Structure and Rank of a Cyclic Code over a Class of Nonchain Rings

Jain, Nikita; Dutt, Sucheta; Sehmi, Ranjeet

doi:https://doi.org/10.1155/2024/8817721

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Abstract Introduction Preliminaries Conclusion Data Availability Disclosure Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 8817721 | https://doi.org/10.1155/2024/8817721

Structure and Rank of a Cyclic Code over a Class of Nonchain Rings

Nikita Jain,¹Sucheta Dutt,¹and Ranjeet Sehmi¹

Academic Editor: Faranak Farshadifar

Received15 May 2023

Revised05 Dec 2023

Accepted16 Dec 2023

Published18 Jan 2024

Abstract

The rings have been classified into chain rings and nonchain rings based on the values of . In this paper, the structure of a cyclic code of arbitrary length over the rings for those values of for which these are nonchain rings has been established. A unique form of generators for a cyclic code over these rings has also been obtained. Furthermore, the rank and cardinality of a cyclic code over these rings have been established by finding a minimal spanning set for the code.

1. Introduction

From a mathematical point of view, one of the main aims of algebraic coding theory is to construct codes that can detect and correct the maximum number of errors during data transmission. To construct such codes, it is important to know the structure of a code.

The class of cyclic codes is one of the significant classes of codes, as these codes offer efficient encoding and decoding of the data using shift registers. These codes have good error-detecting and error-correcting capabilities. The theory of cyclic codes over finite fields is well established. The study of cyclic codes over rings started after the remarkable work done by Calderbank et al. [1], wherein a Gray map was introduced to show that some nonlinear binary codes can be viewed as binary images of linear codes over .

Recent research involves various approaches to determine the generators of cyclic codes over various finite commutative rings. A vast literature is available on cyclic codes over integer residue rings [2–4], Galois rings [5, 6], and finite chain rings [7, 8].

The generators of a cyclic code of arbitrary length over finite chain rings of the type and have been obtained by Abualrub and Siap [9]. The same approach is used to find the generators of a cyclic code over the ring by Ashker and Hamoudeh [10] and by Abhay Kumar and Kewat [11].

The study of cyclic codes over nonchain rings can lead to better performance in terms of error-correcting capabilities and efficiency compared to codes over chain rings. The algebraic structure of cyclic codes over nonchain rings can be more complex than the structure of cyclic codes over chain rings, which can lead to improved code properties.

The structure of linear and cyclic codes of odd length over a finite nonchain ring has been determined by Yildiz and Karadeniz [12, 13]. A unique set of generators of a cyclic code over the ring has been obtained by Sobhani and Molakarimi [14]. The structure of a cyclic code over the ring has been obtained by Dougherty et al. [15]. The structure of a cyclic code of arbitrary length over the ring has been determined by Parmod Kumar Kewat et al. [16]. Linear and cyclic codes over the nonchain ring , were first introduced by Yildiz et al. [17, 18]. They have found some good linear codes over as the Gray images of cyclic codes over . The structure of a cyclic code of arbitrary length over , has been studied by Bandi and Bhaintwal [19]. Cyclic and some constacyclic codes of odd length over the nonchain ring , have been studied by Ozen et al. [20].

In most of the studies, the structural properties of a cyclic code over nonchain rings have been established when the square of the indeterminate coefficient, i.e., , is equal to zero. We make advancements to this study in the direction of the structure of a cyclic code of arbitrary length when takes nonzero values also.

The rings with have been classified into chain rings and nonchain rings by Adel Alahmadi et al. [21]. They have proved that is a chain ring for and is a nonchain ring for . This motivated us to investigate the algebraic structure of a cyclic code of arbitrary length over the nonchain rings , where is other than 0 which is not under consideration until now.

We noticed that among the eight nonchain rings , some exhibit isomorphism with each other. Specifically, the ring , and the ring , are confirmed to be isomorphic. Consequently, we can skip the examination of the structure of the ring with the condition . Additionally, it is worth noting that the ring , is isomorphic to the rings , where . Similarly, the ring , where , is isomorphic to the ring , where . Consequently, it is enough to focus on the structure of a cyclic code of arbitrary length over the rings , where .

In this paper, a unique form of generators of a cyclic code of arbitrary length over nonchain rings of the type , has been determined. Furthermore, the rank and cardinality of a cyclic code over these rings have been obtained.

2. Preliminaries

Let be a ring with unity. A subset of is called a code of length over . A linear code of length is a submodule of over the ring . An element of a linear code is termed a codeword. If a codeword of , is also a codeword of , then is called a cyclic code of length over . There is a one-to-one correspondence between the cyclic codes of length over and the ideals of the ring . The rank of a cyclic code, denoted by , is the number of elements in the minimal (linear) spanning set of code over . A finite commutative ring is a chain ring if all its ideals form a chain under the inclusion relation; otherwise, is a nonchain ring.

Throughout this article, we will denote the nonchain rings for by . Define

Lemma 1. The map defined as is a ring homomorphism for .

Proof. Case 1: When . For , . Let and be arbitrary elements of , where .
We have, = = .
Again, .
Case 2: When . For , . Let and , where with .
Now, .
Also, .
Thus, is a ring homomorphism for .

The following lemma by Abualrub and Siap [22] determines the structure of cyclic codes of arbitrary length over .

Lemma 2 (see [22]). Let be a cyclic code of arbitrary length over . Then, , where , and are binary polynomials such that and either or with deg deg .

3. Structure of a Cyclic Code of Arbitrary Length over

In this section, we establish the structure of a cyclic code of arbitrary length over the nonchain rings , .

Theorem 3. Let be a cyclic code of arbitrary length over the rings . Then,, where + , + , + , such that the polynomials are in for . Furthermore,

Proof. Let be a cyclic code of length over , . Let with a ring homomorphism as defined in Lemma 1 for . Clearly, is a cyclic code of length over . Using Lemma 2, we get = , where and either or with deg deg .
Let . It is easy to see that is times a cyclic code of length over . Therefore, using Lemma 2, we obtain = , where and either or with deg deg .
It follows that , where , , , such that the polynomials are in for and satisfy conditions (2)–(5).
Let be a cyclic code of length over , generated by the polynomials as obtained in Theorem 3. Define Residue and Torsion of asClearly, Res and Tor are the ideals of the ring . Furthermore, define = Res(Res ) = mod = Tor(Res ) = mod = Res(Tor ) = mod = Tor(Tor ) = It is easy to see that are ideals of the ring generated by the unique minimal degree polynomials , respectively, as defined in Theorem 3.

Theorem 4. Let be a cyclic code of arbitrary length over the ring , where , are polynomials as defined in Theorem 3. Then, there exists a set of generators of , where ,,, and such that the polynomials are in satisfy conditions (2)–(5) as defined in Theorem 3 and are unique minimal degree polynomial generators of . Also, either or deg deg for.

Proof. Clearly, , , , and are the generators of such that either or deg deg and either or deg deg . Furthermore, if either or deg deg for all , then we get the required result. Otherwise, let us suppose that deg deg for some and . Assume that deg deg for and , i.e., deg deg . By division algorithm, there exist some such that , where either or deg deg . Consider . Furthermore, deg deg . Again by division algorithm, there exist some such that , where either or deg deg . Now, consider the polynomial = + . Clearly, . Also, we have that either or deg deg and either or deg deg . Since is a linear combination of , and , we have = . Furthermore, using similar arguments, we can find polynomials and satisfying the required properties such that and complete the proof of the theorem.
In the following theorem, a unique form of the generators of a cyclic code of arbitrary length over has been determined.

Theorem 5. Let be a cyclic code of arbitrary length over the ring , where ,,, and such that the polynomials and satisfy conditions (2)–(5) as defined in Theorem 3 with either or deg degforand are the unique minimal degree polynomial generators of . Then, the polynomials are uniquely determined.

Proof. Consider another set of generators of , where + , + , + , and such that the polynomials are in and satisfy conditions (2)–(5) as defined in Theorem 3 with either or deg deg for and are the unique minimal degree polynomial generators of .
Clearly, , for . Consider + + . This implies that . Also deg deg which is a contradiction because is a minimal degree polynomial in . Hence, . It follows that + which implies that . As deg deg , we must have .
Subsequently, implying that . This together with the fact that deg deg implies that .
In a similar manner, we can prove that , , and . Hence, the uniqueness of the polynomials , and is established.

The following theorem which gives some divisibility properties of polynomials , in can be proved through simple calculations. These properties will be required to prove the results of Section 4.

Theorem 6. Let be a cyclic code of arbitrary length over the ring , where the generators + + , + , + , and are in the unique form as in Theorem 5. Then, the following divisibility relations hold over the ring .(i)(ii)(iii)(iv)(v)(vi)(vii),,for(viii)for, andfor.

4. Rank and Cardinality of a Cyclic Code of Arbitrary Length over

In this section, the rank and cardinality of a cyclic code of arbitrary length over have been obtained by determining a minimal spanning set of a cyclic code over .

Definition 7. A set of elements of a cyclic code over a finite commutative ring is called a spanning set of if each element of can be written as a linear combination of elements of with coefficients in .

Definition 8. A spanning set of a cyclic code is called a minimal spanning set of if no proper subset of spans .

Definition 9. The rank of a cyclic code is the number of elements in the minimal spanning set of .
Obviously, the minimal spanning set of a cyclic code is not unique. However, the number of elements in any minimal spanning set of remains the same. We prove this in the following theorem.

Theorem 10. Let and be two minimal spanning sets of a cyclic code over a finite commutative ring . Then, .

Proof. Without loss of generality, we may assume that is the least positive number such that no set with less than elements spans . Clearly, . Suppose, if possible, . It is easy to see that is not a zero divisor for some . Without loss of generality, we may assume that is not a zero divisor. Since spans and , we can find such that . Furthermore, it is easy to see that at least one is not a zero divisor. Without loss of generality, we may assume that is not a zero divisor. It follows thatand hence, the set also spans . Consequently, we have that , where is the cyclic code spanned by the set . It is easy to see that is a minimal spanning set for . Clearly, is also a minimal spanning set of . Repeating the same arguments on the sets and , we get that the set also spans and therefore spans . Thus, , where and are minimal spanning sets of . Furthermore, repeating the above process a number of times, we obtain that the set also spans . This is a contradiction to the fact that is a minimal spanning set of . Therefore, must be equal to .
In the following theorem, the rank of a cyclic code of arbitrary length over has been obtained.

Theorem 11. Let be a cyclic code of arbitrary length over the ring , where the generators + + , + , + , and are in the unique form as given in Theorem 5. Then, is, where = deg for and .

Proof. It can be easily seen that the set , …, , …, , …, is a spanning set of .
To prove that is , it is sufficient to show that the set is a minimal spanning set of , where .
To prove that the set spans , it is enough to show that . First, let us suppose that . As in , there exists some with deg such that = . Multiplying both sides by , we getwhich implies that . Next, suppose that . Using the divisibilities for and for from Theorem 6, it can be proved that by working on the same lines as above. Thus, we have , where . By taking as a divisor and applying the division algorithm for and , respectively, we can show that and . Thus, is a spanning set of .
Now to prove that the set is a minimal spanning set, it is enough to show that none of and can be written as a linear combination of other elements of . Suppose, if possible, that can be written as a linear combination of other elements of , i.e.,where deg , deg , deg , and deg . Multiplying equation (9) on both sides by for , we getMultiplying equation (9) on both sides by for , we getEquations (10) and (11) are not possible as degrees of left-hand side and right-hand side in each of these equations do not match. Thus, cannot be written as a linear combination of other elements of . Using a similar argument, it can be shown that none of and can be written as a linear combination of other elements of . Hence, is a minimal spanning set of . Furthermore, number of elements in + + , where .
Corollary 12 follows immediately from the above theorem.

Corollary 12. Let be a cyclic code of arbitrary length over the rings , where the generators + + , + , + , and . Then, cardinality of is defined as follows:where deg for and .

The following examples illustrate the above results.

Example 1. Let be a cyclic code of length 4 over the ring for . Here, . Using Theorem 11, minimal spanning set of is. Hence, and .

Example 2. Let be a cyclic code of length 4 over the ring for . Here, . Using Theorem 11, minimal spanning set of is. Hence, and .

Example 3. Let , , be a cyclic code of length 6 over the ring for . Here,. Using Theorem 11, minimal spanning set of is {, ), }. Hence, and .

Example 4. Let , , be a cyclic code of length 6 over the ring for. Here,. Using Theorem 11, minimal spanning set of is {, , , }. Hence, and .

5. Conclusion and Future Scope

In this paper, the structure of a cyclic code of arbitrary length over the rings for those values of for which these are nonchain rings has been established. A unique form of the generators of these codes has been obtained. Furthermore, formulae for rank and cardinality of a cyclic code over these rings have been established by finding their minimal spanning sets. This study can be used to find some new and good codes over . Also, the structural properties of cyclic codes of arbitrary length over the rings where for can be established.

Data Availability

Data sharing is not applicable to this article.

Disclosure

This paper has previously been archived in [23].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Nikita Jain and Ranjeet Sehmi are contributing authors.

Acknowledgments

The first author would like to thank the Council of Scientific and Industrial Research (CSIR), India, for providing fellowship (Grant no. 08/423(0003)/2020-EMR-I).

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Copyright © 2024 Nikita Jain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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