Cyclic Codes Research Articles

New upper bounds on the number of non-zero weights of constacyclic codes

For any simple-root constacyclic code C over a finite field Fq, as far as we know, the group G generated by the multiplier, the constacyclic shift and the scalar multiplications is the largest subgroup of the automorphism group Aut(C) of C. In this paper, by calculating the number of G-orbits of C﹨{0}, we give an explicit upper bound on the number of non-zero weights of C and present a necessary and sufficient condition for C to meet the upper bound. Some examples in this paper show that our upper bound is tight and better than the upper bounds in Zhang and Cao (2024) [26]. In particular, our main results provide a new method to construct few-weight constacyclic codes. Furthermore, for the constacyclic code C belonging to two special types, we obtain a smaller upper bound on the number of non-zero weights of C by substituting G with a larger subgroup of Aut(C). The results derived in this paper generalize the main results in Chen et al. (2024) [9].

Self dual and MHDR dual cyclic codes over finite chain rings

Self dual and MHDR dual cyclic codes over finite chain rings

Design and Use of a Mobile Game Developed to Raise Environmental Awareness in Secondary Schools

This research aims to develop, improve, and evaluate a mobile game to improve students' environmental awareness and investigate the game's impact on the development of environmental awareness. The research was conducted with 74 sixth-grade students and two teachers at a secondary school in Bursa during the 2022-2023 academic year. The environmental awareness scale, observation notes, and semi-structured interview forms were used as data collection tools. The results showed a significant difference between the scores in the pre-test and post-test in favour of the post-test. The developed mobile game significantly affects environmental awareness. The observation and interview data were analyzed using cyclic coding and thematic analysis methods. Three themes named “the effects on motivational processes,” “the effects of structuring of knowledge,” and “the effects of environmental awareness” emerged. Findings showed that the game mechanics included in the developed mobile game positively affected the “autonomy and self-efficacy” of the intrinsic motivation elements of the students to achieve their individual goals. Findings also revealed that the mobile game improves students' feelings of attention, interest, satisfaction, and confidence. In addition, it has been found that students structure their knowledge by resorting to reflection with the score table and instant feedback mechanisms in the mobile game.

Two classes of LCD BCH codes over finite fields

BCH codes form a special subclass of cyclic codes and have been extensively studied in the past decades. Determining the parameters of BCH codes, however, has been an important but difficult problem. Recently, in order to further investigate the dual codes of BCH codes, the concept of dually-BCH codes was proposed. In this paper, we study BCH codes of lengths qm+1q+1 and qm+1 over the finite field Fq, both of which are LCD codes. The dimensions of narrow-sense BCH codes of length qm+1q+1 with designed distance δ=ℓqm−12+1 are determined, where q>2 and 2≤ℓ≤q−1. Lower bounds on the minimum distances of the dual codes of narrow-sense BCH codes of length qm+1 are developed for odd q, which are good in some cases. Moreover, sufficient and necessary conditions for the even-like subcodes of narrow-sense BCH codes of length qm+1 being dually-BCH codes are presented, where q is odd and m≢0(mod4).

Quantum and LCD codes from skew constacyclic codes over a general class of non-chain rings

Quantum and LCD codes from skew constacyclic codes over a general class of non-chain rings

$$\mathbb {F}_{q}\mathbb {F}_{q}[u]$$-additive constacyclic codes are asymptotically good

$$\mathbb {F}_{q}\mathbb {F}_{q}[u]$$-additive constacyclic codes are asymptotically good

Hulls of cyclic codes with respect to the regular permutation inner product

Hulls of cyclic codes with respect to the regular permutation inner product

A class of BCH codes of length [formula omitted] and their duals

BCH codes are a special subclass of cyclic codes. In many cases, BCH codes are best cyclic codes and they have wide applications in communication and storage systems. In this paper, we investigate the parameters of a class of narrow-sense BCH codes over GF(q) of length 2(q2m−1)q+1 with small and large dimensions. We study the q-cyclotomic cosets modulo 2(q2m−1)q+1, determine the dimensions of these BCH codes and give the lower bounds on their minimum distances. Furthermore, we present the lower bounds on the minimum distances of their dual codes.

On a class of constacyclic codes of length 4ps over 𝔽pm[u] 〈u3〉

Let [Formula: see text] be a prime such that [Formula: see text], where [Formula: see text] is a positive integer. For any nonzero element [Formula: see text] of [Formula: see text], we determine the algebraic structure of all [Formula: see text]-constacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text], where [Formula: see text] and [Formula: see text] is a positive integer. If the unit [Formula: see text] is a square, [Formula: see text], each [Formula: see text]-constacyclic code of length [Formula: see text] is expressed as a direct sum of an [Formula: see text]-constacyclic code and an [Formula: see text]- constacyclic code of length [Formula: see text]. In the main case that the unit [Formula: see text] is not a square, it is shown that any nonzero polynomial of degree at most [Formula: see text] over [Formula: see text] is invertible in the ambient ring [Formula: see text]. It is also proven that the ambient ring [Formula: see text] is a local ring with the unique maximal ideal [Formula: see text], where [Formula: see text]. Such [Formula: see text]-constacyclic codes are then classified into eight distinct types of ideals, and the detailed structures of ideals in each type are provided. Among other results, the number of codewords, and the dual of each [Formula: see text]-constacyclic code are obtained. The non-existence of self-dual and isodual [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text], when the unit [Formula: see text] is not a square, is likewise proved.

Enumeration formulae for self-orthogonal, self-dual and complementary-dual additive cyclic codes over finite commutative chain rings

Enumeration formulae for self-orthogonal, self-dual and complementary-dual additive cyclic codes over finite commutative chain rings

On reversible DNA codes over the ring $${\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle$$ based on the deletion distance

AbstractLet $${\mathfrak {R}}= {\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle$$ R = Z 4 [ u , v ] / ⟨ u 2 - 2 , u v - 2 , v 2 , 2 u , 2 v ⟩ be a ring, where $${\mathbb {Z}}_{4}$$ Z 4 is a ring of integers modulo 4. This ring $${\mathfrak {R}}$$ R is a local non-chain ring of characteristic 4. The main objective of this article is to construct reversible cyclic codes of odd length n over the ring $${\mathfrak {R}}.$$ R . Employing these reversible cyclic codes, we obtain reversible cyclic DNA codes of length n, based on the deletion distance over the ring $${\mathfrak {R}}.$$ R . We also construct a bijection $$\Gamma$$ Γ between the elements of the ring $${\mathfrak {R}}$$ R and $$S_{D_{16}}.$$ S D 16 . As an application of $$\Gamma ,$$ Γ , the reversibility problem which occurs in DNA k-bases has been solved. Moreover, we introduce a Gray map $$\Psi _{\hom }:{\mathfrak {R}}^{n}\rightarrow {\mathbb {F}}_{2}^{8n}$$ Ψ hom : R n → F 2 8 n with respect to homogeneous weight $$w_{\hom }$$ w hom over the ring $${\mathfrak {R}}$$ R . Further, we discuss the GC-content of DNA cyclic codes and their deletion distance. Moreover, we provide some examples of reversible DNA cyclic codes.

Application of (σ,δ)-cyclic codes in DNA codes over a non-chain ring

For a prime [Formula: see text] and a positive integer [Formula: see text], let [Formula: see text] be the finite field of characteristic [Formula: see text], and [Formula: see text] be a non-chain ring for a positive integer [Formula: see text]. In this paper, we study the [Formula: see text]-cyclic codes over [Formula: see text]. Further, we study the application of these codes in finding DNA codes. Toward this, we first define a Gray map to find classical codes over [Formula: see text] using codes over the ring [Formula: see text]. Later, we find the conditions for a [Formula: see text]-cyclic code to be reversible. Finally, we provide DNA codes using this algebraic method with some classical codes of better parameters.

Self-Dual Negacyclic Codes With Variable Lengths and Square-Root-Like Lower Bounds on the Minimum Distances

Self-Dual Negacyclic Codes With Variable Lengths and Square-Root-Like Lower Bounds on the Minimum Distances

Two Classes of Optimal Ternary Cyclic Codes with Minimum Distance Four

Two Classes of Optimal Ternary Cyclic Codes with Minimum Distance Four

New Lower Bounds for the Minimum Distance of Cyclic Codes and Applications to Locally Repairable Codes

New Lower Bounds for the Minimum Distance of Cyclic Codes and Applications to Locally Repairable Codes

The Extended Codes of a Family of Reversible MDS Cyclic codes

The Extended Codes of a Family of Reversible MDS Cyclic codes

Several families of q-ary cyclic codes with length $$q^m-1$$

Several families of q-ary cyclic codes with length $$q^m-1$$

A novel irradiation module for ANICCA fuel cycle code based on multi-task learning

This paper introduces an alternative approach to irradiation modelling within the context of nuclear fuel cycle codes like ANICCA, the fuel cycle code developed by the Belgian Nuclear Research Centre (SCK CEN). The focus lies on upgrading the irradiation module by substituting the CRAM-based depletion calculations for a more flexible and innovative approach based on Multi-Task Learning (MTL). Utilizing data generated with SERPENT2 Monte Carlo simulations, two MTL neural networks are developed to surrogate irradiation processes of Uranium Oxide (UOX) and Mixed Oxide (MOX) fuels, respectively. MTL enables simultaneous learning of the evolution of the inventories for different observables – i.e., transuranium elements, fission products, minor actinides and fertile materials – offering improved predictive capabilities compared to non-MTL neural networks, as demonstrated in the cross-validation tests.Hyperparameters of the models were found using Bayesian optimization, resulting in superior model performance compared to the old CRAM-based model. Verification against SERPENT2, used as a reference code for comparison, demonstrates the stability and accuracy of the MTL-based models, outperforming the original CRAM method for the majority of predicted isotopes.

Cyclic Codes over a Non-Commutative Non-Unital Ring

In this paper, we investigate cyclic codes over the ring E of order 4 and characteristic 2 defined by generators and relations as E=⟨a,b∣2a=2b=0,a2=a,b2=b,ab=a,ba=b⟩. This is the first time that cyclic codes over the ring E are studied. Each cyclic code of length n over E is identified uniquely by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, left self-dual, right self-dual, and linear complementary dual (LCD) cyclic codes over E. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over E and quasi-cyclic codes of length 2n over F2 is studied. Motivated by DNA computing, conditions for reversibility and invariance under complementation are derived.

Quantum variations of cyclotomic cosets for cyclic stabiliser codes construction

Cyclotomic cosets are intrinsically linked with the design and construction of classical cyclic codes whose properties can be inferred from the coset structures. This paper proposes some new quantum variations of cyclotomic cosets for cyclic stabiliser construction. These variations are governed by several parameters which are devoted to designing the two essential parts: the error part and the position part of these cosets. Criteria on these cosets in generating additive cyclic stabiliser are extensively studied, followed by actual implementation on stabiliser codes construction of several selected classes of length.