensional QC codes over R.

Note that R is isomorphic to Z4 + uZ4. We begin with the following lemma.

Lemma 2.2. Suppose that C′ is a two-dimensional cyclic code of length 3m over R. Then {pi(x,y) | i =
1, . . . , 6 } is a generating set of C′, where

p1(x,y) =α01(x) + (u + 1)α11(x) + (β01(x) + (u + 1)β11(x))y

+ (γ01(x) + (u + 1)γ11(x))y
2,

p2(x,y) =(u + 1)α12(x) + (β02(x) + (u + 1)β12(x))y + (γ02(x) + (u + 1)γ12(x))y
2,

p3(x,y) =(β03(x) + (u + 1)β13(x))y + (γ03(x) + (u + 1)γ13(x))y
2,

p4(x,y) =(u + 1)β14(x)y + (γ04(x) + (u + 1)γ14(x))y
2,

p5(x,y) =(γ05(x) + (u + 1)γ15(x))y
2,

p6(x,y) =(u + 1)γ16(x)y
2,

and α0i(x), α1i(x), β0i(x), β1i(x),γ0i(x) and γ1i(x) are generator polynomials of cyclic codes over Z4 for
each i = 1, . . . , 6.

Proof. Suppose that I is an ideal of R′ and that f(x,y) is an arbitrary element of I. So it can be
written uniquely as the following form

f(x,y) = f0(x) + (u + 1)f1(x) + (f
′
0(x) + (u + 1)f

′
1(x))y + (f

′′
0 (x) + (u + 1)f

′′
1 (x))y

2,

where f0(x), f′0(x), f
′′
0 (x), f1(x), f

′
1(x) and f

′′
1 (x) are polynomials in S. The main strategy employed in

our proof is to introduce six auxiliary ideals in S. To achieve this, we break our proof into six steps as
follows:

Step 1: Set I0 := {g0(x) ∈ S : there exists g(x,y) ∈ I such that

g(x,y) =g0(x) + (u + 1)g1(x) + (g
′
0(x) + (u + 1)g

′
1(x))y + (g

′′
0 (x) + (u + 1)g

′′
1 (x))y

2,

where g1(x),g
′
0(x),g

′
1(x),g

′′
0 (x),g

′′
1 (x) ∈ S}.

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A. Ghajari et. al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 57–70

It is not hard to see that the set I0 is an ideal of S. Since m is odd, S is a principal ideal ring. Thus,
there exists a polynomial α01(x) in S such that I0 = 〈α01(x)〉. Since α01 is an element of I0, according
to the definition of I0, there exists p1(x,y) ∈ I with

p1(x,y) = α01(x) + (u + 1)α11(x) + (β01(x) + (u + 1)β11(x))y

+ (γ01(x) + (u + 1)γ11(x))y
2,

where α01(x), α11(x), β01(x), β11(x), γ01(x), γ11(x) ∈ S. It is clear that f0(x) ∈ I0. Hence there exists
t0(x) ∈ Z4[x] such that

f0(x) = α01(x)t0(x).

Set

h1(x,y) :=f(x,y) −p1(x,y)t0(x)
=(u + 1)h01(x) + (h

′
01(x) + (u + 1)h

′
11(x))y + (h

′′
01(x) + (u + 1)h

′′
11(x))y

2,

where h01(x), h′01(x), h
′
11(x), h

′′
01(x), h

′′
11(x) ∈ S. Since f(x,y) and p1(x,y) are in I and I is an ideal of

R, h1(x,y) is again in I.

Step 2: Put I′0 := {g1(x) ∈ S : there exists g(x,y) ∈ I such that

g(x,y) =(u + 1)g1(x) + (g
′
0(x) + (u + 1)g

′
1(x))y + (g

′′
0 (x) + (u + 1)g

′′
1 (x))y

2,

where g′0(x),g
′
1(x),g

′′
0 (x),g

′′
1 (x) ∈ S}.

Clearly, I′0 is an ideal of S. Thus, there exists a polynomial α12(x) ∈ S such that I′0 = 〈α12(x)〉. There
exists p2(x,y) ∈ I such that

p2(x,y) = (u + 1)α12(x) + (β02(x) + (u + 1)β12(x))y + (γ02(x) + (u + 1)γ12(x))y
2,

where β02(x), β12(x), γ02(x), γ12(x) ∈ S. According to the definition of I′0, h01(x) ∈ I′0, and so h01(x) =
α12(x)t1(x) for some t1(x) ∈ Z4[x]. Set

h2(x,y) :=h1(x,y) −p2(x,y)t1(x)
=(h′02(x) + (u + 1)h

′
12(x))y + (h

′′
02(x) + (u + 1)h

′′
12(x))y

2,

where h′02(x), h
′
12(x), h

′′
02(x), h

′′
12(x) ∈ S. Since h1(x,y) and p2(x,y) are polynomials in I we have that

h2(x,y) ∈ I.
Step 3: Set I1 := {g′0(x) ∈ S : there exists g(x,y) ∈ I such that

g(x,y) =(g′0(x) + (u + 1)g
′
1(x))y + (g

′′
0 (x) + (u + 1)g

′′
1 (x))y

2, where

g′1(x),g
′′
0 (x),g

′′
1 (x) ∈ S}.

Obviously, I1 is an ideal of S, and so there exists a polynomial β03(x) in S such that I1 = 〈β03(x)〉. There
exists a polynomial p3(x,y) ∈ I such that

p3(x,y) = (β03(x) + (u + 1)β13(x))y + (γ03(x) + (u + 1)γ13(x))y
2,

where β13(x), γ03(x), γ13(x) ∈ S. According to the definition of I1, h′02(x) in I1. Hence h′02(x) =
β03(x)t2(x) for some t2(x) ∈ Z4[x]. Put

h3(x,y) :=h2(x,y) −p3(x,y)t2(x)
=(u + 1)h′13(x)y + (h

′′
03(x) + (u + 1)h

′′
13(x))y

2,

where h′13(x), h
′′
03(x), h

′′
13(x) ∈ S. Similar to the previous discussion h3(x,y) ∈ I.

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A. Ghajari et. al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 57–70

Step 4: Set I′1 := {g′1(x) ∈ S : there exists g(x,y) ∈ I such that

g(x,y) = (u + 1)g′1(x)y + (g
′′
0 (x) + (u + 1)g

′′
1 (x))y

2, where g′′0 (x),g
′′
1 (x) ∈ S}.

It is clear that I′1 is an ideal of S. Thus, there exists a polynomial β13(x) ∈ S such that I′1 = 〈β13〉. Since
β13(x) in I′1, according to definition I

′
1, we have a polynomial p4(x,y) ∈ I, where

p4(x,y) = (u + 1)β13(x)y + (γ03(x) + (u + 1)γ13(x))y
2,

and γ03(x), γ13(x) ∈ S. Obviously, h′13(x) ∈ I′1. So, we get h′13(x) = β13(x)t3(x) for some t3(x) ∈ Z4[x].
Put

h4(x,y) := h3(x,y) −p4(x,y)t3(x) = (h04(x) + (u + 1)h14(x))y2,

where h04(x), h14(x) ∈ S. It is clear that h4(x,y) ∈ I.
Step 5: Set

I2 := {g′′0 (x) ∈ S : there exists g(x,y) ∈ I such that
g(x,y) = (g′′0 (x) + (u + 1)g

′′
1 (x))y

2, where g′′1 (x) ∈ S}.

Clearly, I2 is an ideal of S. Therefore, there exists γ05(x) ∈ S such that

I2 = 〈γ05(x)〉.

Besides, γ05(x) in I2, and so we have a polynomial p5(x,y) ∈ I, where

p5(x,y) = (γ05(x) + (u + 1)γ15(x))y
2,

where γ15(x) ∈ S. Obviously, h04(x) ∈ I2, and so we obtain that h04(x) = γ05(x)t4(x) for some t4(x) ∈
Z4[x]. Put h5(x,y) := h4(x,y)−p5(x,y)t4(x) = (u + 1)h05(x)y2, where h05(x) ∈ S. Similarly, h5(x,y) in
I.

Step 6: Put

I′2 := {g
′′
1 (x) ∈ S : there exists g(x,y) ∈ I such that g(x,y) = (u + 1)g

′′
1 (x)y

2}.

It is clear that I′2 is an ideal of S. Thus there exists γ15(x) ∈ S such that I′2 =< γ15(x) >. Also there
exists a polynomial p6(x,y) ∈ I such that p6(x,y) = (u + 1)γ15(x)y2. Now, since h05 ∈ I′2, there exists
t5(x) ∈ S such that h05(x) = γ15(x)t5(x). Therefore, h5(x,y) = (u + 1)γ15(x)t5(x)y2 = p6(x,y)t5(x).
Now, we get

f(x,y) = h1(x,y) + p1(x,y)t0(x),

h1(x,y) = h2(x,y) + p2(x,y)t1(x),

h2(x,y) = h3(x,y) + p3(x,y)t2(x),

h3(x,y) = h4(x,y) + p4(x,y)t3(x),

h4(x,y) = h5(x,y) + p5(x,y)t4(x),

h5(x,y) = p6(x,y)t5(x).

These equality imply that

f(x,y) = p1(x,y)t0(x) + p2(x,y)t1(x) + p3(x,y)t2(x) + p4(x,y)t3(x)

+ p5(x,y)t4(x) + p6(x,y)t5(x).

Thus

I = 〈p1(x,y),p2(x,y),p3(x,y),p4(x,y),p5(x,y),p6(x,y)〉.

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A. Ghajari et. al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 57–70

Now, we state an important lemma.

Lemma 2.3. Let C be a 1-generator two-dimensional QC code of length n = 3m` which is generated
by G(x,y) = (G1(x,y),G2(x,y), . . . ,G`(x,y)) ∈ R, where Gi(x,y) ∈ R′ for all i with 1 ≤ i ≤ `. Then
Gi(x,y) ∈ Ci, where Ci is a two-dimensional cyclic code of length m over R. Furthermore, if m is odd,
then Gi(x,y) can be selected as the form

Gi(x,y) = ϕ0i(x) + (u + 1)ϕ1i(x) + (ψ0i(x) + (u + 1)ψ1i(x))y + (θ0i(x) + (u + 1)θ1i(x))y
2,

where ϕ0i(x), ϕ1i(x), ψ0i(x), ψ1i(x), θ0i(x) and θ1i(x) are polynomials in R[x] for all i with 1 ≤ i ≤ `
and moreover, ϕ0i(x), ψ0i(x) and θ0i(x) are monic polynomials for all i = 1, · · · ,`.

Proof. Consider the projection map ψi : R→ R′ given by

ψi(k1(x,y), . . . ,k`(x,y)) = ki(x,y),

where ki(x,y) ∈ R′ for all i = 1, · · · ,`. It is clear that the set ψi(C) is a two-dimensional cyclic code
over R for all i with 1 ≤ i ≤ `. Now, in view of Lemma 2.2, for all 1 ≤ i ≤ `, one can obtain a generator
for ψi(C) as follows

ψi(C) = 〈p1i(x,y),p2i(x,y),p3i(x,y),p4i(x,y),p5i(x,y),p6i(x,y)〉,

where, for each j = 1, . . . , 6, pji(x,y) are polynomials as described in Lemma 2.2 . Since Gi(x,y) ∈ ψi(C)
for all 1 ≤ i ≤ `, there exists a polynomial fi(x,y) ∈ R[x,y] such that

Gi(x,y) = fi(x,y)(α
i
01(x) + (u + 1)α

i
11(x) + (β

i
01(x) + (u + 1)β

i
11(x))y

+ (γi01(x) + (u + 1)γ
i
11(x))y

2)

= ϕ0i(x) + (u + 1)ϕ1i(x) + (ψ0i(x) + (u + 1)ψ1i(x))y

+ (θ0i(x) + (u + 1)θ1i(x))y
2,

where αi01(x), α
i
11(x), β

i
01(x), β

i
11(x), γ

i
01(x) and γ

i
11(x) are generator polynomials of cyclic codes over Z4

and, for all 1 ≤ i ≤ `, ϕ0i(x), ϕ1i(x), ψ0i(x), ψ1i(x), θ0i(x) and θ1i(x) are polynomials in R[x]. Moreover,
ϕ0i(x), ψ0i(x) and θ0i(x) are monic polynomials for all i = 1, · · · ,`.

In the light of the above two lemmas, we will obtain the minimal generating sets for 1-generator
two-dimensional QC codes.

Theorem 2.4. Let C be a 1-generator two-dimensional QC code of length n = 3m` over R which is
generated by G = (G1(x,y),G2(x,y), . . . ,G`(x,y)), with m is odd and

Gi(x,y) = ϕ0i(x) + (u + 1)ϕ1i(x) + (ψ0i(x) + (u + 1)ψ1i(x))y

+ (θ0i(x) + (u + 1)θ1i(x))y
2,

where ϕ0i(x), ϕ1i(x), ψ0i(x), ψ1i(x), θ0i(x) and θ1i(x) are polynomials in R[x] for all i with 1 ≤ i ≤ `
and moreover, ϕ0i(x), ψ0i(x) and θ0i(x) are monic polynomials for all 1 ≤ i ≤ `.

Assume that

deg(ϕ0i(x)) >deg(ϕ1i(x)),

deg(ψ0i(x)) >deg(ψ1i(x)) and

deg(θ0i(x)) >deg(θ1i(x)), for all 1 ≤ i ≤ `,

and that the polynomials ϕ0i(x) + (u + 1)ϕ1i(x), (ψ0i(x) + (u + 1)ψ1i(x))y and (θ0i(x) + (u + 1)θ1i(x))y2
are not zero divisor in R′. Assume that

g0(x) = gcd{ϕ01(x),ϕ02(x), . . . ,ϕ0`(x)}, q0(x) = gcd{ϕ11(x),ϕ12(x), . . . ,ϕ1`(x)},
g1(x) = gcd{ψ01(x),ψ02(x), . . . ,ψ0`(x)}, q1(x) = gcd{ψ11(x),ψ12(x), . . . ,ψ1`(x)},
g2(x) = gcd{θ01(x),θ02(x), . . . ,θ0`(x)}, q2(x) = gcd{θ11(x),θ12(x), . . . ,θ1`(x)}

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A. Ghajari et. al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 57–70

and, for k = 0, 1, 2, gk(x)|xm − 1 and qk(x)|xm − 1. Let

S1 =

r0−1⋃
j=0

{xj(ϕ01(x) + (u + 1)ϕ11(x), . . . ,ϕ0`(x) + (u + 1)ϕ1`(x))},

S2 =

r1−1⋃
j=0

{xj((ψ01(x) + (u + 1)ψ11(x))y, . . . , (ψ0`(x) + (u + 1)ψ1`(x))y)},

S3 =

r2−1⋃
j=0

{xj((θ01(x) + (u + 1)θ11(x))y2, . . . , (θ0`(x) + (u + 1)θ1`(x))y2)},

S4 =

t0−1⋃
j=0

{xj((u + 1)h0ϕ11(x), . . . , (u + 1)h0ϕ1`(x))},

S5 =

t1−1⋃
j=0

{xj((u + 1)h1ψ11(x)y, . . . , (u + 1)h1ψ1`(x)y)},

S6 =

t2−1⋃
j=0

{xj((u + 1)h2θ11(x)y2, . . . , (u + 1)h2θ1`(x)y2)},

where, for all k = 0, 1, 2, we set rk := deg(x
m−1
gk(x)

) and tk := deg(x
m−1
qk(x)

). Then S1 ∪S2 ∪S3 ∪S4 ∪S5 ∪S6
is a minimal generating set for C. Moreover, | C |= 16r0+r1+r2 4t0+t1+t2 for all 1 ≤ i ≤ `.

Proof. For k = 0, 1, 2, put hk(x) := x
m−1
gk(x)

and δk(x) := x
m−1
qk(x)

, and let c(x,y) = f(x,y)G be a codeword
in C, where f(x,y) = f0(x) + f1(x)y + f2(x)y2, with fi(x) ∈ R[x] for all 1 ≤ i ≤ 3. For simplicity of
presentation, in our proof, we will use the notion f instead of f(x). By the division algorithm, we get
the unique polynomials Q0(x), Q1(x), Q2(x), R0(x), R1(x), R2(x) in R[x] such that

f0 = h0Q0 + R0, where R0 = 0 or deg(R0) < r0,
f1 = h1Q1 + R1, where R1 = 0 or deg(R1) < r1,
f2 = h2Q2 + R2, where R2 = 0 or deg(R2) < r2.

There exist polynomials ai, a′i, a
′′
i ∈ Z4[x] such that h0ϕ0i = h0g0ai = 0,

h1ψ0i = h1g1a
′
i = 0, h2θ0i = h2g2a

′′
i = 0 for all 1 ≤ i ≤ `. We have

c(x,y) = f(x,y)G = (h0Q0 + R0)(ϕ01 + (u + 1)ϕ11, . . . ,ϕ1` + (u + 1)ϕ1`)

+ (h1Q1 + R1)((ψ01 + (u + 1)ψ11)y, . . . , (ψ0` + (u + 1)ψ1`)y)

+ (h2Q2 + R2)((θ01 + (u + 1)θ11)y
2, . . . , (θ0` + (u + 1)θ1`)y

2)

= Q0h0((u + 1)ϕ11, . . . , (u + 1)ϕ1`)

+ R0(ϕ01 + (u + 1)ϕ11, . . . ,ϕ0` + (u + 1)ϕ1`)

+ Q1h1((u + 1)ψ11y, . . . , (u + 1)ψ1`y)

+ R1((ψ01 + (u + 1)ψ11)y, . . . , (ψ0` + (u + 1)ψ1`)y)

+ Q2h2((u + 1)θ11y
2, . . . , (u + 1)θ1`y

2)

+ R2((θ01 + (u + 1)θ11)y
2, . . . , (θ0` + (u + 1)θ1`)y

2).

They are not difficult to verify that

R0(ϕ01 + (u + 1)ϕ11, . . . ,ϕ0` + (u + 1)ϕ1`) ∈Span(S0),
R1((ψ01 + (u + 1)ψ11)y, . . . , (ψ0` + (u + 1)ψ1`)y) ∈Span(S1), and
R2((θ01 + (u + 1)θ11)y

2, . . . , (θ0` + (u + 1)θ1`)y
2) ∈Span(S2).

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A. Ghajari et. al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 57–70

Again, using the division algorithm, we get the unique polynomials Q′0(x),Q
′
1(x),

Q′2(x),R
′
0(x),R

′
1(x), R

′
2(x) ∈ R[x] such that

Q0 = δ0Q
′
0 + R

′
0, where R

′
0 = 0 or deg(R

′
0) < t0,

Q1 = δ1Q
′
1 + R

′
1, where R

′
1 = 0 or deg(R

′
1) < t1, and

Q2 = δ2Q
′
2 + R

′
2, where R

′
2 = 0 or deg(R

′
2) < t2.

There exist polynomials bi, b′i, b
′′
i ∈ Z4[x] such that

δ0Q
′
0(u + 1)h0ϕ1i = (u + 1)Q

′
0q0δ0bi = 0,

δ1Q
′
1(u + 1)h1ψ1i = (u + 1)Q

′
1q1δ1b

′
i = 0, and

δ2Q
′
2(u + 1)h2θ1i = (u + 1)Q

′
2q2δ2b

′′
i = 0

in R′ for all 1 ≤ i ≤ `. It is not hard to see that

Q0((u + 1)h0ϕ11, . . . , (u + 1)h0ϕ1`) = R
′
0((u + 1)h0ϕ11, . . . , (u + 1)h0ϕ1`) ∈ Span(S4),

Q1((u + 1)h1ψ11y, . . . , (u + 1)h1ψ1`y) = R
′
1((u + 1)h1ψ11y, . . . , (u + 1)h1ψ1`y) ∈ Span(S5), and

Q2((u + 1)h2θ11y
2, . . . , (u + 1)h2θ1`y

2) = R′2((u + 1)h2θ11y
2, . . . , (u + 1)h2θ1`y

2) ∈ Span(S6).
Thus S1∪S2∪S3∪S4∪S5∪S6 is a Spanning set for C. Also, it is clear S1∩S2∩S3∩S4∩S5∩S6 = {0}.

With the aid of the above theorem, we obtain the following corollary.

Corollary 2.5. If ` is a positive integer and ϕ1i(x) = xm −1, ψ1i(x) = xm −1 and θ1i(x) = xm −1 are
polynomials over R, for all i with 1 ≤ i ≤ `, then C is a free two-dimensional QC code of rank r0 +r1 +r2
over R and its minimal generating set is S1 ∪S2 ∪S3 such that

S1 =

r0−1⋃
j=0

{xj(ϕ01(x), . . . ,ϕ0`(x))},

S2 =

r1−1⋃
j=0

{xj(ψ01(x)y, . . . ,ψ0`(x)y)},

S3 =

r2−1⋃
j=0

{xj(θ01(x)y2, . . . ,θ0`(x)y2)},

where, for all k = 0, 1, 2, we set rk := deg(x
m−1
gk(x)

). Furthermore, | C |= 16r0+r1+r2 .

Proof. By Theorem 2.4, if ϕ1i(x) = xm − 1, ψ1i(x) = xm − 1 and θ1i(x) = xm − 1 are polynomials
over R, for all i with 1 ≤ i ≤ `, then

q0 =gcd{ϕ11(x), . . . ,ϕ1`(x),xm − 1} = xm − 1,
q1 =gcd{ψ11(x), . . . ,ψ1`(x),xm − 1} = xm − 1, and
q2 =gcd{θ11(x), . . . ,θ1`(x),xm − 1} = xm − 1.

Hence δ0 = 1, δ1 = 1 and δ2 = 1. Clearly, S1 ∩ S2 ∩ S3 = {0}. Therefore, its minimal generating
set is S1 ∪ S2 ∪ S3. This means that C is a free two-dimensional QC code of rank r0 + r1 + r2. Thus
| C |= 16r0+r1+r2.

In the next theorem, we provide a lower bound on minimum distance of the free 1-generator two-
dimensional QC codes over R.

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A. Ghajari et. al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 57–70

Theorem 2.6. Let C be a free 1-generator two-dimensional QC code of length n = 3m` over R as in
Corollary 2.5. Suppose that

h0i =(x
m − 1)/ϕ0i(x), h1i = (xm − 1)/ψ0i(x),

h2i =(x
m − 1)/θ0i(x), h0 = lcm{h01, . . . ,h0`},

h1 =lcm{h11, . . . ,h1`} and h2 = lcm{h21, . . . ,h2`}

for all i with 1 ≤ i ≤ `. Then we have the following statements.

(i) dmin(C) ≥
∑
i/∈A d0i +

∑
j/∈B d1j +

∑
t/∈D d2t for all 1 ≤ i,j,t ≤ `, where A,B,C ⊆{1, 2, . . . ,`} are

sets from maximum size for which

lcm{h0i, i ∈ A} 6= h0, lcm{h1j,j ∈ B} 6= h1 and
lcm{h2t, t ∈ D} 6= h2.

(ii) If h01 = h02 = . . . = h0`, h11 = h12 = . . . = h1` and h21 = h22 = . . . = h2`, then

dmin(C) ≥
∑̀
i=1

d0i +
∑̀
i=1

d1i +
∑̀
i=1

d2i.

Proof. (i) Consider the projection map ψi : R→ R′ given by

ψi(k1(x,y), . . . ,k`(x,y)) = ki(x,y),

where ki(x,y) ∈ R′ for all i with 1 ≤ i ≤ `. It is easy to show that ψi(C) is a two-dimensional code
over R. Let c(x,y) = f(x,y)G be a nonzero codeword in C, where f(x,y) ∈ R[x,y]. Since C is a free
1-generator two-dimensional QC code, we have that ϕ1i(x) = xm − 1, ψ1i(x) = xm − 1, θ1i(x) = xm − 1
for all 1 ≤ i ≤ `. So, the i-th component is zero if and only if (xm − 1) | f(x,y)G. This means that
(xm−1) | f0(x)ϕ0i(x), (xm−1) | f1(x)ψ0i(x) and (xm−1) | f2(x)θ0i(x), that is, if and only if h0i | f0(x),
h1i | f1(x), h2i | f2(x) for all 1 ≤ i ≤ `. Thus c(x,y) = 0 if and only if h0 | f0(x), h1 | f1(x) and
h2 | f2(x). Therefore, c(x,y) 6= 0 if and only if h0 - f0(x) or h1 - f1(x) or h2 - f2(x). Thus, c(x,y) 6= 0
have the most number of zero blocks whenever

h0 6= lcm{h0i, i ∈ A}, where lcm{h0i, i ∈ A} | f0(x),
h1 6= lcm{h1j,j ∈ B}, where lcm{h1j,j ∈ B} | f1(x),
h2 6= lcm{h2t, t ∈ D}, where lcm{h2t, t ∈ D} | f2(x),

where A, B and D are a maximal subset of {1, 2, . . . ,`} having this property. Thus

dmin(C) ≥
∑
i/∈A

d0i +
∑
i/∈B

d1i +
∑
i/∈D

d2i.

(ii) Now, we know that A = ∅ if and only if h01 = h02 = . . . = h0` and also, B = ∅ if and
only if h11 = h12 = . . . = h1`. Moreover, D = ∅ if and only if h21 = h22 = . . . = h2`. Thus,
dmin(C) ≥

∑`
i=1 d0i +

∑`
i=1 d1i +

∑`
i=1 d2i.

Corollary 2.7. Let C be a 1-generator two-dimensional QC code of length n = 3m` over R which is
generated by

G = (ϕ01(x) + (u + 1)ϕ11(x) + (ψ01(x) + (u + 1)ψ11(x))y+

(θ01(x) + (u + 1)θ11(x))y
2, . . . ,ϕ0`(x) + (u + 1)ϕ1`(x)+

(ψ0`(x) + (u + 1)ψ1`)y + (θ0`(x) + (u + 1)θ1`(x))y
2),

65



A. Ghajari et. al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 57–70

where m is odd. Assume that ϕ1i(x) = xm−1, ψ1i(x) = xm−1 and θ1i(x) = xm−1 for each i = 1, 2, . . . ,`.
Let h0i = (xm − 1)/ϕ0i(x), h1i = (xm − 1)/ψ0i(x) and h2i = (xm − 1)/θ0i(x), for all i with 1 ≤ i ≤ `,
and that

h0 = lcm{h01,h02, . . . ,h0`},
h1 = lcm{h11,h12, . . . ,h1`}, and
h2 = lcm{h21,h22, . . . ,h2`}.

Then

(i) C is a free two-dimensional QC code from deg(h0)+ deg(h1)+ deg(h2). Moreover, | C |=
16deg(h0)+deg(h1)+deg(h2).

(ii) dmin(C) ≥
∑
i/∈A d0i +

∑
i/∈B d1i +

∑
i/∈D d2i, where A,B,D ⊆{1, 2, . . . ,`} are set from maximum

size for which

lcm{h0i, i ∈ A} 6= h0, lcm{h1j,j ∈ B} 6= h1 and
lcm{h2t, t ∈ D} 6= h2.

(iii) If h01 = h02 = . . . = h0`, h11 = h12 = . . . = h1` and h21 = h22 = . . . = h2`, then

dmin(C) ≥
∑̀
i=1

d0i +
∑̀
i=1

d1i +
∑̀
i=1

d2i.

Proof. Let c(x,y) = f(x,y)G be a codeword in C such that f(x,y) = f0(x) + f1(x)y + f2(x)y2,
where fi(x) ∈ R[x] for i = 0, 1, 2. By the division algorithm, we can find unique polynomials
Q1(x),Q2(x),Q3(x),R1(x),R2(x), R3(x) ∈ R[x] such that

f0(x) = h0Q1(x) + R1(x), where R1(x) = 0, or degR1(x) < deg(h0),
f1(x) = h1Q2(x) + R2(x), where R2(x) = 0, or degR2(x) < deg(h1),
f2(x) = h2Q2(x) + R3(x), where R3(x) = 0, or degR3(x) < deg(h2).

Now, we have

c(x,y) = f(x,y)G

= (h0Q1(x) + R1(x))(ϕ01(x), . . . ,ϕ0`(x))

+ (h1Q2(x) + R2(x))(ψ01(x)y, . . . ,ψ0`(x)y)

+ (h2Q3(x) + R3(x))(θ01(x)y
2, . . . ,θ0`(x)y

2).

We know that h0ϕ0i(x) = h1ψ0i(x) = h2θ0i(x) = xm − 1. Therefore, we obtain

R1(x)(ϕ01(x), . . . ,ϕ0`(x)) ∈ Span(S1),
R2(x)(ψ01(x)y, . . . ,ψ0`(x)y) ∈ Span(S2) and
R3(x)(θ01(x)y

2, . . . ,θ0`(x)y
2 ∈ Span(S3).

Thus t0 = 0, t1 = 0 and t2 = 0 which implies that S4 = S5 = S6 = ∅. Using the definition of free
module, we obtain C is a free two-dimensional QC code of rank deg(h0) + deg(h1) + deg(h2). Therefore,
| C |= 16deg(h0)+deg(h1)+deg(h2).

The statements (ii) and (iii) follow from Theorem 2.6.

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A. Ghajari et. al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 57–70

3. 1-generator two-dimensional GQC codes

In this section, we study two-dimensional GQC codes over R. At first, we recall the definition of
1-generator two-dimensional GQC codes over R.

Definition 3.1. Let m1, m2, . . . ,m` be positive integers and

Ri = R[x,y]/〈xmi − 1,y3 − 1〉

for all i with 1 ≤ i ≤ `. Any ideal of R = R1 ×R2 × . . .×R` is called a two-dimensional GQC code of
length (m1,m2, . . . ,m`) with index ` over R.

If C is a two-dimensional GQC code of length (m1,m2, . . . ,m`) with m = m1 = . . . = m`, then C is
a two-dimensional QC code with length n = 3m`.

Lemma 3.2. Let C be a 1-generator two-dimensional GQC code of length (m1, . . . ,m`) and G′(x,y) =
(G′1(x,y),G

′
2(x,y), . . . ,G

′
`(x,y)) ∈ R be a generator of C, where G

′
i(x,y) ∈ Ri for all i with 1 ≤ i ≤ `.

Then G′i(x,y) ∈ Ci, where Ci is a two-dimensional cyclic code of length mi over R for i = 1, · · · ,`.
Also, if mi is odd, then G′i(x,y) can be selected to be of the form G

′
i(x,y) = (ϕ0i(x) + (u+ 1)ϕ1i(x) +

(ψ0i(x) + (u + 1)ψ1i(x))y + (θ0i(x) + (u + 1)θ1i(x))y
2) where ϕ0i(x), ϕ1i(x), ψ0i(x), ψ1i(x), θ0i(x) and

θ1i(x) are polynomials in R[x] for all i with 1 ≤ i ≤ `. Furthermore, ϕ0i(x), ψ0i(x) and θ0i(x) are monic
polynomials for all 1 ≤ i ≤ `.

By using a method similar to that we used in the proof of Theorem 2.4, one can obtain the next
theorem which gives the minimal generating set of 1-generator two-dimensional GQC codes over R.

Theorem 3.3. Let C be a 1-generator two-dimensional GQC code of length (m1,m2, . . . ,m`) over R
which is generated by G′(x,y) = (G′1(x,y),G

′
2(x,y), . . . ,G

′
`(x,y)), where mi is odd for all i with 1 ≤ i ≤ `.

Then G′i(x,y) = ϕ0i(x) + (u+ 1)ϕ1i(x) + (ψ0i(x) + (u+ 1)ψ1i)y + (θ0i(x) + (u+ 1)θ1i(x)y
2), where ϕ0i(x),

ϕ1i(x), ψ0i(x), ψ1i(x), θ0i(x) and θ1i(x) are polynomials in R[x] for all i with 1 ≤ i ≤ `. Furthermore,
ϕ0i(x), ψ0i(x) and θ1i(x) are monic polynomials for all 1 ≤ i ≤ `.

Assume that deg(ϕ0i(x)) ≥ deg(ϕ1i(x)), deg(ψ0i(x)) ≥ deg(ψ1i(x)) and that deg(θ0i(x)) ≥
deg(θ1i(x)). Suppose that polynomials ϕ0i(x) + (u + 1)ϕ1i(x), (ψ0i(x) + (u + 1)ψ1i)y, (θ0i(x) + (u +
1)θ1i(x))y

2 are not zero-divisor of Ri. Let

h0i = (x
mi − 1)/gcd(ϕ0i(x),xmi − 1),

h0 = lcm(h01, . . . ,h0l), deg(h0) = r0,

δ0,i = (x
mi − 1)/gcd(h0ϕ1i(x),xmi − 1),

δ0 = lcm(δ01, . . . ,δ0`) and deg(δ0) = t0.

Let

h1i = (x
mi − 1)/gcd(ψ0i(x),xmi − 1),

h1 = lcm(h11,h12, . . . ,h1`), deg(h1) = r1,

δ1i = (x
mi − 1)/gcd(h1ψ1i(x),xmi − 1),

δ1 = lcm(δ11,δ12, . . . ,δ1`) and deg(δ1) = t1.

Suppose that

h2i = (x
mi − 1)/gcd(θ0i(x),xmi − 1),

h2 = lcm(h21, . . . ,h2`), deg(h2) = r2,

δ2i = (x
mi − 1)/gcd(h2θ1i(x),xmi − 1),

δ2 = lcm(δ21,δ22, . . . ,δ2`) and deg(δ2) = t2.

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A. Ghajari et. al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 57–70

Then the minimal generating set of C is S1 ∪S2 ∪S3 ∪S4 ∪S5 ∪S6, where

S1 =

r0−1⋃
j=0

{xj(ϕ01(x) + (u + 1)ϕ11(x), . . . ,ϕ0`(x) + (u + 1)ϕ1`(x))},

S2 =

r1−1⋃
j=0

{xj((ψ01(x) + (u + 1)ψ1i(x))y, . . . , (ψ0`(x) + (u + 1)ψ1`(x))y)},

S3 =

r2−1⋃
j=0

{xj((θ01(x) + (u + 1)θ11(x))y2, . . . , (θ0`(x) + (u + 1)θ1`(x))y2)},

S4 =

t0−1⋃
j=0

{xj((u + 1)h0ϕ11(x), . . . , (u + 1)h0ϕ1`(x))},

S5 =

t1−1⋃
j=0

{xj((u + 1)h1ψ11(x)y, . . . , (u + 1)h1ψ1`(x)y)},

S6 =

t2−1⋃
j=0

{xj((u + 1)h2θ11(x)y2, . . . , (u + 1)h2θ1`(x)y2)}.

Thus | C |= 16r0+r1+r2 4t0+t1+t2 .

According to Theorem 3.3, we have the following corollary.

Corollary 3.4. If ` is a positive integer and ϕ1i(x) = xmi −1, ψ1i(x) = xmi −1 and θ1i(x) = xmi −1 are
polynomials over R for all i with 1 ≤ i ≤ `, then C is a free two-dimensional GQC code of rank r0 +r1 +r2
over R and its minimal generating set is S1 ∪S2 ∪S3. Furthermore, C has 16r0+r1+r2 codewords.

In the following theorem, we give a lower bound on the minimum distance of free 1-generator two-
dimensional GQC codes over R. Its proof is exactly the same as the proof of Theorem 2.6, so we delete
it.

Theorem 3.5. Let C be a free 1-generator two-dimensional GQC code of length (m1,m2, . . . ,m`) over
R as in Corollary 3.4. Let

h0i = (x
mi − 1)/ϕ0i(x), h0 = lcm{h01, . . . ,h0`},

h1i = (x
mi − 1)/ψ0i(x), h1 = lcm{h11, . . . ,h1`},

h2i = (x
mi − 1)/θ0i(x) and h2 = lcm{h21, . . . ,h2`}.

Then

(i) dmin(C) ≥
∑
i/∈A d0i +

∑
i/∈B d1i +

∑
i/∈D d2i, where A,B,D ⊆ {1, 2, . . . ,`} are sets of maximum

size for which

lcm{h0i, i ∈ A} 6= h0, lcm{h1j,j ∈ B} 6= h1 and
lcm{h2t, t ∈ D} 6= h2.

(ii) If h01 = h02 = . . . = h0`, h11 = h12 = . . . = h1` and h21 = h22 = . . . = h2`, then

dmin(C) ≥
∑̀
i=1

d0i +
∑̀
i=1

d1i +
∑̀
i=1

d2i.

According to Corollary 3.4 and Theorem 3.5 we obtain the following corollary.

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A. Ghajari et. al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 57–70

Corollary 3.6. Assume that ` is a positive integer and ϕ1i(x) = xmi − 1, ψ1i(x) = xmi − 1 and
θ1i(x) = x

mi − 1 are polynomials over R for all i with 1 ≤ i ≤ `. Let

h0i = (x
mi − 1)/ϕ0i(x), h0 = lcm{hoi, . . . ,h0`},

h1i = (x
mi − 1)/ψ0i(x), h1 = lcm{h11, . . . ,h1`},

h2i = (x
mi − 1)/θ0i(x) and h2 = lcm{h21, . . . ,h2`}

for all 1 ≤ i ≤ `. Then we have the following statements.

(i) C is a free two-dimensional code of rank deg(h0) + deg(h1) + deg(h2). Moreover, | C |=
16deg(h0)+deg(h1)+deg(h2),

(ii) dmin(C) ≥
∑
i/∈A d0i +

∑
j/∈B d1j +

∑
k/∈D d2k, where A, B, D ⊆{1, . . . ,`},

(iii) Let h01 = . . . = h0` = h0, h11 = . . . = h1` = h1 and h21 = . . . = h2` = h2. Then we have
dmin(C) ≥

∑`
i=1 d0i +

∑`
i=1 d1i +

∑`
i=1 d2i.

4. Conclusion

This paper is devoted to the study of two-dimensional quasi-cyclic codes and two-dimensional gen-
eralized quasi-cyclic codes of length 3m` which are a natural generalization of quasi-cyclic codes and
generalized quasi-cyclic codes over the ring R = Z4[u]/〈u2 − 1〉 with u2 = 1. We first determine the
generator polynomials of two-dimensional cyclic codes over R. Then we find the generator polynomi-
als of two-dimensional quasi-cyclic codes and two-dimensional generalized quasi-cyclic codes over R and
give their minimal generating sets. Moreover, we study the minimum distances of the family of the free
1-generator two-dimensional quasi-cyclic codes and two-dimensional generalized quasi-cyclic codes.

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	Introduction
	Generator polynomials
	1-generator two-dimensional GQC codes
	Conclusion
	References