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Self-dual cyclic codes over 𝔽
𝟐+ 𝒖𝔽
𝟐+ 𝒗𝔽
𝟐+ 𝒖𝒗𝔽
𝟐Leilei Gao
Department of Mathematics , School of Mathematics and Statistics, Shandong University of Technology, Zibo , China.
E-mail address: [email protected] ABSTRACT
In this work, we describe the algebraic structure of self-dual cyclic codes over the ring 𝔽2+ 𝑢𝔽2+ 𝑣𝔽2+ 𝑢𝑣𝔽2, where 𝑢2= 0, 𝑣2= 0 and 𝑢𝑣 = 𝑣𝑢. We provide a necessary and sufficient condition for the existence of self-dual cyclic codes of odd length 𝑛 over the ring 𝔽2+ 𝑢𝔽2+ 𝑣𝔽2+ 𝑢𝑣𝔽2. Further, by the Gray map, we construct self- dual codes over the ring 𝔽2+ 𝑢𝔽2.
Keywords: Self-dual cyclic codes Gray map self-dual codes.
1. Introduction
Codes over finite rings have been studied since the early 1970s. There are a lot of works on codes over finite rings after the discovery that certain the hidden linear structures behind well-known nonlinear codes such as Kerdock and Preparata codes as the Gray images of linear codes over ℤ4 [2].
Rings of order 16 are of importance in many areas. For example, the smallest local finite Frobenius commutative non-chain ring is of order 16 [8]. Recently, there are some works on linear codes over the ring ℤ4+ 𝑢ℤ4, one of 16 elements rings, such as [9, 13, 6, 7]. In [6], the MacWilliams identities of linear codes and constructing formally self- dual codes over ℤ4+ 𝑢ℤ4 are discussed. Later, some structural properties of cyclic codes over ℤ4+ 𝑢ℤ4 and constructing new ℤ4-linear codes are considered in [7]. Recently, Luo and Parampalli introduce algebraic structures of self-dual cyclic codes of odd length 𝑛 over ℤ4+ 𝑢ℤ4 and provide a necessary and sufficient condition for the existence of self-dual cyclic codes of odd length [9].
The ring Δ = 𝔽2+ 𝑢𝔽2+ 𝑣𝔽2+ 𝑢𝑣𝔽2, where 𝑢2= 0 , 𝑣2= 0 and 𝑢𝑣 = 𝑣𝑢 , is another 16 elements ring.
Recently, there are also some works on linear codes over this ring, such as [10, 3, 4, 5]. To the best of our knowledge, there have no any research on self-dual cyclic codes over Δ. We will do this issue in this paper.
The paper is organized as follows. In Section 2, we recall some results on the ring Δ. In Section 3, we consider some structural properties of cyclic codes over Δ. Then we describe the structures of self-dual cyclic codes and provide a necessary and sufficient condition for the existence of self-dual cyclic codes of odd length over Δ. In Section 4, by the Gray map, some examples of constructing self-dual codes over the ring Λ = 𝔽2+ 𝑢𝔽2 are given.
2. Preliminaries
Let Δ = 𝔽2+ 𝑢𝔽2+ 𝑣𝔽2+ 𝑢𝑣𝔽2, where 𝑢2= 𝑣2= 0 and 𝑢𝑣 = 𝑣𝑢 . Then Δ is a commutative ring with 16 elements and characteristic 2. Any element of Δ can be expressed uniquely as 𝑎 + 𝑏𝑢 + 𝑐𝑣 + 𝑑𝑢𝑣, where 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝔽2. There are 5 different nontrivial ideals of Δ ( see [4]), which are described as follows
𝐼𝑢𝑣= 〈𝑢𝑣〉 = {0, 𝑢𝑣}, 𝐼𝑢= 〈𝑢〉 = {0, 𝑢, 𝑢𝑣, 𝑢 + 𝑢𝑣}, 𝐼𝑣= 〈𝑣〉 = {0, 𝑣, 𝑢𝑣, 𝑣 + 𝑢𝑣}, 𝐼𝑢+𝑣= 〈𝑢 + 𝑣〉 = {0, 𝑢 + 𝑣, 𝑢𝑣, 𝑢 + 𝑣 + 𝑢𝑣},
𝐼𝑢,𝑣= 〈𝑢, 𝑣〉 = {0, 𝑢, 𝑣, 𝑢 + 𝑣, 𝑢𝑣, 𝑢 + 𝑢𝑣, 𝑣 + 𝑢𝑣, 𝑢 + 𝑣 + 𝑢𝑣}.
Obviously, the ring Δ is not a finite chain ring. Observe that 𝐼𝑢,𝑣 is the unique maximal ideal of Δ. Therefore, the local ring Δ is not principle either. Further, Δ is a local Frobenius ring [8].
Let Λ = 𝔽2+ 𝑢𝔽2, where 𝑢2= 0. Now we define the Gray map 𝜃 from Δ to Λ as follows
𝜃: Δ ⟶ Λ
𝑝 + 𝑞𝑣 ⟼ (𝑞, 𝑝 + 𝑞),
where 𝑝, 𝑞 ∈ Λ. It is well known that the Lee weights of elements in Λ are defined as 𝑤𝐿(0) = 0, 𝑤𝐿(1) = 1, 𝑤𝐿(𝑢) = 2, 𝑤𝐿(1 + 𝑢) = 1. For any 𝛼 = 𝑝 + 𝑞𝑣 ∈ Δ, its Gray weight is defined as
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𝑤𝐺(𝛼) = 𝑤𝐿(𝑞) + 𝑤𝐿(𝑝 + 𝑞), where 𝑝, 𝑞 ∈ Λ.
Define a Gray weight of a vector 𝑐 = (𝑐0, 𝑐1, … , 𝑐𝑛−1) ∈ Δ𝑛 to be the rational sum of the Gray weight of its component, i.e.
𝑤𝐺(𝑐) = 𝑤𝐺(𝑐0) + 𝑤𝐺(𝑐1) + ⋯ + 𝑤𝐺(𝑐𝑛−1).
For any elements 𝑐1, 𝑐2∈ Δ𝑛, the Gray distance is given by 𝑑𝐺(𝑐1, 𝑐2) = 𝑤𝐺(𝑐1− 𝑐2). A code 𝐶 of length 𝑛 over Δ is a subset of Δ𝑛. 𝐶 is a linear code if and only if 𝐶 is an Δ-submodule of Δ𝑛. The minimum Gray distance of 𝐶 is the smallest nonzero Gray distance between all pairs of distinct codewords. The minimum Gray weight of 𝐶 is the smallest nonzero Gray weight among all codewords. If 𝐶 is a linear code, then the minimum Gray distance is the same as the minimum Gray weight.
Now we extend the Gray map Θ to Δ𝑛 as follows
Θ: Δ𝑛 ⟶ Λ2𝑛
(𝑐0, 𝑐1, … , 𝑐𝑛−1) ⟼ (𝑞0, 𝑝0+ 𝑞0, … , 𝑞𝑛−1, 𝑝𝑛−1+ 𝑞𝑛−1),
where 𝑐𝑖= 𝑝𝑖+ 𝑞𝑖𝑣, 𝑖 = 0,1, … , 𝑛 − 1. The Gray map Θ is a distance-preserving map from Δ𝑛 (Gray distance)to Λ2𝑛 (Lee distance) and it is also Λ-linear.
Proof. For any 𝑐1, 𝑐2∈ Δ𝑛 and 𝑘1, 𝑘2∈ Λ, we have Θ(𝑘1𝑐1+ 𝑘2𝑐2) = 𝑘1Θ(𝑐1) + 𝑘2Θ(𝑐2), which implies that Θ is Λ-linear. Let 𝑐1= (𝑐1,0, 𝑐1,1, … , 𝑐1,𝑛−1) and 𝑐2= (𝑐2,0, 𝑐2,1, … , 𝑐2,𝑛−1) be elements of Δ𝑛, where 𝑐𝑖,𝑗= 𝑝𝑖,𝑗+ 𝑞𝑖,𝑗𝑣, 𝑖 = 1,2, 𝑗 = 0,1, … , 𝑛 − 1 . Then 𝑐1− 𝑐2= (𝑐1,0− 𝑐2,0, … , 𝑐1,𝑛−1− 𝑐2,𝑛−1) and Θ(𝑐1− 𝑐2) = Θ(𝑐1) − Θ(𝑐2). Therefore, 𝑑𝐺(𝑐1, 𝑐2) = 𝑤𝐺(𝑐1− 𝑐2) = 𝑤𝐿(Θ(𝑐1) − Θ(𝑐2)) = 𝑑𝐿(Θ(𝑐1), Θ(𝑐2)). The second equality holds because of the definition is the Gray weight of the element in Λ.
Let 𝐶 be a (𝑛, 𝑀, 𝑑) linear code over Δ, where 𝑛, 𝑀, 𝑑 are respectively the length, the number of the codewords and the minimum Gray distance of 𝐶. Then Θ(𝐶) is a (2𝑛, 𝑀, 𝑑) linear code over Δ.
Proof. According to Proposition 1, we know that Θ is Λ-linear, which implies that Θ(𝐶) is a Λ-linear code.
From the definition of the Gray map Θ, Θ(𝐶) is with length 2𝑛. Moreover, Θ is a bijective map from Δ𝑛 to Λ2𝑛 implying that Θ(𝐶) has 𝑀 codewords. At last, the preserving distance of Θ leads to Θ(𝐶) has the minimum Lee distance 𝑑.
3. Structure of self-dual cyclic codes
Cyclic codes over the ring Δ are defined in a natural way. Let 𝐶 be a linear code of length 𝑛 over the ring Δ. If for any codeword 𝑐 = (𝑐0, 𝑐1, … , 𝑐𝑛−1) ∈ 𝐶, the vector (𝑐𝑛−1, 𝑐0, 𝑐1, … , 𝑐𝑛−2) is also in 𝐶, then the code 𝐶 is called a cyclic code over the ring Δ.
Let 𝑅 = Δ[𝑥]
〈𝑥𝑛−1〉. We present any vector (𝑐0, 𝑐1, … , 𝑐𝑛−1) ∈ Δ𝑛 by the residue class of the polynomial 𝑐0+ 𝑐1𝑥 + ⋯ + 𝑐𝑛−1𝑥𝑛−1 of 𝑅. Then we have a Δ-module isomorphism 𝜑 as follows
𝜑: Δ𝑛 ⟶ 𝑅
(𝑎0, 𝑎1, … , 𝑎𝑛−1) ⟼ 𝑎0+ 𝑎1𝑥 + ⋯ + 𝑎𝑛−1𝑥𝑛−1+ 〈𝑥𝑛− 1〉.
It is easy to see that a linear code 𝐶 of length 𝑛 is a cyclic code over Δ if and only if 𝜑(𝐶) is an ideal of 𝑅. In this paper, we identity cyclic codes over Δ with ideals of 𝑅.
Let 𝑓(𝑥) be a basic irreducible polynomial in Λ[𝑥] and 𝑅𝑓 = Λ[𝑥]
〈𝑓(𝑥)〉. Then 𝑅𝑓 is a chain ring. Further, its ideals can be given similarly to that of ℤ4 as follows. [1] 𝑓(𝑥) is a basic irreducible polynomial in Λ[𝑥], then the only ideals of 𝑅𝑓 are 0, 𝑅𝑓 and 𝑢𝑅𝑓. Let 𝑓(𝑥) be a basic irreducible polynomial in Λ[𝑥] and 𝑔(𝑥) be a non-zero polynomial in 𝑅𝑓, then 𝑔(𝑥) is either a unit in the ring 𝑅𝑓 or an element of 𝑢𝑅𝑓.
Proof. Let 𝜙 be a map from Λ[𝑥] to 𝔽2[𝑥] which sends 0, 𝑢 to 0; 1, 1 + 𝑢 to 1 and 𝑥 to 𝑥. Since 𝑓(𝑥) is a basic irreducible polynomial in Λ[𝑥], then we can obtain gcd(𝜙(𝑔(𝑥)), 𝜙(𝑓(𝑥))) = 1𝑜𝑟𝜙(𝑓(𝑥)).
Case 1. gcd(𝜙(𝑔(𝑥)), 𝜙(𝑓(𝑥))) = 1. This means that 𝜙(𝑔(𝑥)) and 𝜙(𝑓(𝑥)) are coprime in 𝔽2[𝑥]. Then there exist polynomial 𝜆1(𝑥) and 𝜆2(𝑥) in Λ[𝑥] such that
𝜙(𝜆1(𝑥))𝜙(𝑔(𝑥)) + 𝜙(𝜆2(𝑥))𝜙(𝑓(𝑥)) = 1.
Thus
𝜆1(𝑥)𝑔(𝑥) + 𝜆2(𝑥)𝑓(𝑥) = 1 + 𝑢𝑘(𝑥), where 𝑘(𝑥) ∈ Λ[𝑥]. Multiplying the above equation by 𝑢𝑘(𝑥), we have
𝑢𝑘(𝑥)𝜆1(𝑥)𝑔(𝑥) + 𝑢𝑘(𝑥)𝜆2(𝑥)𝑓(𝑥) = 𝑢𝑘(𝑥).
Then we obtain
(1 − 𝑢𝑘(𝑥))𝜆1(𝑥)𝑔(𝑥) + (1 − 𝑢𝑘(𝑥))𝜆2(𝑥)𝑓(𝑥) = 1.
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Therefore, 𝑔(𝑥) and 𝑓(𝑥) are comprime in Λ[𝑥]. Hence, it is easy to see that 𝑔(𝑥) is a unit in the ring 𝑅𝑓. Case 2. gcd(𝜙(𝑔(𝑥)), 𝜙(𝑓(𝑥))) = 𝜙(𝑓(𝑥)) . Since 𝑔(𝑥) is a non-zero polynomial in 𝑅𝑓 , then d𝑒𝑔(𝑔(𝑥)) < d𝑒𝑔(𝑓(𝑥)). Since 𝑓(𝑥) is a basic irreducible polynomial, then d𝑒𝑔(𝜙(𝑔(𝑥))) < d𝑒𝑔𝜙((𝑓(𝑥))). This means that 𝜙(𝑔(𝑥)) = 0. So we can attain that 𝑔(𝑥) is an element of 𝑢𝑅𝑓.
If 𝑓(𝑥) is a basic irreducible polynomial in Λ[𝑥], then the only ideals of 𝑅 are the elements in 𝑆 = {0, 𝑣𝑅𝑓, 𝑢𝑣𝑅𝑓, 𝑅𝑓+ 𝑣𝑅𝑓, 𝑢𝑅𝑓+ 𝑣𝑅𝑓, 𝑢𝑅𝑓+ 𝑢𝑣𝑅𝑓, (𝑢 + 𝑣 ∑𝑖∈𝐴𝑥𝑖)𝑅𝑓}, where 𝐴 is a subset of {0,1, … , 𝑛 − 1}.
Proof. Let 𝐼 be an arbitrary nontrivial ideal of Δ[𝑥]
〈𝑓(𝑥)〉. If 𝐼 ⊆ 𝑣𝑅𝑓, by Lemma 1, it is easy to see that 𝐼 is one of 𝑣𝑅𝑓 and 𝑢𝑣𝑅𝑓.
Assume that 𝐼 ⊈ 𝑣𝑅𝑓. Let 𝐼𝑛= {𝑎(𝑥) ∈ 𝑅𝑓|∃𝑏(𝑥) ∈ 𝑅𝑓𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑎(𝑥) + 𝑣𝑏(𝑥) ∈ 𝐼}. Obviously, 𝐼𝑛 is an ideal of the ring 𝑅𝑓. By Lemma 1, we have
(i) 𝐼𝑛= 𝑅𝑓, then there exists a polynomial 𝑏(𝑥) ∈ 𝑅𝑓 such that 1 + 𝑣𝑏(𝑥) ∈ 𝐼 . Therefore, (1 + 𝑣𝑏(𝑥))2= 1 is in 𝐼. It follows that 𝐼 = 𝑅𝑓+ 𝑣𝑅𝑓.
(ii) 𝐼𝑛= 𝑢𝑅𝑓, then there exists an element 𝑏(𝑥) ∈ 𝑅𝑓 such that 𝑢 + 𝑣𝑏(𝑥) ∈ 𝐼, which implies that 𝑢𝑣 = (𝑢 + 𝑣𝑏(𝑥))𝑣 is in 𝐼. So, we get 𝑢𝑣𝑅𝑓 ⊆ 𝐼.
(ii-1) 𝑏(𝑥) = 0, then 𝑢 is in 𝐼. Hence, 𝑢𝑅𝑓⊆ 𝐼. Therefore, 𝑢𝑅𝑓+ 𝑢𝑣𝑅𝑓 ⊆ 𝐼. Moreover, we have (ii-1-1) 𝐼 = 𝑢𝑅𝑓+ 𝑢𝑣𝑅𝑓.
(ii-1-2) 𝑢𝑅𝑓+ 𝑢𝑣𝑅𝑓⊈ 𝐼 . Then there exists 𝛼(𝑥) ∈ 𝐼\(𝑢𝑅𝑓+ 𝑢𝑣𝑅𝑓) . Hence, there are polynomials 𝑔(𝑥), 𝑎(𝑥), 𝑏(𝑥) ∈ 𝑅𝑓 such that 𝛼(𝑥) = 𝑢𝑎(𝑥) + 𝑢𝑣𝑏(𝑥) + 𝑣𝑔(𝑥) ∈ 𝐼, which implies that 𝑣𝑔(𝑥) ∈ 𝐼. Observe that 𝛼(𝑥) is not in 𝑢𝑅𝑓+ 𝑢𝑣𝑅𝑓, we get that 𝑔(𝑥) is not in 𝑢𝑅𝑓. By Lemma 2, we know 𝑔(𝑥) is a unit in 𝑅𝑓. So there exists polynomial ℎ(𝑥) in 𝑅𝑓 such that 𝑣 = 𝑣𝑔(𝑥)ℎ(𝑥) ∈ 𝐼. This means that 𝐼 = 𝑢𝑅𝑓+ 𝑣𝑅𝑓.
(ii-2) 𝑏(𝑥) ≠ 0, then 𝑏(𝑥) = ∑𝑖∈𝐴𝑥𝑖+ 𝑢 ∑𝑗∈𝐵𝑥𝑗, where 𝐴, 𝐵 are two subsets of {0,1, … , 𝑛 − 1}. Since 𝑢 + 𝑣𝑏(𝑥) = 𝑢 + 𝑣(∑𝑖∈𝐴𝑥𝑖) + 𝑢𝑣(∑𝑗∈𝐵 𝑥𝑗) ∈ 𝐼, then we can gain 𝑢 + 𝑣 ∑𝑖∈𝐴𝑥𝑖∈ 𝐼. Hence, 〈𝑢 + 𝑣 ∑𝑖∈𝐴𝑥𝑖〉 ⊆ 𝐼.
(ii-2-1) 𝐼 = 〈𝑢 + 𝑣 ∑𝑖∈𝐴𝑥𝑖〉 = (𝑢 + 𝑣 ∑𝑖∈𝐴𝑥𝑖)𝑅𝑓+ 𝑢𝑣𝑅𝑓. Since ∑𝑖∈𝐴𝑥𝑖 is a unit in 𝑅𝑓 and 𝑢𝑣 = 𝑢(𝑢 + 𝑣 ∑𝑖∈𝐴𝑥𝑖)(∑𝑖∈𝐴𝑥𝑖)−1, then 𝐼 = (𝑢 + 𝑣 ∑𝑖∈𝐴𝑥𝑖)𝑅𝑓.
(ii-2-2) 〈𝑢 + 𝑣 ∑𝑖∈𝐴𝑥𝑖〉 ⫋ 𝐼 . Then there exists 𝑐(𝑥) ∈ 𝐼\〈𝑢 + 𝑣 ∑𝑖∈𝐴𝑥𝑖〉 . Therefore, there exist polynomials 𝑎(𝑥), ℎ(𝑥) ∈ 𝑅𝑓 such that 𝑣ℎ(𝑥) = 𝑐(𝑥) − (𝑢 + 𝑣 ∑𝑖∈𝐴𝑥𝑖)𝑎(𝑥) ∈ 𝐼. Since 𝑐(𝑥) ∉ 〈𝑢 + 𝑣 ∑𝑖∈𝐴𝑥𝑖〉, it follows that ℎ(𝑥) ∉ 𝑢𝑅𝑓. By Lemma 2, we have ℎ(𝑥) is a unit in 𝑅𝑓, then 𝑣 is in 𝐼 . Let 𝑎(𝑥) = 𝑢𝑎1(𝑥) + 𝑣𝑎2(𝑥) = (𝑢 + 𝑣 ∑𝑖∈𝐴𝑥𝑖)𝑎1(𝑥) + 𝑣(𝑎2(𝑥) − 𝑎1(𝑥) ∑𝑖∈𝐴𝑥𝑖) ∈ 〈𝑢 + 𝑣 ∑𝑖∈𝐴𝑥𝑖〉 + 𝑣𝑅𝑓= 𝑢𝑅𝑓+ 𝑣𝑅𝑓.
Let 𝑥𝑛− 1 = 𝑓1𝑓2⋯ 𝑓𝑚 be a representation of as a product of basic irreducible pairwise-coprime polynomials in Λ[𝑥] and 𝑆𝑖 be the set of ideals of Δ[𝑥]
〈𝑓𝑖(𝑥)〉. Then 𝑆𝑖= {0, 𝑣𝑅𝑓𝑖, 𝑢𝑣𝑅𝑓𝑖, 𝑅𝑓𝑖+ 𝑣𝑅𝑓𝑖, 𝑢𝑅𝑓𝑖+ 𝑣𝑅𝑓𝑖, 𝑢𝑅𝑓+ 𝑢𝑣𝑅𝑓𝑖, (𝑢 + 𝑣 ∑𝑖∈𝐴𝑥𝑖)𝑅𝑓𝑖} = {∑ (〈𝑢𝑗𝑖𝑣𝑘𝑖+ 〈𝑓𝑖(𝑥)〉〉)} ∪ {〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗) + 〈𝑓𝑖(𝑥)〉〉}, where 0 ≤ 𝑗𝑖, 𝑘𝑖≤ 2, 1 ≤ 𝑖 ≤ 𝑚. Let 𝑓1, 𝑓2, … , 𝑓𝑚 be a product of basic irreducible pairwise-coprime polynomials of 𝑥𝑛− 1 in Λ[𝑥]. Let 𝑓̂𝑖 denote the polynomial 𝑥𝑛−1
𝑓𝑖(𝑥). Then any ideal in 𝑅 is a sum of ideals of the form 〈𝑢𝑗𝑖𝑣𝑘𝑖𝑓̂𝑖+ 〈𝑥𝑛− 1〉〉 and 〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗)𝑓̂𝑖+ 〈𝑥𝑛− 1〉〉), where 0 ≤ 𝑗𝑖, 𝑘𝑖≤ 2, 1 ≤ 𝑖 ≤ 𝑚 and 𝐴 is a subset of {0,1, … , 𝑛 − 1}.
Proof. Observe that 𝑓𝑖(𝑥) and 𝑓̂𝑖(𝑥) are coprime for 𝑖 = 1,2, … , 𝑛 − 1. Then there exist polynomials 𝑏𝑖(𝑥) and 𝑐𝑖(𝑥) in Δ[𝑥] such that
𝑏𝑖(𝑥)𝑓̂𝑖(𝑥) + 𝑐𝑖(𝑥)𝑓𝑖(𝑥) = 1.
Let 𝑒𝑖(𝑥) = 𝑏𝑖(𝑥)𝑓̂𝑖(𝑥) + 〈𝑥𝑛− 1〉. By the Chinese Remainder Theorem, we have 𝑅 =⊕𝑖=1
𝑚
𝑅𝑒𝑖(𝑥) and
𝑅𝑒𝑖(𝑥) ≅ Δ[𝑥]
〈𝑓𝑖(𝑥)〉,
for 𝑖 = 1,2, … , 𝑛 − 1. By Proposition 3, an ideal of 𝑅𝑓𝑖 is one of the elements of 𝑆𝑖. Then 𝐼𝑖 corresponds to the form
〈𝑢𝑗𝑖𝑣𝑘𝑖𝑓̂𝑖+ 〈𝑥𝑛− 1〉〉 or 〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗)𝑓̂𝑖+ 〈𝑥𝑛− 1〉〉) in 𝑅𝑒𝑖(𝑥) , where 0 ≤ 𝑗𝑖, 𝑘𝑖≤ 2 and 1 ≤ 𝑖 ≤ 𝑚 . Therefore, 𝐼 is a direct sum of ideals of the form 〈𝑢𝑗𝑖𝑣𝑘𝑖𝑓̂𝑖+ 〈𝑥𝑛− 1〉〉 or 〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗)𝑓̂𝑖+ 〈𝑥𝑛− 1〉〉) in 𝑅, where 0 ≤ 𝑗𝑖, 𝑘𝑖≤ 2 and 1 ≤ 𝑖 ≤ 𝑚.
By Proposition 4, we certify that any ideal in 𝑅 is a sum of ideals of the form 〈𝑢𝑗𝑖𝑣𝑘𝑖𝑓̂𝑖+ 〈𝑥𝑛− 1〉〉 and
50
〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗)𝑓̂𝑖+ 〈𝑥𝑛− 1〉〉), where 0 ≤ 𝑗𝑖, 𝑘𝑖≤ 2, 1 ≤ 𝑖 ≤ 𝑚 and 𝐴 is a subset of {0,1, … , 𝑛 − 1}. Through the analysis of the front, we know that a linear code 𝐶 of length 𝑛 is a cyclic code over Δ if and only if 𝜑(𝐶) is an ideal of 𝑅. Therefore, we can gain the algebraic structure of cyclic codes over Δ. In order to simply the algebraic structure of cyclic codes, we need the following lemma first. Let 𝑔1(𝑥), 𝑔2(𝑥), … , 𝑔𝑚(𝑥) be monic polynomials in Λ[𝑥]. Then we have
〈𝑔1(𝑥)〉 + 〈𝑔2(𝑥)〉 + ⋯ + 〈𝑔𝑚(𝑥)〉 = 〈𝒦(𝑥)〉, where 𝒦(𝑥) = g𝑐𝑑(𝑔1(𝑥), 𝑔2(𝑥), … , 𝑔𝑚(𝑥)).
Proof. Since 𝒦(𝑥) = g𝑐𝑑(𝑔1(𝑥), 𝑔2(𝑥), … , 𝑔𝑚(𝑥)) , then 𝒦(𝑥)|𝑔𝑖(𝑥) , for 𝑖 = 1,2, … , 𝑚 . Hence,
〈𝑔𝑖(𝑥)〉 ⊆ 〈𝒦(𝑥)〉. Therefore, 〈𝑔1(𝑥)〉 + 〈𝑔2(𝑥)〉 + ⋯ + 〈𝑔𝑚(𝑥)〉 ⊆ 〈𝒦(𝑥)〉.
Since 𝒦(𝑥) = g𝑐𝑑(𝑔1(𝑥), 𝑔2(𝑥), … , 𝑔𝑚(𝑥)), then there have polynomials 𝑎1(𝑥), … , 𝑎𝑚(𝑥) in Λ[𝑥] such that
𝑎1(𝑥)𝑔1(𝑥) + 𝑎2(𝑥)𝑔2(𝑥) + ⋯ + 𝑎𝑚(𝑥)𝑔𝑚(𝑥) = 𝒦(𝑥).
Therefore, 〈𝒦(𝑥)〉 = 〈𝑎1(𝑥)𝑔1(𝑥) + 𝑎2(𝑥)𝑔2(𝑥) + ⋯ + 𝑎𝑚(𝑥)𝑔𝑚(𝑥)〉 ⊆ 〈𝑔1(𝑥)〉 + 〈𝑔2(𝑥)〉 + ⋯ + 〈𝑔𝑚(𝑥)〉 . Hence, 〈𝑔1(𝑥)〉 + 〈𝑔2(𝑥)〉 + ⋯ + 〈𝑔𝑚(𝑥)〉 = 〈𝒦(𝑥)〉.
Let 𝐶 be a cyclic code of odd length 𝑛 over Δ. Then
𝐶 = 〈𝐹̂1〉 ⊕ 〈𝑢𝐹̂2〉 ⊕ 〈𝑣𝐹̂3〉 ⊕ 〈𝑢𝑣𝐹̂4〉 ⊕ 〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗)𝐹̂5〉 ⊕ (〈𝑢𝐹̂6〉 + 〈𝑣𝐹̂6〉).
where 𝐴 is a subset of {0,1, … , 𝑛 − 1} , 𝐹0, 𝐹1, … , 𝐹6 in Λ[𝑥] are pairwise coprime monic polynomials and 𝐹0𝐹1⋯ 𝐹6= 𝑥𝑛− 1.
Proof. Let 𝑓1𝑓2⋯ 𝑓𝑚 be a product of basic irreducible pairwise-coprime polynomials of 𝑥𝑛− 1 in Λ[𝑥]. By Proposition 4, 𝐶 is a direct sum of ideals of the form 〈𝑢𝑗𝑖𝑣𝑘𝑖𝑓̂𝑖〉 and 〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗)𝑓̂𝑖〉), where 0 ≤ 𝑗𝑖, 𝑘𝑖≤ 2 and 1 ≤ 𝑖 ≤ 𝑚. After arranging if necessary, we assume that
𝐶 = 〈𝑓̂𝑘1+1〉 ⊕ ⋯ ⊕ 〈𝑓̂𝑘1+𝑘2〉
⊕ 〈𝑢𝑓̂𝑘1+𝑘2+1〉 ⊕ ⋯ ⊕ 〈𝑢𝑓̂𝑘1+𝑘2+𝑘3〉
⊕ 〈𝑣𝑓̂𝑘1+𝑘2+𝑘3+1〉 ⊕ ⋯ ⊕ 〈𝑣𝑓̂𝑘1+𝑘2+𝑘3+𝑘4〉
⊕ 〈𝑢𝑣𝑓̂𝑘1+𝑘2+𝑘3+𝑘4+1〉 ⊕ ⋯ ⊕ 〈𝑢𝑣𝑓̂𝑘1+𝑘2+𝑘3+𝑘4+𝑘5〉
⊕ 〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗)𝑓̂𝑘1+𝑘2+𝑘3+𝑘4+𝑘5+1〉
⊕ ⋯ ⊕ 〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗)𝑓̂𝑘1+𝑘2+𝑘3+𝑘4+𝑘5+𝑘6〉
⊕ 〈𝑢𝑓̂𝑘1+𝑘2+𝑘3+𝑘4+𝑘5+𝑘6+1〉 + 〈𝑣𝑓̂𝑘1+𝑘2+𝑘3+𝑘4+𝑘5+𝑘6+1〉
⊕ ⋯ ⊕ 〈𝑢𝑓̂𝑚〉 + 〈𝑣𝑓̂𝑚〉, where 𝑘1, … , 𝑘6≥ 0 and 𝑘1+ ⋯ + 𝑘6+ 1 ≤ 𝑚.
Then we define that
𝐹0= 𝑓1⋯ 𝑓𝑘1, 𝐹1= 𝑓𝑘1+1⋯ 𝑓𝑘1+𝑘2,
𝐹2= 𝑓𝑘1+𝑘2+1⋯ 𝑓𝑘1+𝑘2+𝑘3, 𝐹3= 𝑓𝑘1+𝑘2+𝑘3+1⋯ 𝑓𝑘1+𝑘2+𝑘3+𝑘4, 𝐹4= 𝑓𝑘1+𝑘2+𝑘3+𝑘4+1⋯ 𝑓𝑘1+𝑘2+𝑘3+𝑘4+𝑘5,
𝐹5= 𝑓𝑘1+𝑘2+𝑘3+𝑘4+𝑘5+1⋯ 𝑓𝑘1+𝑘2+𝑘3+𝑘4+𝑘5+𝑘6, 𝐹6= 𝑓𝑘1+𝑘2+𝑘3+𝑘4+𝑘5+𝑘6+1⋯ 𝑓𝑚.
Obviously, 𝐹0, 𝐹1,...,𝐹6 are pairwise coprime, 𝐹0𝐹1⋯ 𝐹6= 𝑥𝑛− 1.
By lemma 3, It is clear that
〈𝐹̂1〉 = 〈𝑓̂𝑘1+1〉 ⊕ ⋯ ⊕ 〈𝑓̂𝑘1+𝑘2〉, 〈𝑢𝐹̂2〉 = 〈𝑢𝑓̂𝑘1+𝑘2+1〉 ⊕ ⋯ ⊕ 〈𝑢𝑓̂𝑘1+𝑘2+𝑘3〉,
〈𝑣𝐹̂3〉 = 〈𝑣𝑓̂𝑘1+𝑘2+𝑘3+1〉 ⊕ ⋯ ⊕ 〈𝑣𝑓̂𝑘1+𝑘2+𝑘3+𝑘4〉,
〈𝑢𝑣𝐹̂4〉 = 〈𝑢𝑣𝑓̂𝑘1+𝑘2+𝑘3+𝑘4+1〉 ⊕ ⋯ ⊕ 〈𝑢𝑣𝑓̂𝑘1+𝑘2+𝑘3+𝑘4+𝑘5〉,
〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗)𝐹̂5〉 = 〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗)𝑓̂𝑘1+𝑘2+𝑘3+𝑘4+𝑘5+1〉
⊕ ⋯ ⊕ 〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗)𝑓̂𝑘1+𝑘2+𝑘3+𝑘4+𝑘5+𝑘6〉,
〈𝑢𝐹̂6〉 + 〈𝑣𝐹̂6〉 = (〈𝑢𝑓̂𝑘1+𝑘2+𝑘3+𝑘4+𝑘5+𝑘6+1〉 + 〈𝑣𝑓̂𝑘1+𝑘2+𝑘3+𝑘4+𝑘5+𝑘6+1〉)
⊕ ⋯ ⊕ (〈𝑢𝑓̂𝑚〉 + 〈𝑣𝑓̂𝑚〉).
Hence, we obtain
𝐶 = 〈𝐹̂1〉 ⊕ 〈𝑢𝐹̂2〉 ⊕ 〈𝑣𝐹̂3〉 ⊕ 〈𝑢𝑣𝐹̂4〉 ⊕ 〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗)𝐹̂5〉 ⊕ (〈𝑢𝐹̂6〉 + 〈𝑣𝐹̂6〉).
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In the following content, we study the algebraic structure of self-dual cyclic codes of odd length 𝑛 over Δ.
Let 𝑎 = (𝑎0, 𝑎1, … , 𝑎𝑛−1), 𝑏 = (𝑏0, 𝑏1, … , 𝑏𝑛−1) be two vectors in Δ𝑛. The vectors 𝑎 and 𝑏 are called orthogonal if 𝑎 ⋅ 𝑏 = 𝑎0𝑏0+ 𝑎1𝑏1+ ⋯ + 𝑎𝑛−1𝑏𝑛−1= 0. For a linear code 𝐶 over Δ, its dual code 𝐶⊥= {𝑎 ∈ Δ𝑛|𝑎 ⋅ 𝑏 = 0, ∀𝑏 ∈ 𝐶}. If 𝐶 = 𝐶⊥, then 𝐶 is called a self-dual code. If the number of codewords in any linear code 𝐶 over Δ is 2𝑘, for some integer 𝑘 ∈ {0,1, … ,4𝑛}, then its dual code 𝐶⊥ has 2ℓ codewords, where 𝑘 + ℓ = 4𝑛.
Let 𝐶 be a cyclic code of odd length 𝑛 with notation as in Theorem 1. Then 𝐶⊥= 〈𝐹̂0∗〉 ⊕ 〈𝑢𝐹̂2∗〉 ⊕
〈𝑣𝐹̂3∗〉 ⊕ (〈𝑢𝐹̂4∗〉 + 〈𝑣𝐹̂4∗〉) ⊕ 〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗)𝐹̂5∗〉 ⊕ 〈𝑢𝑣𝐹̂6∗〉, where 𝐹𝑖∗ denotes the reciprocal polynomial of 𝐹𝑖, 𝑖 = 0,1, … ,6.
Proof. Let 𝐶∗= 〈𝐹̂0∗〉 ⊕ 〈𝑢𝐹̂2∗〉 ⊕ 〈𝑣𝐹̂3∗〉 ⊕ (〈𝑢𝐹̂4∗〉 + 〈𝑣𝐹̂4∗〉) ⊕ 〈(𝑢 + 𝑣 ∑𝑗∈𝐴𝑥𝑗)𝐹̂5∗〉 ⊕ 〈𝑢𝑣𝐹̂6∗〉. For 𝑖, 𝑗 ∈ {0,1, … ,6}, if 𝑖 ≠ 𝑗, then 𝑥𝑛− 1|𝐹̂𝑖(𝐹̂𝑗∗)∗. Therefore, 𝐹̂𝑖(𝐹̂𝑗∗)∗= 0. Hence, 𝐶∗⊆ 𝐶⊥.
Since |𝑢𝑣Δ| = 2, |𝑢Δ| = |𝑣Δ| = |(𝑢 + 𝑣)Δ| = 22, |𝑢Δ| + |𝑣Δ| = 23 and |Δ| = 24, let 𝑘 = 4d𝑒𝑔𝐹1+ 2d𝑒𝑔𝐹2+ 2d𝑒𝑔𝐹3+ d𝑒𝑔𝐹4+ 2d𝑒𝑔𝐹5+ 3d𝑒𝑔𝐹6 and ℓ = 4d𝑒𝑔𝐹0+ 2d𝑒𝑔𝐹2+ 2d𝑒𝑔𝐹3+ 3d𝑒𝑔𝐹4+ 2d𝑒𝑔𝐹5+ d𝑒𝑔𝐹6, then |𝐶| = 2𝑘 and |𝐶⊥| = 2ℓ. Observe that ℓ + 𝑘 = 4𝑛. Therefore, |𝐶⊥| = |𝐶∗|. Hence, 𝐶⊥= 𝐶∗.
In Theorem 1 and Theorem 2, we describe the algebraic structure of cyclic codes and their dual codes over the ring Δ. In the following, we provide a necessary and sufficient condition for the existence of self-dual cyclic codes as the main result of this paper. Let 𝐶 be a cyclic code of odd length 𝑛 with notation as in Theorem 2. Then 𝐶 is self-dual if and only if 〈𝐹0∗〉 = 〈𝐹1〉, 〈𝐹2∗〉 = 〈𝐹2〉, 〈𝐹3∗〉 = 〈𝐹3〉, 〈𝐹4∗〉 = 〈𝐹6〉 and 〈𝐹5∗〉 = 〈𝐹5〉.
Proof. By checking generators of 𝐶 and 𝐶⊥, it is easy to see that if 𝐶 is a self-dual code, then it need to meet 〈𝐹0∗〉 = 〈𝐹1〉, 〈𝐹2∗〉 = 〈𝐹2〉, 〈𝐹3∗〉 = 〈𝐹3〉, 〈𝐹4∗〉 = 〈𝐹6〉 and 〈𝐹5∗〉 = 〈𝐹5〉.
At the end of the paper, we will show that the Gray map images of self-dual codes over the ring Δ is also self-dual over the ring Λ. Let 𝐶 be a linear code of length 𝑛 over Δ. Then Θ(𝐶)⊥= Θ(𝐶⊥). Moreover, if 𝐶 is self- dual over Δ, then Θ(𝐶) is also self-dual over Λ.
Proof. For all 𝑐1= (𝑐1,0, 𝑐1,1, … , 𝑐1,𝑛−1) ∈ 𝐶 and 𝑐2= (𝑐2,0, 𝑐2,1, … , 𝑐2,𝑛−1) ∈ 𝐶⊥, where 𝑐𝑖,𝑗 = 𝑝𝑖,𝑗+ 𝑞𝑖,𝑗𝑣, 𝑝𝑖,𝑗, 𝑞𝑖,𝑗 ∈ Λ , 𝑖 = 1,2 and 𝑗 = 0,1, … , 𝑛 − 1 . Since 𝑐1⋅ 𝑐2= 0 , then we have ∑𝑛−1𝑗=0 𝑐1,𝑗𝑐2,𝑗= ∑𝑛−1𝑗=0 𝑝1,𝑗𝑝2,𝑗+
∑𝑛−1𝑗=0 (𝑝1,𝑗𝑞2,𝑗+ 𝑝2,𝑗𝑞1,𝑗)𝑣 = 0 implying ∑𝑛−1𝑗=0 𝑝1,𝑗𝑝2,𝑗= 0 and ∑𝑛−1𝑗=0 (𝑝1,𝑗𝑞2,𝑗+ 𝑝2,𝑗𝑞1,𝑗) = 0. Therefore, Θ(𝑐1) ⋅ Θ(𝑐2) = ∑𝑛−1𝑗=0 (𝑝1,𝑗𝑝2,𝑗+ 𝑝1,𝑗𝑞2,𝑗+ 𝑝2,𝑗𝑞1,𝑗) = 0. Thus, Θ(𝐶)⊥ ⊆ Θ(𝐶)⊥. From Proposition 2, we have |Θ(𝐶)⊥| =
|Θ(𝐶)⊥|, which implies that Θ(𝐶)⊥= Θ(𝐶⊥). Clearly, Θ(𝐶) is self-orthogonal if 𝐶 is self-dual. Since |Θ(𝐶)| = |𝐶|, then Θ(𝐶) is self-dual.
4. Example
In this section, we show some examples of self-dual cyclic codes of odd length 𝑛 over Δ applying Theorem 3.
Moreover, by Proposition 5, we can construct some self-dual codes of length 2𝑛 over Λ. We compute the minimum distance of the codes below by the computational algebra system Magma [12]. Let 𝑛 = 5. Then
𝑥5− 1 = (1 + 𝑥)(1 + 𝑥 + 𝑥2+ 𝑥3+ 𝑥4)
over Λ. Suppose that 𝐹2= 1 + 𝑥 and 𝐹3= 1 + 𝑥 + 𝑥2+ 𝑥3+ 𝑥4, where 〈𝐹2∗〉 = 〈𝐹2〉 and 〈𝐹3〉 = 〈𝐹3∗〉. Let 𝐶 =
〈𝑢(1 + 𝑥 + 𝑥2+ 𝑥3+ 𝑥4), 𝑣(1 + 𝑥)〉. By Theorem 3, 𝐶 is a self-dual cyclic code of length 5 over Δ with parameter (5, 4422, 4). By and Proposition 2 and Proposition 5, we know Θ(𝐶) is a self-dual code over Λ with parameter (10, 210, 4). This code is optimal [11]. Let 𝑛 = 7. Then 𝑥7− 1 = (1 + 𝑥)(1 + 𝑥 + 𝑥3)(1 + 𝑥2+ 𝑥3) = 𝑓1𝑓2𝑓3
over Λ and 〈𝑓1∗〉 = 〈𝑓1〉, 〈𝑓2∗〉 = 〈𝑓3〉, 〈𝑓3∗〉 = 〈𝑓2〉. By Theorem 3, some self-dual cyclic codes of length 14 over Λ are obtained in Table 1.
52
Table 1: Self-dual cyclic codes of length 14 over Λ.
Generators 𝑑𝐶 Θ(𝐶)
〈𝑓1𝑓2, 𝑢𝑓2𝑓3〉 4 (14, 47, 4)
〈𝑓1𝑓2, 𝑣𝑓2𝑓3〉 4 (14, 47, 4)
〈𝑓1𝑓2, (𝑢 + 𝑣)𝑓2𝑓3〉
4 (14, 47, 4)
〈𝑓1𝑓3, 𝑢𝑓1𝑓3〉 4 (14, 47, 4)
〈𝑓1𝑓3, 𝑣𝑓1𝑓3〉 4 (14, 47, 4)
〈𝑓1𝑓3, (𝑢 + 𝑣)𝑓1𝑓3〉
4 (14, 47, 4)
〈𝑢𝑣𝑓1𝑓3, 𝑢𝑓1𝑓2, 𝑣𝑓1𝑓2, 𝑢𝑓2𝑓34 〉 (14, 47, 4)
〈𝑢𝑣𝑓1𝑓3, 𝑢𝑓1𝑓2, 𝑣𝑓1𝑓2, 𝑣𝑓2𝑓3 4 〉 (14, 47, 4)
〈𝑢𝑣𝑓1𝑓3, 𝑢𝑓1𝑓2, 𝑣𝑓1𝑓2, (𝑢 + 𝑣)𝑓2𝑓3〉
4 (14, 47, 4)
〈𝑢𝑣𝑓1𝑓2, 𝑢𝑓1𝑓3, 𝑣𝑓1𝑓3, 𝑢𝑓2𝑓3 4 〉 (14, 47, 4)
〈𝑢𝑣𝑓1𝑓2, 𝑢𝑓1𝑓3, 𝑣𝑓1𝑓3, 𝑣𝑓2𝑓3 4 〉 (14, 47, 4)
〈𝑢𝑣𝑓1𝑓2, 𝑢𝑓1𝑓3, 𝑣𝑓1𝑓3, (𝑢 + 𝑣)𝑓2𝑓3〉
4 (14, 47, 4)
Let 𝑛 = 15 . Then 𝑥15− 1 = (1 + 𝑥)(1 + 𝑥 + 𝑥2)(1 + 𝑥 + 𝑥4)(1 + 𝑥3+ 𝑥4)(1 + 𝑥 + 𝑥2+ 𝑥3+ 𝑥4) = 𝑓1𝑓2𝑓3𝑓4𝑓5 over Λ. Observe that 〈𝑓1∗〉 = 〈𝑓1〉, 〈𝑓2∗〉 = 〈𝑓2〉, 〈𝑓5∗〉 = 〈𝑓5〉, 〈𝑓3∗〉 = 〈𝑓4〉. By Theorem 3, some self-dual cyclic codes of length 30 over Λ are shown in Table 2.
Table 2: Self-dual cyclic codes of length 30 over Λ.
Generators 𝑑𝐶 Θ(𝐶)
〈𝑓1𝑓2𝑓4𝑓5, 𝑢𝑓3𝑓4𝑓5, 𝑣𝑓1𝑓2𝑓3𝑓 8 4〉 (30, 415, 8)
〈𝑓1𝑓2𝑓4𝑓5, 𝑢𝑓3𝑓4𝑓5, (𝑢 + 𝑣)𝑓1𝑓2𝑓3𝑓4〉
8 (30, 415, 8)
〈𝑓1𝑓2𝑓4𝑓5, 𝑢𝑓3𝑓4〉 6 (30, 415, 6)
〈𝑓1𝑓2𝑓4𝑓5, 𝑣𝑓3𝑓4〉 6 (30, 415, 6)
〈𝑓1𝑓2𝑓4𝑓5, (𝑢 + 𝑣)𝑓3𝑓4〉
6 (30, 415, 6)
〈𝑓1𝑓2𝑓4𝑓5, (𝑢 + 𝑣)𝑓3𝑓4𝑓5, 𝑣𝑓1𝑓2𝑓3𝑓4〉
6 (30, 415, 6)
〈𝑢𝑣𝑓1𝑓2𝑓4𝑓5, 𝑢𝑓3, 𝑣𝑓1𝑓2𝑓3〉 4 (30, 415, 4)
〈𝑢𝑣𝑓1𝑓2𝑓4𝑓5, 𝑣𝑓3, 𝑢𝑓1𝑓2𝑓3〉 4 (30, 415, 4)
〈𝑢𝑣𝑓1𝑓2𝑓4𝑓5, 𝑢𝑓3𝑓5, 𝑣𝑓1𝑓2𝑓3 4 〉 (30, 415, 4)
53 References
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