Unique Generators for Cyclic Codes of Arbitrary Length over and Their Applications

Abstract

Cyclic codes play a very important role in the history of coding theory since they have good algebraic structures that can be widely used in coding and decoding. However, generators for repeated-root cyclic codes of arbitrary length over are not unique in previous works for , and hence it is impossible to determine their dual codes. In this work, we propose unique generators for cyclic codes of arbitrary length over . As its applications, we derive the numbers of their codewords, as well as generators for their dual codes. Furthermore, we propose necessary and sufficient conditions for their self-dualities.

1. Introduction

In 1994, Hammons et al. [1] discovered that many efficient nonlinear binary codes are closely related to linear codes over integer residue ring . After that, codes over finite rings have got much attention [2–6]. Among various finite rings, is interesting which is because that it shares some good properties of and the finite field , and cyclic codes over such a ring can be used to construct optimal binary codes [7]. From then on, much work has been done on this kind of rings, especially on cyclic codes over for integers , and positive prime number . Qian et al. [8] determined the structure of simple-root cyclic codes over , which are principal ideals over . However, structures of repeated-root cyclic codes over are more complicated than simple-root cyclic codes since some repeated-root cyclic codes are not principal ideals.

Abualrub and Saip [9] studied generators for cyclic codes of arbitrary length over and and proposed generators for the dual codes of cyclic codes with arbitrary length over . Later, the results about generators in [9] were generalized to cyclic codes of arbitrary length over , , and in a similar way [10–12]. Recently, Cao et al. [13] determined the number of codewords of a cyclic code with oddly even length over and investigated the dual code of each cyclic code and self-dual cyclic codes with such a length over .

When , we find that generators for cyclic codes of arbitrary length in [9–12] are not unique. This makes the authors unable to determine generators in [9–12] or give wrong generators in [11] for the dual code of a cyclic code with an arbitrary length over . Therefore, the dual code of a cyclic code and self-dual cyclic codes over are still open. In this paper, we give unique generators for cyclic codes of arbitrary length over and then determine their numbers of codewords, dual codes, and self-dualities. The remainder of this paper is organized as follows. After a brief introduction of some notations and existing results in Section 2, we give generators for the dual code of cyclic codes with arbitrary length over in Section 3. We then propose necessary and sufficient conditions for the self-duality of cyclic codes over in Section 4 and conclude in Section 5.

2. Preliminaries

Let and . The cyclic shift on is the shift , where and for nonnegative integers and a prime integer . Let be a code over , then is said to be a cyclic code over if . We will identify a codeword with its polynomial representation . With this identification, cyclic codes over can be identified as ideals in the ring . Similarly, cyclic codes over can be identified as ideals in the ring . Definitions 1 and 2 are similar to [10–12].

Definition 1. Let be a cyclic code of arbitrary length over and let , then the map is defined by , which is a ring homomorphism that can be extended to a homomorphism defined by for .
From Definition 1, we know that is an ideal of , and hence is a cyclic code over . The kernel of iswhich is an ideal of .

Definition 2. Let be a cyclic code of arbitrary length over and let , then the map is defined by , which is a ring homomorphism that can be extended to a homomorphism defined by for .
Definition 2 shows that is an ideal of , and hence is a cyclic code over . Therefore, we let in the following. Besides, the kernel of iswhich is an ideal of .

Proposition 1 (see [9–12]). Let be a cyclic code of arbitrary length over , then there exist polynomials such thatwhere , , , , ,, , and .
Next, we give an example to show that the generators for in Proposition 1 are not unique.

Example 1. In , we have , where and . Let and be cyclic codes of length over . Clearly, and are in the form of Proposition 1 but

Definition 3. For any , we say c and b are orthogonal if . The dual code of a cyclic code with length over is defined by .
It is well known that is also a cyclic code over and . If is a polynomial of degree , then the reciprocal of is . Now, we define the annihilator of as , then is a cyclic code over and .

3. Unique Generators for Cyclic Codes of Arbitrary Length over

In this section, we specify a choice of generator polynomials for such that its generators are unique, and its basic idea is inspired by Ding et al. [3].

Lemma 1 (see [11]). A cyclic code of arbitrary length over can be written uniquely as , where and monic are polynomials in such that , in , and .

Lemma 2. Let be a cyclic code of arbitrary length over , where and monic are polynomials in such that in and . Then, if and only if in .

Proof. Assume that . In , we have , which implies . Consequently, there are polynomials such that .
Let , satisfyingThen, in . Since in , then in . This completes the proof of the necessary part. Next, we will consider the sufficient part.
Suppose that in . Since and in , then . On the other hand, suppose for , then we only need to consider the following three cases to construct in . Case 1:  Case 2:  Case 3: From Cases 1 to 3 and the conditions and in , we derive that if , which implies . Thus, .

Theorem 1. Let be a cyclic code of arbitrary length over ; then, there exist unique polynomials such thatwhere , and are monic, in , ,, , , and .

Proof. According to Lemmas 1 and 2 and the note after Definition 2, we know that is a cyclic code over , and it can be written uniquely aswhere and monic are polynomials in such that in , , and . Therefore, we only need to prove the uniqueness of , and .
Since , then is an ideal of , which implies that monic polynomial with in is unique.
Now, we consider the uniqueness of and . By equation (7), it is easy to check that can be written aswhere , , , , , and . Consequently,This means . Suppose there are such that , , andthenwhich implies . Since and , then we have , which means is unique. Similarly, we can get and hence give the proof of the uniqueness of .
Next, we will consider the necessary and sufficient conditions for the constraints and in Theorem 1, which will be proposed in Lemmas 3 and 4, respectively.

Lemma 3. Let be a cyclic code of arbitrary length over , and the constraints on corresponding polynomials are the same as that in Theorem 1. Then, we have the following results (1) to (5) in :(1)(2)(3)(4)(5)

Proof. Since and , then in from Lemma 2. On the other hand, we have the following in . Case 1:  Case 2:  Case 3:  Case 4: These cases mean that the results (2) to (5) are true in , which are also true in through a similar discussion to Lemma 2.

Lemma 4. Let be a cyclic code of arbitrary length over , where and monic are polynomials in such that in , , , and . Then, and if the corresponding polynomials satisfy all conditions (1) to (5) in Lemma 3.

Proof. It is straight forward that from Lemmas 2 and 3 (1). Next, we will construct for . On the one hand, we use one generator of . Case 1: since , then we have for in . Case 2: since , then from Lemma 3 (1) and in . Case 3: since , then from Lemma 3 (3). Case 4: since , then we have from in . Case 5: since , then we have from Lemma 3 (2). Case 6: . On the other hand, we use two generators of . Indeed, we only need to consider and , since . Case 7: from Lemma 3 (4), we have and hence . Case 8: from Lemma 3 (5), we haveFrom Cases 1 to 8, we have if , which implies . Since , then . Thus, we have .
Finally, we consider the number of codewords of a given cyclic code over . Let . If we denote for , then . Consequently, the -th torsion code of is defined by , which is a cyclic code of length over .

Lemma 5. Let be a cyclic code of arbitrary length over , where and monic are polynomials in such that in , , , and . Then, and if and only if and .

Proof. Let , thenTherefore, if and only ifOn the other hand,Hence, if and only if

Theorem 2. Let be a cyclic code of arbitrary length over , then there exist unique polynomials , such thatwhere , and are monic, , ,, , , in ,, , and . Besides, the number of codewords of is

Proof. From Theorem 1 and Lemmas 3 and 4, the uniqueness of generators for is clear. Consequently, we only need to calculate the number of codewords of . It is easy to check that from the definition of torsion code. On the other hand, and from Lemma 5. Therefore, by Han et al. [14], we haveIf is a cyclic code of arbitrary length over and the corresponding polynomials satisfy the conditions in Theorem 2, then we denote byFrom Theorem 2, the following corollary is straightforward.

Corollary 1. Let be a cyclic code of length over , then there exit unique polynomials and unique integers such thatwhere , , , , and , , , , and in . Besides, the number of codewords of is .

4. The Dual Codes of Cyclic Codes of Arbitrary Length over

Lemma 6. Let be a cyclic code of arbitrary length over , thenwhere .

Proof. From Theorem 2, we know that , and are polynomials in and all their degrees are not more than . In , we haveBesides, it is easy to check thatwhich impliesSimilarly, we can prove and .

Lemma 7. Let be a cyclic code of arbitrary length over , thenwhere .

Proof. Denote bywhere , , , and . It is straightforward that from Lemma 6. Next, we will proveFrom Theorem 2, we have , . Besides, the polynomials , , are monic, , , and . Case 1: since in , then in . Case 2: since , then in . Case 3: since , then in . Case 4: since , then in . Case 5: since , then in , we have Case 6: sincethen in , we have , which is becauseFrom the above discussion and Theorem 2, we haveand hence . Since , then . This gives the proof.

Theorem 3. Let be a cyclic code of arbitrary length over . Degrees of are denoted by , respectively. Degrees of are denoted by , respectively. Constant terms of are written as , respectively. Then,wherefor

Proof. From Theorem 2, we know that are monic polynomials in and in , which implies that . From Lemma 7, we have From the definitions of torsion code, we haveSincethenwhich implies thatfrom Theorem 1 and Lemma 5.

Lemma 8. Let be a monic divisor of in and be the constant term of . If in , then is a positive even number and .

Proof. Suppose for nonnegative and . Let be the factorization of into a product of monic basic irreducible pairwise coprime polynomials in , then and . Since is a monic divisor of in , then we let for and . Thus, . Since and in , then we have , which implies is a positive even number and .

Theorem 4. Let be a cyclic code of arbitrary length over . Degrees of are denoted by , respectively. Degrees of are denoted by , respectively. Constant terms of are written as , respectively. Then, is a self-dual cyclic code of length over if and only if is a positive even number, , , , , ,andfor .

Proof. According to Theorem 3, we haveBesides,Since is a monic divisor of and , then is a positive even number and from Lemma 8. This gives the proof.

Corollary 2. Let be a cyclic code of length over . Degrees of are denoted by , respectively. Then, is a self-dual cyclic code of over if and only if , , , , and for .

Proof. Let in Theorem 4, then and .
Observe that , , and if and only if and . So, , and . Thus,where .

Example 2. In , we have . Letbe a cyclic code of length over . It is easy to check that is in the unique form in Corollary 1, and hence . From the unique form in Corollary 1, we get that is a new cyclic code of length over comparing with codes in [10]. Furthermore,Thus, , , and . Besides,which implies that , and from Corollary 2.
Since , then , which implies that . SincethenThus, is a self-dual cyclic code of length over .

Example 3. In , we have . Suppose , , , , , and . From Theorem 2, we get is a cyclic code of length over and .
Besides, degrees of are , respectively. Constant terms of are , and hence . Since , then . It is easy to check that ,, and .
Since , then is a self-dual cyclic code of length over from Theorem 4.

5. Conclusion

In this paper, we obtain unique generators for cyclic codes with arbitrary length over , as well as their numbers of codewords. Based on these, we determine the dual of cyclic codes of arbitrary length and propose necessary and sufficient conditions for the self-dualities of cyclic codes. It is interesting to consider the number of self-dual cyclic codes of even length over .

Data Availability

No data was used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by University Natural Science Research Project of Anhui Province under Grant nos. KJ2020A0779 and KJ2020A0780 and in part by Natural Science Research Project of Anhui Xinhua University under Grant no. 2020zr010.