MDS and MHDR Cyclic Codes over Finite Chain Rings

Dalal, Monika; Dutt, Sucheta; Sehmi, Ranjeet

doi:https://doi.org/10.1155/2024/4540992

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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 4540992 | https://doi.org/10.1155/2024/4540992

MDS and MHDR Cyclic Codes over Finite Chain Rings

Monika Dalal,¹Sucheta Dutt,¹and Ranjeet Sehmi¹

Academic Editor: Xiaogang Liu

Received18 Apr 2023

Revised17 Nov 2023

Accepted20 Nov 2023

Published25 Jan 2024

Abstract

This work establishes a unique set of generators for a cyclic code over a finite chain ring. Towards this, we first determine the minimal spanning set and rank of the code. Furthermore, sufficient as well as necessary conditions for a cyclic code to be an MDS code and for a cyclic code to be an MHDR code are obtained. Finally, to support our results, some examples of optimal cyclic codes are presented.

1. Introduction

Coding theory aims to provide optimal codes for detecting and correcting a maximum number of errors during data transmission through noisy channels. Cyclic codes have been in focus due to their rich algebraic structure which enables easy encoding and decoding of data through the process of channel coding. Cyclic codes over rings have gained a lot of importance after the remarkable breakthrough given by Hammons et al. in reference [1]. A vast literature is available on the structure of cyclic codes over fields, integer residue rings, Galois rings, finite chain rings, and some finite nonchain rings [2–29]. Cyclic codes over finite chain rings with length coprime to the characteristic of residue field have been investigated in references [2, 16, 22]. Islam and Prakash have established a unique set of generators for cyclic codes over in reference [4] and for cyclic codes over in reference [5]. A. Sharma and T. Sidana have studied cyclic codes of length over finite chain rings in reference [15], thereby extending the results of Kiah et al. on cyclic codes over Galois rings [14]. Dinh explored the structure and properties of cyclic codes of length over finite chain rings with nilpotency index 2 [13]. However, in most of the studies, there have been some limitations on either the length of code or the nilpotency index of the ring. We do not impose any such restriction in this paper. Salagean made use of the existence of a Grobner basis for an ideal of a polynomial ring to establish a unique set of generators for a cyclic code over a finite chain ring with arbitrary parameters [18]. Al-Ashker et al. have also worked in the same direction in the paper [28] by extending the novel approach given by Siap and Abualrub [12] which pulls back the generators of a cyclic code over to establish the structure of cyclic codes over the ring , . They have also extended this approach over the finite chain ring , [24]. Monika and Sehmi have given a constructive approach to establish a generating set for a cyclic code over a finite chain ring by making use of minimal degree polynomials of certain subsets of the code [20]. We make some advancements to this study by establishing a unique set of generators for a cyclic code over a finite chain ring with arbitrary parameters. It is noted that this unique set of generators retains all the properties of generators obtained in reference [20].

The paper is organised as follows: In Section 2, we state some preliminary results. In Section 3, we establish a unique set of generators for a cyclic code over a finite chain ring. In Section 4, we establish a minimal spanning set and rank of the cyclic code. We give sufficient as well as necessary conditions for a cyclic code to be an MDS code. We establish sufficient as well as necessary conditions for a cyclic code of length which is not coprime to the characteristic of residue field of the ring to be an MHDR code. Finally, we provide a few examples of MDS and MHDR cyclic codes over some finite chain rings.

2. Preliminaries

Let be a finite commutative chain ring. Let be the unique maximal ideal of and be the nilpotency index of . Let be the residue field of , where for a prime and a positive integer .

The following is a well-known result (for reference, see [15]).

Proposition 1. Let be a finite commutative chain ring. Then, we have the following:(i), where and (ii)There exists an element with multiplicative order . The set is called the Teichmller set of (iii)Every can be uniquely expressed as , where for . Also, is a unit in if and only if

Remark 2. Let , where for is a polynomial of degree in . By using Proposition 1(iii), can be expressed aswhere for .
We define a map by for . Clearly, is a natural onto homomorphism, and therefore, , where denotes the image of under . This map can be naturally extended from to by , where for .
Let us now recall some basic definitions and known results.
A linear code with length over a finite commutative chain ring is said to be a cyclic code if for every . It is well established that can be viewed as an ideal of . The Hamming weight of is defined as the number of integers such that for . The Hamming distance of a code over is given by . is said to be an MDS (maximum distance separable) code with respect to the Hamming metric if . The rank of is defined as the total number of elements in the minimal spanning set of . is said to be an MHDR (maximum Hamming distance with respect to rank) code if . The torsion code of is defined as , where . Then, for all , is a principally generated cyclic code over the residue field of . The degree of the generator polynomial of is called the torsional degree of . A polynomial in is said to be monic if its leading coefficient, i.e., the coefficient of its leading term is a unit in . The leading coefficient of a polynomial in is denoted by .

3. Unique Set of Generators

In this section, a unique set of generators for a cyclic code of arbitrary length over has been established. For this, let us first recall the construction given by Monika et al. to obtain a generating set for a cyclic code over a finite chain ring [20]. Let be minimal degree polynomials of certain subsets of such that and the leading coefficient of is equal to , where is some unit in , , , and is the smallest of such power. If , then is a monic polynomial and we have .

Lemma 3 (see [20]). Let be a cyclic code having a length over and , , be polynomials as defined above. Then, we have the following:(i) is generated by the set (ii)For , , where is a monic polynomial over the finite commutative chain ring having nilpotency index and maximal ideal (iii) forms a Grobner basis for

The following results are straightforward generalisations of reference [19] for cyclic codes over the class of Galois rings to finite chain rings and have been communicated in reference [21]. These results are required to proceed further.

Lemma 4 (see [21]). We consider a cyclic code of arbitrary length over generated by as defined above. Then, for every , . Also, and is the torsional degree of .

Remark 5 (see [21]). Let be a cyclic code having a length over , where for are polynomials as defined above. Then, we have(i)(ii)(iii)

Remark 6. For a cyclic code with a generating set as defined above, the abovementioned remark implies that(i)for , the torsional degree of is (ii)for and , the torsional degree of is

Theorem 7 (see [21]). Let be a cyclic code having an arbitrary length over generated by polynomials as defined earlier. If , then we havewhere for are the torsional degrees of , , and for .

Theorem 8 (see [21]). Let be a cyclic code as defined above. Then, .

Remark 9. Let such that , , and is monic. Let be the leading coefficient of and , respectively. Then, we havefor some such that . If , then by applying a similar argument as mentioned above on , we havefor some such that . Again, if , then repeatedly apply the abovementioned argument a finite number of times to obtain polynomials in with such thatwhere and . Now, by back substituting all these values of one by one, we finally get for , , and .
In the following theorem, a unique set of generators for a cyclic code over a finite chain ring has been obtained, which retains all the properties as that of the generating set obtained in reference [20].
For a positive integer , define .

Theorem 10. Let be a cyclic code having an arbitrary length over as defined above. Then, there exist polynomials in such that for , we havewhere for , , such that is the generator polynomial of torsion code of and . Furthermore, for , for and , and for . Also, is generated by the set which retains all the properties as that of the generating set and are unique in this form.

Proof. Let be a cyclic code over such that are polynomials as defined above. By construction, it is clear that is unique in . Therefore, . Now, we consider thatwhere for , such that is the generator polynomial of torsion code of , and and for . If for , then is of the desired form. Otherwise, suppose to be the least nonnegative integer such that . Then, . By Remark 9, we havefor some polynomials such that . Let , where for . We substitute this in equation (9) and then back substitute the value of in equation (8) to getThis implies thatClearly, the term with content on the right-hand side of the abovementioned equation now belongs to . Following the same arguments as mentioned above for every , where , we can obtain a polynomial say in by subtracting a suitable multiple of from which will satisfy all the desired properties, i.e., , , for and for such that .
Now consider the following polynomial:where for , , and and for . Furthermore, if for and for , then is of the desired form. Otherwise, let there exist least positive integers and such that and . By using Remark 9 for and , we have such that and the degrees of and are strictly less than that of and , respectively. Let and for every in . Then, for and for . Using this to obtain the value of and and then back substituting these values in the summand for , we get . Clearly, on the right-hand side of this equation, the term with content now has a degree that is strictly less than that of and the term with content has a degree that is strictly less than that of . Following the similar arguments as mentioned above for every and , we can finally obtain a polynomial in by subtracting a suitable multiple of and from and satisfies all the desired properties. Similarly, for every , we can obtain a polynomial in by subtracting suitable multiples of from , such that is of the desired form and . It is clear from the abovementioned arguments that these have the same structural properties as those of , for every .
Then, we show that the polynomials , obtained above are unique in this form. Let , where and , such that for , for the generator polynomial of the torsion code of , and . Furthermore, for , for and , and for . Clearly, by the abovementioned construction. For , we consider the following polynomial:Let us denote the polynomials by for . Then, we havesince , i.e., . We have that . From Remark 5 and Lemma 4, we have that Tor Tor for . Therefore, but which implies that . By substituting this in equation (15) and applying the same arguments a finite number of times, we get for . By substituting this in equation (15), we haveWe have . By using Lemma 4, we get that . Then, , since . By using this in equation (15), we getAgain, we have . By using Remark 5 and Lemma 4, we get that for . Then, , since . By substituting this in equation (15) and repeatedly applying the same argument a finite number of times, we get for . Working in a similar manner for every , we can finally conclude that . Hence, the generator polynomials for are unique in .

Remark 11. It is observed that the unique set of generators obtained in Theorem 7 forms a Grobner basis for over .

4. MDS and MHDR Cyclic Codes over a Finite Chain Ring

In this section, the minimal spanning set and rank of a cyclic code over a finite chain ring have been established. Sufficient as well as necessary conditions for a cyclic code to be an MDS code and for a cyclic code to be an MHDR code have been obtained. Finally, to support our results, some examples of optimal cyclic codes have been presented.

Theorem 12. Let be a cyclic code having an arbitrary length over a finite chain ring . Then, , where is the degree of minimal degree polynomial in .

Proof. Let be a cyclic code having an arbitrary length n over . Let be a unique set of generators for as obtained above. Clearly, the set spans . Now, we shall prove that also spans . For this, we need to prove that for are in . We shall show this by induction on . First, we prove that . Clearly, is a polynomial of degree in . Then, we have for some with a degree less than which implies that . Therefore, we have . We suppose that for . Now, we will show that . Clearly, is a polynomial of degree in . Then, we have ; and , where and for all , . This implies that for , which further implies that . Therefore, we have for all , .
Then, we prove the linear independence of . Let if possible, there exist such thatThis implies that , where and for . Clearly, and for all , . Then, by multiplying equation (18) by , we getThen, the degree of of equation (19) is but that of is atmost which is a contradiction. Therefore, cannot be expressed as a linear combination of elements of . We can apply similar arguments to prove that none of can be expressed as a linear combination of elements of . Therefore, we get that is linearly independent, and hence, it is a set for . It follows that .
The following theorem determines all the MDS cyclic codes of arbitrary length over a finite chain ring .

Theorem 13. A cyclic code having a length over is an MDS if and only if it is principally generated by a monic polynomial and Tor is an MDS cyclic code having a length over with respect to Hamming metric.

Proof. Let be an MDS cyclic code having a length over such that , are polynomials as in Theorem 10. Since is an MDS, therefore, . By using Theorem 7, we have which implies that . Thus, we can conclude that for and because and . This implies that is principally generated by a monic polynomial and . By using Theorems 7 and 8, we have . Thus, is an MDS cyclic code over the residue field
Conversely, suppose a cyclic code having a length over is principally generated by a monic polynomial, say as obtained in Theorem 10 and is an MDS code over . Then, this means that and . By using Theorems 7 and 8, we can conclude that , i.e., is an MDS cyclic code over .
The following lemma by Sharma and Sidana determines the of a cyclic code of length , , and over a finite chain ring as given in reference [27].

Lemma 14 (see [27]). Let be a cyclic code having a length for and over . Then, we have

We use Lemma 14 mentioned above to determine all MHDR cyclic codes of length , and over in Theorems 15 and 16.

Theorem 15. A cyclic code of length , over a finite chain ring is an MHDR code.

Proof. Let be a cyclic code of length , over . By Lemma 14, we havewhich implies that for by using Theorem 12. Hence, a cyclic code of length , over is always an MHDR code.

Theorem 16. Let be a cyclic code having a length over . Then, is an MHDR if and only if .

Proof. By Lemma 14, we have the following:(i)for , the Hamming distance of is 1, which is the same as by using Theorem 12. So, is an MHDR code.(ii)for with , the Hamming distance of is . Here, is an MHDR if and only if , i.e., by using Theorem 12. Then, would imply , i.e., . It follows that , which implies , since . Then, is an MHDR if and only if .(iii)for , , the Hamming distance of is . is an MHDR code if and only if by using Theorem 12. Then, we have . It follows that , which implies that , since . Then, is an MHDR for . It can be easily seen that for other values of , is not an MHDR code.

Theorem 17. Let be an MDS cyclic code having an arbitrary length over . Then, is also an MHDR code over .

Proof. Let be an MDS cyclic code having an arbitrary length over . By Theorem 13, is principally generated by a monic polynomial over say with degrees and and Tor is also an MDS code over . Then, we haveAlso, from Theorem 7, we haveEquations (22) and (23) together with Theorem 12 imply that . Therefore, is an MHDR cyclic code over .

However, Example 1 shows that the converse of the abovementioned statement is not true.

Example 1. Let . Let be a cyclic code having a length over . Here, , , , and . By using Theorem 16 we see that is an MHDR cyclic code over . However, is not an MDS code, since it is not principally generated (using Theorem 17).

Example 2. Let . Let be a cyclic code having a length over . Here, , , , and . By using Theorem 16 we see that is an MHDR cyclic code over . Also, is an MDS code, since it is principally generated by a monic polynomial and (using Theorem 13).

Example 3. Let . Let be a cyclic code having a length over . Here, , , , and . It is principally generated by a monic polynomial and , so we see that is an MDS code over by using Theorem 13. Also, from Theorem 15, we see that is also an MHDR code.

Example 4. Let . Let be a cyclic code having a length over . Here, , , , and . It is not generated by a monic polynomial, so by Theorem 13, is not an MDS code. Also, from Theorem 15, we see that is not an MHDR code.

Example 5. Let . Let be a cyclic code having a length over . Here, , , , , and . Since is not generated by a monic polynomial, so by Theorem 13, it is not an MDS code. Also, from Theorem 16, we see that is not an MHDR code.

5. Conclusion

In this work, a unique set of generators for a cyclic code having an arbitrary length over a finite chain ring with an arbitrary nilpotency index has been established. The minimal spanning set and rank of the code have also been determined. Furthermore, sufficient as well as necessary conditions for a cyclic code having an arbitrary length to be an MDS code and for a cyclic code having a length which is not coprime to the characteristic of the residue field of the ring to be an MHDR code have been obtained. Some examples of optimal cyclic codes have also been presented.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Disclosure

A preprint has previously been published [30].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Monika Dalal and Ranjeet Sehmi equally contributed to this work.

Acknowledgments

This research was supported by the Council of Scientific and Industrial Research (CSIR), India, in the form of research fellowship to the first author.

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Copyright

Copyright © 2024 Monika Dalal 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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