1. Introduction
In the early 1970s, linear codes over finite rings gained much attention in the field of algebraic coding theory after the introduction of Gray maps. In 1994, Hammons et al. [
1] calculated non-linear binary codes over
under the Gray map. The research of linear codes over finite rings has since gained more interest than the binary field, and numerous families of codes have been examined in [
2,
3,
4,
5], such as over
,
;
,
;
where
p is a prime. Observe that cyclic codes are linear block codes in which each codeword’s cyclic shift is again a codeword. Because of their rich algebraic structure, these error-correcting codes are a significant family of linear codes. Numerous findings and new codes were discovered when cyclic codes were investigated across different finite rings in [
4,
6,
7].
A reversible code is a linear code in which its components form a symmetry, i.e., it remains unchanged when the digits of codewords are reversed. By virtue of this symmetry, it is one of the many well-known kinds of cyclic codes. These codes are employed in data storage, retrieval systems, and DNA computing. It is important to note that reversible cyclic codes of length n over a finite field
are closely related to Linear complementary dual codes shortened as LCD codes and can be used for application in cryptography as well when
(see [
8,
9]). In 1964, James L. Massey [
6] first defined the characteristic properties of reversible cyclic codes. After that, the results of the formation of reversible cyclic codes over
were presented by Siap and Abualrub in [
3] in 2007. In 2015, Srinivasulu and Bhaintwal [
10] investigated reversible cyclic codes over
and their implications for DNA codes. In 2021, Prakash et al. [
11] examined the reversible cyclic codes and their applications over the ring
with
Motivated by this work, we examine reversible cyclic codes of any length
n over
.
The paper is structured as follows: we give some preliminaries in
Section 2, while in
Section 3, we provide the structure of cyclic codes of arbitrary length
n over the ring
.
Section 4 has some crucial results on reversible cyclic codes over
.
Section 5 discusses the dual of reversible cyclic code. The distance of the code is discussed in
Section 6. Finally,
Section 7 gives some examples to support our results.
2. Preliminary Results
A code of length n over is a non-empty subset of , and its element is called a codeword. A code is linear over if it is an -submodule of . A linear code over is called a cyclic code if for any , Additionally, a cyclic code over can be viewed as an ideal in identifying by .
The number of the nonzero component in is referred to as the Hamming weight. The Hamming distance between any two codewords is defined such that is the number of components in which these codewords differ. We define the Hamming distance of the code as .
Every polynomial with , the reciprocal of is defined to be the poynomial . Notice that , and if , then . We say that is self reciprocal if and only if . Additionally, for any two polynomials , satisfying ∣ implies ∣.
3. Construction of Cyclic Codes over
Throughout this paper, we assume that where . The structure of the generators of the cyclic codes over depends on the ring where .
Now, let
be a cyclic code in
. Define
by
is a ring homomorphism that can be extended to a homomorphism
defined by
where
Let
. Notice that
J is a cyclic code in
since it is an ideal in
. So,
and
with
, i.e.,
divides
.
Further, let
be a cyclic code in
. Define
by
,
is a ring homomorphism. Extend
to a homomorphism
defined by
where
with
, i.e.,
divides
. The image of
is also an ideal and hence a binary cyclic code generated by
with
. So,
for some binary polynomial
.
Note that because implies = . Since ⇒ .
In the rest of the article, we use , , , and as mentioned above.
Abualrub and Siap [
3], in their paper, classify all of the cyclic codes in
as follows.
Theorem 1 ([
3], Theorem 2)
. Let be a cyclic code in , where .- (1)
If n is odd, then is a principal ideal ring and where are binary polynomials with , i.e., , and .
- (2)
If n is not odd, then
- (a)
where , in T and in and . Or,
- (b)
where , in T, , divides and and . Or,
- (c)
with , and and . Moreover, , and .
4. Reversible Cyclic Code over
In this part, we focus on reversible codes for arbitrary lengths and identify the necessary and sufficient criteria for the reversibility of cyclic codes
over
. Some cases for even length have been discussed in [
12]. For any codeword
, the reverse of the codeword (symmetry of the codeword) is represented by
, where
Definition 1. A linear code of length n over a ring is said to be reversible if the symmetry of each codeword is in , i.e., , for all .
In [
6], Massey-characterized cyclic codes tend to be reversible over the finite field, as follows:
Theorem 2 ([
6] Theorem 1)
. The cyclic code over generated by the monic polynomial is reversible if and only if is self-reciprocal. We mention some of the results of Mostafanasab, and of Yousefian Darani [
12], which are necessary to prove our results.
Lemma 1 ([
12] Propsition 2.5)
. Let be a cyclic code of length n over a commutative ring and . Then, if and only if . Lemma 2 ([
12] Corollary 2.6)
. Let be a linear code. Then, is reversible if and only if for all . Lemma 3 ([
12] Lemma 2.7)
. Let for . If then for . Lemma 4 ([
12] Lemma 2.11)
. Let for . Suppose that , and where . Then, Lemma 5. A reversible cyclic code of length n over the ring under the homomorphism (as defined in the Section 3) is also reversible. Proof. Let
, where
. Hence,
. Now as
is a reversible cyclic code, we have
. Now, consider
so
is reversible. □
Lemma 6. For any reversible cyclic code over , the generated polymials , and are also reversible cyclic codes over .
Proof. We know that from
Section 3,
and using Lemma 5,
is reversible code over
. Hence,
is reversible cyclic code over
.
As and , it is sufficient to show that is reversible.
Let . Then, is a polynomial in , which gives . Moreover, i.e., . Hence, is a reversible cyclic code.
In a similar way, is a polynomial in with coefficient in } and ; it is sufficient to show that is reversible.
Let be arbitrary. Then, is a polynomial in . Since is reversible cyclic code in , is also in . Thus, , i.e., . As a consequence, we obtain the desired outcome. □
Theorem 3. Let be a linear cyclic code of odd length n over , where , and are binary polynomials with . Then, is reversible if and only if , , and are self reciprocal.
Proof. Assume that is a reversible cyclic code over . By using Lemma 6, we have that , , and are self reciprocal.
For the remaining part, we assume that , , and are binary self-reciprocal polynomials. Let , i.e., for some polynomials , and over . is reversible if and only if
For this consider
where
, and
are polynomials over
. Therefore,
Thus,
is a reversible cyclic code over
. □
Theorem 4. For where is a cyclic code of length n over such that n is even with with , , and Then, is reversible if and only if
- (1)
and are self reciprocal.
- (2)
- (a)
and . or
- (b)
and
.
Proof. We know that
is an
-module. First, we consider
reversible. Then, using Lemma 4 gives
For
Let
and
where
. Then,
The application of Lemma 3 implies that
,
and
. Now since
and
, where
denotes degree of
, which implies
; hence,
is self-reciprocal. Now,
; equating the degrees on both sides gives that
is either 0 or 1. In a similar manner,
. Similarly, equating the degrees on both sides gives that
is either 0 or 1; hence, the following four cases arise:
Case 1. If , then and .
Case 2. If and , then and .
Case 3. If and , then and .
Case 4. If and , then and .
By assumption, we have
. so
and
. Similarly,
and
.
Moreover,
. Then, the polynomials
, and
exist such that
Therefore,
and
. Since
, we have
and so
, i.e.,
is self reciprocal.
For the converse part, we are using only
and condition
of
holds.
where
,
. Hence,
is reversible. □
5. Dual of Reversible Cyclic Code over
Suppose that is a parity check polynomial over a cyclic -code in and . Then, for the characterization of of cyclic code , we have the following result.
Theorem 5. Let be a cyclic code over , and be its dual. Then, is reversible iff .
Proof. Let . Then, . Conversely, suppose that , then , which means . Hence, is reversible cyclic code. □
Definition 2. For any ideal J in , the annihilator of J in is defined asAnother associated ideal of of cyclic code is Proposition 1. Let be a cyclic code of odd length n over . Then,where is the annihilator of , where . Proof. It is given that
is a cyclic code of odd length over
; then,
where
. Additionally,
,
exists and
such that
,
, and
. Notice that
and
Similarly,
So,
Now, we prove that
. Suppose that
. Then,
From here, we conclude that a polynomial
exists such that
Similarly, one can get
Since
,
and so, there exists polynomial
in
such that
Additionally,
which implies
for some
.
Hence,
Therefore,
□
The implications of Proposition 1 are as follows.
Theorem 6. Let be a cyclic code of odd length n over . Then, Theorem 7. Let be a reversible cyclic code of odd length n over with and Then, is a reversible cyclic code over .
Proof. Assume that is a reversible cyclic code of odd length n over . Hence, by Theorem 3, , , and are self reciprocal. Suppose that , , and
Therefore,
and
This implies
and
Let
. Then,
where
,
, and
, over
. Thus
. Thus, by Theorem 5,
is a reversible cyclic code over
. □
6. Minimum Hamming Distance of a Cyclic Code over
Here, we determine the minimum Hamming distance of a cyclic code of length n over . Assume that is a cyclic code of length n over . Define . Then, is a cyclic code of length n over .
Theorem 8 ([
12] Theorem 4.4)
. If is a cyclic code of length n over . Then, . Theorem 9 ([
12] Theorem 4.4)
. Let be a cyclic code of length n over . Then, , where is the minimum Hamming weight of cyclic code . 7. Examples
Example 1. Let , over with , , and . Here, , then by using Theorem 3, the code generated by is a reversible cyclic code. Additionally, by using Theorem 9 we find that .
Example 2. Let over . Let , where , and . It is easy to check that is self-reciprocal. Additionally, and , & , where denotes degree of . Therefore, is a reversible cyclic code with distance .
8. Conclusions and Future Work
The reversible cyclic codes of any length
n over the ring
, where
, were examined in this article. As ideals for the ring
, we have created a unique set of generators for these codes. Additionally, in
Section 5, we set certain restrictions on the situations in which a reversible cyclic code’s dual is reversible. Finally, we provided helpful examples to support our results. As an application of our research work, we can use these reversible cyclic codes to obtain linear complementary dual codes, which are used in cryptography, in particular, to secure information from so-called, “side-channel assaults (SCA)” or “fault non-invasive attacks”. Further, for future work, we can also find reversible cyclic codes over more general rings such as
Author Contributions
Conceptualization, A.S.A., M.A. and N.u.R.; methodology, M.A. and N.u.R.; software, M.A.; validation, N.u.R.; formal analysis, A.S.A. and M.A.; investigation, M.A. and N.u.R.; resources, A.S.A.; data curation, M.A.; writing—original draft preparation, M.A.; writing—review and editing, A.S.A. and N.u.R.; visualization, M.A. and N.u.R.; supervision, N.u.R.; project administration, A.S.A. and N.u.R.; funding acquisition, A.S.A. All authors have read and agreed to the published version of the manuscript.
Funding
This research was funded by Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
All data required for this paper are included within this paper.
Acknowledgments
The authors are greatly indebted to the referee for his/her valuable suggestions, which immensely improved the article. The authors extend their appreciation to Princess Nourah bint Abdulrahman University for funding this research under Researchers Supporting Project number (PNURSP2022R231), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Hammons, A.; Kumar, P.; Calderbank, A.R.; Sloane, N.J.A.; Solé, P. The -linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 1994, 40, 301–319. [Google Scholar] [CrossRef]
- Abualrub, T.; Oehmke, R. On the generators of cyclic codes of length 2e. IEEE Trans. Inf. Theory 2003, 49, 2126–2133. [Google Scholar] [CrossRef]
- Abualrub, T.; Siap, I. Cyclic codes over the ring and . Des. Codes Cryptogr. 2007, 42, 273–287. [Google Scholar] [CrossRef]
- Abualrub, T.; Siap, I. On the construction of cyclic codes over the ring . In Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, 27–29 May 2006; pp. 430–435. [Google Scholar]
- Ping, L.; Shixin, Z. Cyclic codes of arbitrary lengths over the ring . J. Univ. Sci. Technol. China 2008, 38, 1392–1396. [Google Scholar]
- Massey, J.L. Reversible codes. Inf. Control 1964, 7, 369–380. [Google Scholar] [CrossRef] [Green Version]
- Tzeng, K.; Hartmann, C. On the minimum distance of certain reversible cyclic codes. IEEE Trans. Inf. Theory 1970, 16, 644–646. [Google Scholar] [CrossRef]
- Islam, H.; Prakash, O. New quantum and LCD codes over the finite field of even characteristic. Int. J. Theor. Phys. 2020, 60, 2322–2332. [Google Scholar] [CrossRef]
- Islam, H.; Prakash, O. Construction of LCD and new quantum codes from cyclic codes over a finite non-chain ring. Cryptogr. Commun. 2021, 14, 59–73. [Google Scholar] [CrossRef]
- Srinivasulu, B.; Bhaintwal, M. Reversible cyclic codes over and their applications to DNA codes. In Proceedings of the 7th International Conference on Information Technology and Electrical Engineering, Malang, Indonesia, 2 October 2015; pp. 101–105. [Google Scholar]
- Prakash, O.; Patel, S.; Yadav, S. Reversible cyclic codes over . Comput. Appl. Math. 2021, 40, 1–17. [Google Scholar]
- Mostafanasab, H.; Yousefian Darani, A. On Cyclic DNA Codes Over . Commun. Math. Stat. 2019, 9, 39–52. [Google Scholar] [CrossRef]
| Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).