ISSN 2148-838X J. Algebra Comb. Discrete Appl. 9(3) • 161–174 Received: 11 January 2021 Accepted: 31 March 2022 Journal of Algebra Combinatorics Discrete Structures and Applications A note on two-dimensional cyclic and constacyclic codes Research Article Om Prakash, Shikha Patel Abstract: During the study of the two-dimensional cyclic (TDC) codes of length n = ls over a finite field Fq where s = 2k, Sepasdar and Khashyarmanesh (2016, [11]) arose a problem that the technique used by them to characterize TDC codes of length n = ls does not work for TDC codes of length 3l. It naturally motivates us to study the TDC codes of other lengths together with 3l. Further, (λ1,λ2)- constacyclic codes are the generalization of constacyclic codes. Thus, we study two-dimensional cyclic codes of length 3l and (λ1,λ2)-constacyclic codes of length 2l, respectively over finite fields. Here, the generating set of polynomials for these two-dimensional codes and their duals are obtained. Finally, with the help of our derived results, we have constructed many MDS codes corresponding to the two-dimensional codes. 2010 MSC: 12E20, 94B05, 94B15, 94B60 Keywords: Cyclic codes, Two-dimensional cyclic codes, Constacyclic codes, Dual codes, Generator matrix 1. Introduction Cyclic codes were introduced by Prange [9] in 1957 and have been studied extensively till now. These codes have been studied over several finite rings and produce a huge amount of new and optimal codes, refer [2, 3, 8, 14, 15]. One of the powerful generalizations of cyclic codes over a finite field is constacyclic codes. This class of linear codes has a rich algebraic structure and is easy to recognize and implement. Note that one can specify cyclic codes and λ-constacyclic codes of length n over the finite field Fq by ideals of the polynomial ring R := Fq[u]/〈un −1〉 and R′ := Fq[u]/〈un −λ〉, respectively. Since the above rings are principal ideal rings, the cyclic and constacyclic codes are generated by a unique polynomial. These generator polynomials help us to find the important parameters of the codes. In 1975, Ikai et al. [5] observed the two-dimensional cyclic (TDC) codes as a generalized class of cyclic codes. Moreover, just after 2 years, Imai [4] introduced the concept of binary two-dimensional Om Prakash (Corresponding Author), Shikha Patel; Department of Mathematics, Indian Institute of Technology Patna, Patna–801106, Bihar, India (email: om@iitp.ac.in, shikha_1821ma05@iitp.ac.in). 161 https://orcid.org/0000-0002-6512-4229 https://orcid.org/0000-0001-8443-2596 O. Prakash, S. Patel / J. Algebra Comb. Discrete Appl. 9(3) (2022) 161–174 cyclic codes. The two-dimensional theory has many applications in the analysis and generation of two- dimensional periodic arrays that help us to construct the two-dimensional feedback shift register with a minimum number of storage devices. Therefore, the study of these codes over finite rings has got the attention of many researchers and hence many new techniques have been discovered to produce cyclic codes over the finite commutative rings with better parameters, we refer [7, 10, 11, 14, 15]. In 2014, Xiuli and Hongyan [13] generalized the concept as two-dimensional skew cyclic codes over a finite field. In 2016, Sepasdar and Khashyarmanesh [11] further studied two-dimensional cyclic codes corresponding to the ideals of the ring R′ := F[x,y]/〈xl − 1,y2 k − 1〉. In 2019, Sharma and Bhaintwal [12] studied the structural behaviour of two-dimensional skew cyclic codes over the ring Fq + uFq with u2 = 1. In [11], Sepasdar and Khashyarmanesh studied the algebraic structure of TDC codes of length n = l2k over the finite field. Moreover, they claimed that the method used by them is not applicable for TDC codes of length 3l. These works motivate us to attempt over the problem raised by them in [11] and extend the study for (λ1,λ2)-constacyclic codes as well. Further, we also study the algebraic properties of two-dimensional cyclic and constacyclic codes and their duals. The article is structured as follows: Section 2 contains some basic definitions and notations. Section 3 presents the study of two-dimensional cyclic codes and their duals of length 3l. Here, we obtain the generating polynomials and the generator matrices of these codes. Similarly, in Section 4, we characterize the structure of two-dimensional (λ1,λ2)-constacyclic codes and their duals and calculate the generating polynomials. As an application of our results, some examples are presented in Section 5 while Section 6 concludes the paper. 2. Notation and background For a prime p and an integer m ≥ 1, let Fq be a finite field where q = pm and λ1,λ2 ∈ F∗q. Throughout the paper, R and R′ represent the quotient ring Fq[u,v]/〈ul − 1,v3 − 1〉 and Fq[u,v]/〈ul −λ1,v2 −λ2〉, respectively. Now, we recall some definitions for two-dimensional codes of Flsq . Definition 2.1. A two-dimensional code C ⊆ Flsq is a set of l×s arrays over Fq. These arrays are known as codewords (or code arrays). A two-dimensional code C is said to be linear if it is a subspace of the ls- dimensional linear space Flsq . Later, in 1977, Imai [4] introduced the notion of the binary two-dimensional cyclic codes as follows: Definition 2.2. Let C ⊆ Flsq be a linear code of length n = ls over Fq whose codewords are viewed as l×s arrays, i.e., c ∈C is written as c =   c0,0 c0,1 . . . c0,s−1 c1,0 c1,1 . . . c1,s−1 ... ... ... ... cl−1,0 cl−1,1 . . . cl−1,s−1   . Each codeword c ∈C written in above matrix form is also defined by its polynomial representation c(u,v) =∑l−1 i=0 ∑s−1 j=0 ci,ju ivj = f0(u) + f1(u)v + · · · + fs−1(u)vs−1 ∈ Fq[u,v]/〈ul − 1,vs − 1〉, where fj(u) = c0,j+c1,ju+· · ·+cl−1,jul−1 ∈ Fq[u]/〈ul−1〉. These representations provide an explicit algebraic description for two-dimensional linear codes. • If C is closed under row-shift and column-shift of codewords, then C is called a two-dimensional cyclic code of length ls over Fq, i.e., if for every l×s array c = (cij) ∈C, we have  c0,s−1 c0,0 . . . c0,s−2 c1,s−1 c1,0 . . . c1,s−2 ... ... ... ... cl−1,s−1 cl−1,0 . . . cl−1,s−2   ∈C 162 O. Prakash, S. Patel / J. Algebra Comb. Discrete Appl. 9(3) (2022) 161–174 and   cl−1,0 cl−1,1 . . . cl−1,s−1 c0,0 c0,1 . . . c0,s−1 ... ... ... ... cl−2,0 cl−2,1 . . . cl−2,s−1   ∈C. • Let λ1 ∈ F∗q = Fq \{0}. Then C is said to be a column λ1-constacyclic code of length ls if for every l×s array c = (cij) ∈C, we have  λ1cl−1,0 λ1cl−1,1 . . . λ1cl−1,s−1 c0,0 c0,1 . . . c0,s−1 ... ... ... ... cl−2,0 cl−2,1 . . . cl−2,s−1   ∈C. • Let λ2 ∈ F∗q. Then C is said to be a row λ2-constacyclic code of length ls if for every l × s array c = (cij) ∈C, we have   λ2c0,s−1 c0,0 . . . c0,s−2 λ2c1,s−1 c1,0 . . . c1,s−2 ... ... ... ... λ2cl−1,s−1 cl−1,0 . . . cl−1,s−2   ∈C. • Let λ1,λ2 ∈ F∗q. If C is both column λ1-constacyclic and row λ2-constacyclic, then C is said to be a two-dimensional (λ1,λ2)-constacyclic code of length ls. Clearly, when λ1 = λ2 = 1, then two-dimensional (λ1,λ2)-constacyclic code coincides with two-dimensional cyclic (TDC) code. It is noted that there is a one-one correspondence between cyclic codes of length l over Fq and ideals of the polynomial ring Fq[u]/〈ul−1〉. Similarly, in case of TDC codes, there is also a one-one correspondence between TDC codes of length ls over Fq and ideals of the polynomial ring Fq[u,v]/〈ul−1,vs−1〉. Hence, a TDC code C ⊆ Flsq of length n = ls over the finite field Fq can be viewed as an ideal of the quotient ring Fq[u,v]/〈ul − 1,vs − 1〉. Further, for λ1,λ2 ∈ F∗q, a two-dimensional (λ1,λ2)-constacyclic code C ⊆ Flsq of length n = ls over the finite field Fq is an ideal of the quotient ring Fq[u,v]/〈ul −λ1,vs −λ2〉. Recall that for c = (c0,c1, ...,cn−1) ∈ C ⊆ Fnq , the Hamming weight wH(c) is equal to the number of non-zero components of c and for any two codewords c and c′ of C, the Hamming distance is defined as dH(c,c ′) = wH(c− c′). Also, the Hamming distance for the code C is dC = min {d(c,c′) | c, c′ ∈C,c 6= c′}. Let c = (c0,c1, ...,cn−1) and c′ = (c′0,c ′ 1, . . . ,c ′ n−1) be two elements of C. Then the inner product of c and c′ in Fnq is defined as c · c′ = ∑n−1 j=0 cjc ′ j. The dual code of C is C ⊥ = {c ∈ Fnq | c · c′ = 0, for all c′ ∈C}. It is well known that a code C is self-orthogonal if C ⊆C⊥ and self dual if C = C⊥. Now, we recall the construction of TDC codes of length 2l over Fq from [11], which is as follows: Let M be a non-zero ideal of R = Fq[u,v]/〈ul −1,v2 −1〉 and char(Fq) 6= 2. It is known that Fq[u,v]/〈ul −1,v2 −1〉∼= (Fq[u]/〈ul −1〉)[v]/〈v2 −1〉. Any arbitrary element of M can be uniquely written as g(u,v) = g0(u)+g1(u)v, where gi(u) ∈ Fq[u]/〈ul− 1〉 for i = 0,1. Here, we find a set of generator polynomials for M. To get the generator polynomials, we use the following identity in R. g(u,v) = 2−1 (g(u,v)(1 + v) + g(u,v)(1−v)) (1) where 163 O. Prakash, S. Patel / J. Algebra Comb. Discrete Appl. 9(3) (2022) 161–174 (X) g(u,v)(1 + v) = (g0(u) + g1(u))(1 + v), (Y) g(u,v)(1−v) = (g0(u)−g1(u))(1−v). Next, we consider the following ideals M1 and M2 of Fq[u]/〈ul −1〉, M1 = {f(u) ∈ Fq[u]/〈ul −1〉 | f(u)(1 + v) ∈ M}, M2 = {f(u) ∈ Fq[u]/〈ul −1〉 | f(u)(1−v) ∈ M}. Since Fq[u]/〈ul − 1〉 is a principal ideal ring, M1 and M2 are principal ideals. Thus, there exist unique monic polynomials p1(u) and p2(u) in Fq[u]/〈ul −1〉 such that Mi = 〈pi(u)〉 for i = 1,2. Also, p1(u) and p2(u) are divisors of ul −1. Therefore, there exists p′i(u) such that pi(u)p ′ i(u) = u l −1 for i = 1,2. Now, from (X), we have g0(u) + g1(u) ∈ M1, and hence g0(u) + g1(u) = p ′′ 1(u)p1(u), for some polynomial p′′1(u) ∈ Fq[u]/〈ul −1〉. Also, from (Y ), we have g0(u) −g1(u) ∈ M2, i.e., there exists a polynomial p′′2(u) in Fq[u]/〈ul − 1〉 such that g0(u)−g1(u) = p′′2(u)p2(u). Therefore, from Equation (1), we have g(u,v) = 2−1 ( g(u,v)(1 + v) + g(u,v)(1−v) ) = 2−1 ( (g0(u) + g1(u))(1 + v) + (g0(u)−g1(u))(1−v) ) = 2−1 ( p′′1(u)p1(u)(1 + v) + p ′′ 2(u)p2(u)(1−v) ) . Since g(u,v) is an arbitrary element of M and hence M is an ideal of R generated by the polynomials p1(u)(1 + v) and p2(u)(1−v), i.e., M = 〈p1(u)(1 + v), p2(u)(1−v)〉. (2) These polynomials p1(u)(1 + v) and p2(u)(1 − v) are generators of M or of the corresponding two- dimensional cyclic code of length 2l. The above discussed generating set of polynomials for TDC codes of length 2l will be used in the next Section 3 for the construction of TDC codes of length 3l. 3. Two-dimensional cyclic codes of length 3l In this section, we study two-dimensional cyclic (TDC) codes of length 3l over Fq with char(Fq) 6= 2,3. Here, our main target is to find the generator polynomials for these codes to explore the structural properties and their duals. 3.1. Generator matrix Let I be a non-zero ideal of R = Fq[u,v]/〈ul −1,v3 −1〉. It is known that Fq[u,v]/〈ul −1,v3 −1〉∼= (Fq[u]/〈ul −1〉)[v]/〈v3 −1〉. 164 O. Prakash, S. Patel / J. Algebra Comb. Discrete Appl. 9(3) (2022) 161–174 Therefore, each element g(u,v) ∈ I can be uniquely written as g(u,v) = g0(u) + g1(u)v + g2(u)v2, where gi(u) ∈ Fq[u]/〈ul − 1〉 for i = 0,1,2. In order to obtain the set of generator polynomials of I, we have the following identity in R: g(u,v) = 3−1 ( g(u,v)(1 + v + v2) + g(u,v)(1−v) + g(u,v)(1−v2) ) (3) where (a) g(u,v)(1 + v + v2) = (g0(u) + g1(u) + g2(u))(1 + v + v2); (b) g(u,v)(1−v) = (g0(u)−g2(u) + (g1(u)−g2(u))v)(1−v); (c) g(u,v)(1−v2) = (g0(u)−g2(u) + (g1(u)−g2(u))v)(1−v2). Next, we consider the ideals I1 in Fq[u]/〈ul −1〉 and I2, I3 in Fq[u,v]/〈ul −1,v2 −1〉, as follows: I1 = {f(u) ∈ Fq[u]/〈ul −1〉 | f(u)(1 + v + v2) ∈ I}; I2 = {f(u,v) ∈ Fq[u,v]/〈ul −1,v2 −1〉 | f(u,v)(1−v) ∈ I}; I3 = {f(u,v) ∈ Fq[u,v]/〈ul −1,v2 −1〉 | f(u,v)(1−v2) ∈ I}. Since Fq[u]/〈ul−1〉 is a principal ideal ring, there exists a unique monic polynomial p0(u) ∈ Fq[u]/〈ul−1〉 such that I1 = 〈p0(u)〉. Also, p0(u) is a divisor of ul −1, there exists p′0(u) such that p0(u)p′0(u) = ul −1. Further, in I2 and I3, f(u,v) ∈ Fq[u,v]/〈ul − 1,v2 − 1〉. Hence, from Equation (2), we can find the generators of I2 and I3, i.e., I2 is generated by two polynomials p1(u)(1 + v), p2(u)(1 − v) and I3 is generated by the polynomials p3(u)(1 + v) and p4(u)(1 − v), respectively for some monic polynomials p1(u), p2(u), p3(u) and p4(u) in Fq[u]/〈ul −1〉. Therefore, (a) gives us g0(u) + g1(u) + g2(u) ∈ I1. Hence, g0(u) + g1(u) + g2(u) = p0(u)p ′′ 0(u) for some polynomial p′′0(u) ∈ Fq[u]/〈ul −1〉. Thus, g(u,v)(1 + v + v2) = p0(u)p ′′ 0(u)(1 + v + v 2). (4) Further, from (b), g0(u)−g2(u) + (g1(u)−g2(u))v ∈ I2. Also, g0(u)−g2(u) + (g1(u)−g2(u))v = p1(u)p′′1(u)(1 + v) + p2(u)p ′′ 2(u)(1−v) for some polynomials p′′1(u) and p ′′ 2(u) in Fq[u]/〈ul −1〉. Hence, g(u,v)(1−v) = (p1(u)p′′1(u)(1 + v) + p2(u)p ′′ 2(u)(1−v))(1−v). (5) Again, by (c), g0(u)−g2(u) + (g1(u)−g2(u))v ∈ I3, i.e., g0(u)−g2(u) + (g1(u)−g2(u))v = p3(u)p′′3(u)(1 + v) + p4(u)p ′′ 4(u)(1−v) for some polynomials p′′3(u) and p ′′ 4(u) in Fq[u]/〈ul −1〉. Hence, g(u,v)(1−v2) = (p3(u)p′′3(u)(1 + v) + p4(u)p ′′ 4(u)(1−v))(1−v 2). (6) From Equations (4), (5) and (6), Equation (3) can be viewed as g(u,v) = 3−1 ( p0(u)p ′′ 0(u)(1 + v + v 2) + (p1(u)p ′′ 1(u)(1 + v) + p2(u)p ′′ 2(u)(1−v)) (1−v) + (p3(u)p′′3(u)(1 + v) + p4(u)p ′′ 4(u)(1−v))(1−v 2) ) = 3−1 ( p0(u)p ′′ 0(u)(1 + v + v 2) + p1(u)p ′′ 1(u)(1−v 2) + p2(u)p ′′ 2(u) (1 + v2 −2v) + p3(u)p′′3(u)(1 + v)(1−v 2) + p4(u)p ′′ 4(u)(1−v) (1−v2) ) . 165 O. Prakash, S. Patel / J. Algebra Comb. Discrete Appl. 9(3) (2022) 161–174 As g(u,v) was arbitrary in I and g(u,v) is written as a linear combination of the elements p0(u)(1 + v + v2), p1(u)(1 − v2), p2(u)(1 + v2 − 2v), p3(u)(1 + v)(1 − v2), p4(u)(1 − v)(1 − v2) over Fq[u]/〈ul − 1〉. Therefore, we have I = 〈p0(u)(1 + v + v2),p1(u)(1−v2),p2(u)(1 + v2 −2v),p3(u) (1 + v)(1−v2),p4(u)(1−v)(1−v2)〉. Since p4(u)(1−v)(1−v2) ∈ I, then p4(u)(1−v)(1−v2)(1−v2) ∈ I and this implies p4(u)(1−v)(1−v2)2 = 3p4(u)(1−v2) ∈ I and hence p4(u)(1−v2) ∈ I. Also, from the definition of I3, p4(u) ∈ I3. Hence, there exist polynomials m(u) and m′(u) in Fq[u]/〈ul −1〉 such that p4(u) = m(u)p3(u)(1 + v) + m ′(u)p4(u)(1−v). (7) Comparing the degree of v in Equation (7), we get p4(u) = m(u)p3(u) + m ′(u)p4(u) and m(u)p3(u) = m′(u)p4(u), i.e., p4(u) = 2m(u)p3(u), implies p3(u)|p4(u). As p3(u)(1+v)(1−v2) ∈ I, so p3(u)(1+v)(1−v2)(1+v2) ∈ I and this gives p3(u)(1 − v2) ∈ I. In the same manner, we have p4(u)|p3(u). Therefore, p3(u) = p4(u). Hence, I3 = 〈p3(u)(1 + v),p4(u)(1−v)〉 = 〈p3(u)〉. Thus, I = 〈p0(u)(1 + v + v2),p1(u)(1−v2),p2(u)(1 + v2 −2v),p3(u)(1−v2)〉 Further, we note that p1(u)(1−v2) ∈ I and hence p1(u) ∈ I3. Therefore, there exist polynomials m1(u) and m′1(u) in Fq[u]/〈ul −1〉 such that p1(u) = m1(u)p3(u)(1 + v) + m ′ 1(u)p3(u)(1−v). (8) Comparing the degree of v in Equation (8), we get p1(u) = m1(u)p3(u) + m ′ 1(u)p3(u) and m1(u)p3(u) = m ′ 1(u)p3(u), i.e., p1(u) = 2m1(u)p3(u), and hence p3(u)|p1(u). Next, we have p3(u)(1 − v2) ∈ I. This implies that p3(u)(1−v2)(1 +v2) = p3(u)(1−v) ∈ I, i.e., p3(u) ∈ I2. Thus, there exist polynomials m2(u) and m′2(u) in Fq[u]/〈ul −1〉 such that p3(u) = m2(u)p1(u)(1 + v) + m ′ 2(u)p2(u)(1−v). (9) Comparing the degree of v in Equation (9), we have p3(u) = m2(u)p1(u) + m ′ 2(u)p2(u) and m2(u)p1(u) = m ′ 2(u)p2(u), i.e., p3(u) = 2m2(u)p1(u), implies p1(u)|p3(u) and hence p1(u) = p3(u). Therefore, I = 〈p0(u)(1 + v + v2),p1(u)(1−v2),p2(u)(1 + v2 −2v)〉. Moreover, p2(u)v2(v −1)(1 + v2 −2v) ∈ I as p2(u)(1 + v2 −2v) ∈ I. This implies p2(u)3(1−v) ∈ I and hence p2(u)(1−v) ∈ I. Finally, the set of generator polynomials for I (the associated TDC code) is{ p0(u)(1 + v + v 2),p1(u)(1−v2),p2(u)(1−v) } . Also, any element of I can be written as an Fq[u]/〈ul −1〉-combination of {p0(u)(1 + v + v2),p1(u)(1−v2),p2(u)(1 + v2 −2v)}. In view of the above demonstration, the following theorem determines the generator matrix for the TDC code of length 3l. 166 O. Prakash, S. Patel / J. Algebra Comb. Discrete Appl. 9(3) (2022) 161–174 Theorem 3.1. Suppose C is a TDC code of length n = 3l and its generating set of polynomials is{ p0(u)(1 + v + v 2),p1(u)(1−v2),p2(u)(1−v) } with deg pi(u) = ai for i = 0,1,2. Then {p0(u)(1 + v + v2),up0(u)(1 + v + v2), . . . ,ul−a0−1p0(u)(1 + v + v2),p1(u)(1−v2), up1(u)(1−v2), . . . ,ul−a1−1p1(u)(1−v2),p2(u)(1 + v2 −2v),up2(u)(1 + v2 −2v), . . . ,ul−a2−1 p2(u)(1 + v 2 −2v)}, is an independent set, and hence elements of this set form the rows of the generator matrix of the code C. Proof. Here, it is enough to show that the above set is independent. For this, we consider m0(u)p0(u)(1 + v + v 2) + m1(u)p1(u)(1−v2) + m2(u)p2(u)(1 + v2 −2v) = 0, (10) for some polynomials mi(u) ∈ Fq[u]/〈ul −1〉 with deg mi(u) ≤ l−ai −1 for i = 0,1,2. Claim: mi(u) = 0 for i = 0,1,2. From Equation (10), there exist polynomials a(u,v) and b(u,v) in Fq[u,v] such that m0(u)p0(u)(1 + v + v 2) + m1(u)p1(u)(1−v2) + m2(u)p2(u)(1 + v2 −2v) = a(u,v)(ul −1) + b(u,v)(v3 −1) (11) in Fq[u,v]. Let a(u,v) = ∑t i=0 ai(u)v i and b(u,v) = ∑t′ i=0 bi(u)v i in Fq[u,v] and m′ be the maximum degree of v in the right hand side of Equation (11). Then we have the following table for coefficients of each vi in the right hand side of Equation (11). Index Coefficient i=0 a0(u)(ul −1)− b0(u) i=1 a1(u)(ul −1)− b1(u) i=2 a2(u)(ul −1)− b2(u) i=3 a3(u)(ul −1)− b3(u) + b0(u) i=4 a4(u)(ul −1)− b4(u) + b1(u) i=5 a5(u)(ul −1)− b5(u) + b2(u) ... ... i=m′ −1 am′−1(u)(ul −1) + bm′−4(u) i=m′ am′(u)(ul −1) + bm′−3(u) Since the maximum degree of v in the left hand side of Equation (11) is 2. Therefore, for i ≥ 3, the coefficients of vi in right hand side of Equation (11) must be zero. Finally, we have the following equalities: bm′−3(u) = −am′(u)(ul −1), bm′−4(u) = −am′−1(u)(ul −1), bm′−5(u) = −am′−2(u)(ul −1), bm′−6(u) = −am′−3(u)(ul −1), ... b0(u) = −a3(u)(ul −1) + b3(u). 167 O. Prakash, S. Patel / J. Algebra Comb. Discrete Appl. 9(3) (2022) 161–174 It can be easily seen that the coefficients of each vi in right hand side of Equation (11) is zero or has the factor ul − 1. Hence, comparing the coefficients of vi’s in both side of Equation (11), we get the following equations: m0(u)p0(u) + m1(u)p1(u) + m2(u)p2(u) = 0, m0(u)p0(u)−2m2(u)p2(u) = 0, m0(u)p0(u)−m1(u)p1(u) + m2(u)p2(u) = 0. Since char(Fq) 6= 2,3, the above equations give mi(u) = 0 for i = 0,1,2. Thus, we get the required result. 3.2. Generator matrix of C⊥ Our next aim is to find a generator matrix for the dual of a TDC code. Towards this, we recall the following proposition. Proposition 3.2. [4] The dual of a TDC code is also a TDC code. Since dim(C) + dim(C⊥) = 3l, by Theorem 3.1, dim(C) = 3l −a0 −a1 −a2. Therefore, dim(C⊥) = a0 + a1 + a2. Recall that the reciprocal polynomial f∗(u) of f(u) with deg(f(u)) = k is defined as f∗(u) := ukf(1/u). Further, if g(u) is a generator polynomial of C, then g(u) is a factor of ul −1 and so ul −1 = g(u)h(u) for some polynomial h(u). This polynomial h(u) is called the check polynomial of C and also h∗(u) is a codeword in C⊥. The following theorem gives the generator matrix for the dual C⊥. Theorem 3.3. Let C be a TDC code of length n = 3l with generator polynomials p0(u)(1 + v + v2), p1(u)(1−v2) and p2(u)(1 +v2 −2v) such that deg pi(u) = ai and also, pi(u)p′i(u) = u l −1 for i = 0,1,2. Then H =   (p′0) ∗(u)(1 + v + v2) u(p′0) ∗(u)(1 + v + v2) ... ua0−1(p′0) ∗(u)(1 + v + v2) (p′1) ∗(u)(1−v2) u(p′1) ∗(u)(1−v2) ... ua1−1(p′1) ∗(u)(1−v2) (p′2) ∗(u)(1 + v2 −2v) u(p′2) ∗(u)(1 + v2 −2v) ... ua2−1(p′2) ∗(u)(1 + v2 −2v)   is the generator matrix of C⊥. Proof. With the help of a similar argument used to prove Theorem 3.1, one can easily show that the rows of H are independent. Also, (p′i) ∗(u) ∈ C⊥ and (1 + v + v2)∗ = 1 + v + v2, (1 −v2)∗ = v2 − 1 and (1 + v2 −2v)∗ = 1 + v2 −2v. Hence, (p′0)∗(u)(1 + v + v2), (p′1)∗(u)(1−v2) and (p′2)∗(u)(1 + v2 −2v) are codewords in C⊥. Therefore, the result follows from [6]. 168 O. Prakash, S. Patel / J. Algebra Comb. Discrete Appl. 9(3) (2022) 161–174 4. Two-dimensional (λ1,λ2)-constacyclic codes of length 2l In this section, we study (λ1,λ2)-constacyclic code C of length 2l over Fq with char(Fq) 6= 2. 4.1. Generator matrix of C Suppose J is a non-zero ideal of the ring R′ = Fq[u,v]/〈ul − λ1,v2 − λ2〉 and α ∈ F∗q such that λ2 = α 2 in F∗q. As Fq[u,v]/〈ul −λ1,v2 −λ2〉∼= (Fq[u]/〈ul −λ1〉)[v]/〈v2 −α2〉, an arbitrary element of J can be uniquely written as g(u,v) = g0(u)+g1(u)v where gi(u) ∈ Fq[u]/〈ul−λ1〉 for i = 0,1. First we find a set of generator polynomials for J. In order to get the generator polynomials, we use the following identity in R′. g(u,v) = 2−1α−1 (g(u,v)(α + v) + g(u,v)(α−v)) (12) where (A) g(u,v)(α + v) = (g0(u) + αg1(u))(α + v); (B) g(u,v)(α−v) = (g0(u)−αg1(u))(α−v). Next, we consider the two ideals J1 and J2 of Fq[u]/〈ul −λ1〉 where J1 = {f(u) ∈ Fq[u]/〈ul −λ1〉 | f(u)(α + v) ∈ J}; J2 = {f(u) ∈ Fq[u]/〈ul −λ1〉 | f(u)(α−v) ∈ J}. Since Fq[u]/〈ul −λ1〉 is a principal ideal ring, J1 and J2 are principal ideals. Hence, there exist unique monic polynomials p1(u) and p2(u) in Fq[u]/〈ul −λ1〉 such that Ji = 〈pi(u)〉 for i = 1,2. Also, p1(u) and p2(u) are divisors of ul −λ1. Therefore, there exists p′i(u) such that pi(u)p ′ i(u) = u l −λ1 for i = 1,2. Now, from (A), we have g0(u) + αg1(u) ∈ J1, and hence g0(u) + αg1(u) = p ′′ 1(u)p1(u) for some polynomial p′′1(u) ∈ Fq[u]/〈ul −λ1〉. Also from (B), we have g0(u)−αg1(u) ∈ J2, i.e., there exists a polynomial p′′2(u) in Fq[u]/〈ul −λ1〉 such that g0(u)−αg1(u) = p′′2(u)p2(u). Therefore, we have the following equality in R′ : g(u,v) = 2−1α−1 ( g(u,v)(α + v) + g(u,v)(α−v) ) = 2−1α−1 ( (g0(u) + αg1(u))(α + v) + (g0(u) + αg1(u))(α−v) ) = 2−1α−1 ( p′′1(u)p1(u)(α + v) + p ′′ 2(u)p2(u)(α−v) ) . Since g(u,v) is an arbitrary element of J, hence 169 O. Prakash, S. Patel / J. Algebra Comb. Discrete Appl. 9(3) (2022) 161–174 J = 〈p1(u)(α + v), p2(u)(α−v)〉. The polynomials p1(u)(α+v) and p2(u)(α−v) are generators of J or of the corresponding two-dimensional (λ1,λ2)-constacyclic code C. Further, based on the above construction, we have the following theorem. Theorem 4.1. Let J be an ideal of Fq[u]/〈ul−λ1,v2−λ2〉 with generating set {p1(u)(α+v),p2(u)(α−v)} where p1(u) and p2(u) are monic polynomials in Fq[u]/〈ul −λ1〉 with deg(p1(u)) = c and deg(p2(u)) = d. Then every element of J has the form c1(u)p1(u)(α + v) + c2(u)p2(u)(α−v), where c1(u) and c2(u) are polynomials in the ring Fq[u]/〈ul−λ1〉 with deg(c1(u)) < l−c and deg(c2(u)) < l−d. Lemma 4.2. [6, Theorem 7.2.12] Let l = tpa and λp a 1 = λ1 where p is the characteristic of Fq and does not divide t. Assume ut −λ1 = ∏r i=1 pi(u) where pi(u) are distinct monic irreducible polynomials for i = 1,2, . . . ,r. Then ul − λ1 = (ut − λ1)p a = ∏r i=1 pi(u) pa. Thus, the number of monic divisors of ul −λ1 ∈ Fq[u] is (pa + 1)r. Theorem 4.3. The number of two-dimensional (λ1,λ2)-constacyclic codes of length n = 2l with exactly two generators is ((pa + 1)r) ((pa + 1)r −1) , where ul − λ1 = ∏r i=1 pi(u) pa and λp a 1 = λ1 for some distinct monic irreducible polynomials pi(u) for i = 1,2, . . . ,r. Proof. Initially, we show that for distinct divisors m1(u) and m2(u) of ul − λ1, the polynomi- als m1(u)(α + v) and m2(u)(α − v) are the generator polynomials of some two-dimensional (λ1,λ2)- constacyclic code. For this, we suppose J is an ideal of Fq[u,v]/〈ul − λ1,v2 − λ2〉 generated by {m1(u)(α+v),m2(u)(α−v)} and consider the corresponding two-dimensional (λ1,λ2)-constacyclic code C of Fq[u,v]/〈ul −λ1,v2 −λ2〉. We can assume p1(u)(α + v) and p2(u)(α − v) are the generator polynomials of C as we discussed in the starting of this section. Also, p1(u)(α + v) ∈ J, there exist polynomials g1(u,v) and g2(u,v) in Fq[u,v]/〈ul −λ1,v2 −λ2〉 such that p1(u)(α + v) = g1(u,v)m1(u)(α + v) + g2(u,v)m2(u)(α−v) + a(u,v)(ul −λ1) + b(u,v)(v2 −λ2). Next, consider the evaluation map ψ1 : Fq[u,v] → Fq[u] defined by ψ1(g(u,v)) = g(u,α). We have 2αp1(u) = 2αg1(u,α)m1(u) + a(u,α)(u l −λ1). Since char(Fq) 6= 2 and m1(u) is a divisor of ul −λ1, i.e., m1(u)|p1(u). As J1 = {f(u) ∈ Fq[u]/〈ul −λ1〉 | f(u)(α + v) ∈ J} =< p1(u) >, and m1(u) ∈ J1, so p1(u)|m1(u), and hence p1(u) = m1(u). Similarly, we can prove that p2(u) = m2(u). Moreover, if m1(u) = m2(u), then p1(u) = p2(u) and J =< p1(u) >. Thus, the number of two-dimensional (λ1,λ2)-constacyclic codes of length n = 2l with exactly two generators is ((pa + 1)r) ((pa + 1)r −1) . 170 O. Prakash, S. Patel / J. Algebra Comb. Discrete Appl. 9(3) (2022) 161–174 Now, we provide the generator matrix for a two-dimensional (λ1,λ2)-constacyclic code of length n = 2l. Theorem 4.4. Let C be a two-dimensional (λ1,λ2)-constacyclic code of length n = 2l with generator polynomials p1(u)(α + v) and p2(u)(α−v) where deg(p1(u)) = c and deg(p2(u)) = d. Then G =   p1(u)(α + v) up1(u)(α + v) ... ul−c−1p1(u)(α + v) p2(u)(α−v) up2(u)(α−v) ... ul−d−1p2(u)(α−v)   is the generator matrix of C. Proof. Under the same notations given in Theorem 4.1, suppose c1(u)p1(u)(α+v) +c2(u)p2(u)(α−v) and d1(u)p1(u)(α + v) + d2(u)p2(u)(α−v) are two elements of J such that c1(u)p1(u)(α + v) + c2(u)p2(u)(α−v) = d1(u)p1(u)(α + v) + d2(u)p2(u)(α−v). Then we find the following in Fq[u,v]: (c1(u)−d1(u))p1(u)(α + v) + (c2(u)−d2(u))p2(u)(α−v) = e(u,v)(ul −λ1) + e′(u,v)(v2 −λ2) for some e(u,v), e′(u,v) in Fq[u,v]. Now, we define the evaluation map ψm : Fq[u,v] → Fq[u] by ψm(g(u,v)) = g(u,m) for m = −α,α. Then 2α(c1(u)−d1(u))p1(u) = e(u,α)(ul −λ1), and 2α(c2(u)−d2(u))p2(u) = e(u,−α)(ul −λ1). Since degree of u in the right hand side of the above equalities is at least l whereas the corresponding degree in the left hand side is at most l − 1. Therefore, ci(u) = di(u) for i = 1,2. Hence, J has ql−cql−d elements. Thus, the dimension of the corresponding two-dimensional (λ1,λ2)-constacyclic code is 2l− c−d. 4.2. Generator matrix of C⊥ It is known that dim(C) + dim(C⊥) = 2l, then from Theorem 4.4, we have dim(C) = 2l − c − d. Therefore, dim(C⊥) = c + d. Further, if g(u) is a generator polynomial of a λ1-constacyclic code C, then g(u) is a factor of ul −λ1 and so ul −λ1 = g(u)h(u) for some polynomial h(u) and h∗(u) is a codeword in C⊥. The following theorem gives the generator matrix for the dual C⊥. 171 O. Prakash, S. Patel / J. Algebra Comb. Discrete Appl. 9(3) (2022) 161–174 Theorem 4.5. Let C be a two-dimensional (λ1,λ2)-constacyclic code of length n = 2l with generator polynomials p1(u)(α + v) and p2(u)(α−v) such that deg p1(u) = c, p1(u)p′1(u) = ul −λ1, deg p2(u) = d and p2(u)p′2(u) = u l −λ1. Then H =   (p′1) ∗(u)(1 + vα) u(p′1) ∗(u)(1 + vα) ... uc−1(p′1) ∗(u)(1 + vα) (p′2) ∗(u)(1−vα) u(p′2) ∗(u)(1−vα) ... ud−1(p′1) ∗(u)(1−vα)   is the generator matrix of C⊥. Proof. One can easily check that the rows of H are independent by using the method given in Theorem 4.4. Now, (p′i) ∗(u) ∈C⊥ for i = 1,2; (α+v)∗ = 1+vα and (α−v)∗ = vα−1. Therefore, (p′1)∗(u)(1+vα) and (p′2) ∗(u)(1−vα) are codewords in C⊥. Hence, the result follows from [6]. 5. Examples A linear code C of length n over Fq with minimum distance d and dimension k is represented by [n,k,d]. If C is an [n,k,d] code, then from the Singleton bound, its minimum distance is bounded above by d ≤ n−k + 1. A code meeting the above bound is known as maximum-distance-separable (MDS). We call a code almost MDS if its minimum distance is one unit less than the MDS. A code is called optimal if it has the highest possible minimum distance for its length and dimension. We provide here a few examples in which all computations are carried out by using Magma software [1]. Example 5.1. We consider a two-dimensional cyclic code of length 6 over F7. In F7, we have u2 −1 = (u + 1)(u + 6). Suppose p0(u) = p1(u) = u + 1 and p2(u) = u + 6 = u−1, then from Theorem 3.1, the generator matrix of the TDC code is given by G =   (u + 1)(1 + v + v 2) (u + 1)(1−v2) (u−1)(1 + v2 −2v)   and codewords corresponding to rows of that matrix are c1 = ( 1 1 1 1 1 1 ) , c2 = ( 1 0 −1 1 0 −1 ) , c3 = ( −1 2 −1 1 −2 1 ) . Hence, G =   1 1 1 1 1 11 0 −1 1 0 −1 −1 2 −1 1 −2 1   . Here, the obtained TDC code [6,3,4] is an MDS code. 172 O. Prakash, S. Patel / J. Algebra Comb. Discrete Appl. 9(3) (2022) 161–174 Table 1. Two-dimensional cyclic and (λ1,λ2)-constacyclic codes q (λ1,λ2) (l,n) p0(u) p1(u) p2(u) Obtained TDC/ Remark constacyclic code 7 (1,1) (2,6) u + 1 u + 1 u − 1 [6,3,4] MDS 5 (1,1) (3,9) u − 1 u − 1 1 + u + u2 [9,5,4] Optimal 5 (1,1) (14,42) u + 1 u + 1 u + 4 [42,39,2] Optimal 11 (1,1) (4,12) u + 1 u − 1 u2 + 1 [12,8,4] Almost MDS 9 (1,1) (5,10) u + 2 u2 + w3u + 1 [10,7,4] MDS 5 (3,4) (3,6) u + 3 u2 + 2u + 4 [6,3,4] MDS Example 5.2. We consider a two-dimensional (3,4)-constacyclic code of length 6 over F5. In F5, we have u3 −3 = (u + 3)(u2 + 2u + 4) and also α = 2. Suppose p1(u) = u + 3 and p2(u) = u2 + 2u + 4, then from Theorem 4.4, the generator matrix of the two-dimensional (3,4)-constacyclic code is given by G =   (u + 3)(2 + v)u(u + 3)(2 + v) (u2 + 2u + 4)(2−v)   and codewords corresponding to rows of that matrix are c1 =  1 32 1 0 0  , c2 =  0 01 3 2 1  , c3 =  3 −44 −2 2 −1  . Hence, G =  1 3 2 1 0 00 0 1 3 2 1 3 −4 4 −2 2 −1   . Here, the obtained two-dimensional (3,4)-constacyclic code [6,3,4] is an MDS code. 6. Conclusion In this paper, we have obtained the generating set of polynomials for the ideals corresponding to the two-dimensional cyclic codes of length 3l and (λ1,λ2)-constacyclic codes of length n = 2l, respectively. In future, we would like to develop a technique by which we can find the generating set of polynomials for two-dimensional codes of arbitrary length. 173 O. Prakash, S. Patel / J. Algebra Comb. Discrete Appl. 9(3) (2022) 161–174 Acknowledgment: The authors are thankful to the Department of Science and Technology, Govt. of India for financial support under Ref No. DST/INSPIRE/03/2016/001445 and Indian Institute of Technology Patna for providing the research facilities. References [1] W. Bosma, J. Cannon, Handbook of magma functions, Univ. of Sydney, Sydney (1995). [2] I. F. Blake, Codes over certain rings, Inf. Control. 20(4) (1972) 396–404. [3] I. F. Blake, Codes over integer residue rings, Inf. Control. 29(4) (1975) 295–300. [4] H. Imai, A theory of two-dimensional cyclic codes, Inf. Control. 34(1) (1977) 1–21. [5] T. Ikai, H. Kosako, Y. Kojima, Two-dimensional cyclic codes, Electron. Comm. Japan 57(4) (1974/75) 27–35. [6] S. Ling, C. 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Comput. 306 (2017) 86–91. 174 https://books.google.sk/books?id=xJFHYAAACAAJ https://doi.org/10.1016/S0019-9958(72)90223-9 https://doi.org/10.1016/S0019-9958(75)80001-5 https://doi.org/10.1016/S0019-9958(77)90232-7 https://mathscinet.ams.org/mathscinet-getitem?mr=456921 https://mathscinet.ams.org/mathscinet-getitem?mr=456921 https://doi.org/10.1017/CBO9780511755279 https://doi.org/10.1504/ijicot.2020.10032824 https://doi.org/10.1504/ijicot.2020.10032824 https://doi.org/10.1007/s12095-022-00572-9 https://openlibrary.org/books/OL20255744M/Cyclic_error-correcting_codes_in_two_symbols https://openlibrary.org/books/OL20255744M/Cyclic_error-correcting_codes_in_two_symbols https://doi.org/10.1016/j.ffa.2017.11.008 https://doi.org/10.1016/j.ffa.2017.11.008 https://doi.org/10.1016/j.ffa.2016.06.003 https://doi.org/10.1016/j.ffa.2016.06.003 https://doi.org/10.1007/s00200-019-00388-w https://doi.org/10.1007/s00200-019-00388-w https://doi.org/10.1016/j.ffa.2013.08.001 https://doi.org/10.1016/j.ffa.2013.08.001 https://www.inderscienceonline.com/doi/abs/10.1504/IJICOT.2014.066107 https://www.inderscienceonline.com/doi/abs/10.1504/IJICOT.2014.066107 https://doi.org/10.1016/j.amc.2017.02.017 https://doi.org/10.1016/j.amc.2017.02.017 Introduction Notation and background Two-dimensional cyclic codes of length 3l Two-dimensional (1,2)-constacyclic codes of length 2l Examples Conclusion References