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Cyclic codes over
... Bahattin Yildiz and Suat Karadeniz studied the structure of the ring F2 + uF2 + vF2 + uvF2, where u 2 = v 2 = 0 and uv = vu, and they obtained the structure of linear codes over this ring of any length n as in [7]. In [8] they proved the existence of self dual codes over the ring F2 + uF2 + vF2 + uvF2 of all lengths and obtained some results about their gray images, also they obtained the structure of cyclic codes over the ring F2 + uF2 + vF2 + uvF2 of any length n in [9], and in the light of the study in [9] they obtained the structure of (1 + v)-constacycliccodesovertheringF2 + uF2 + vF2 + uvF2of odd lengths n as in [10]. ...
... Bahattin Yildiz and Suat Karadeniz studied the structure of the ring F2 + uF2 + vF2 + uvF2, where u 2 = v 2 = 0 and uv = vu, and they obtained the structure of linear codes over this ring of any length n as in [7]. In [8] they proved the existence of self dual codes over the ring F2 + uF2 + vF2 + uvF2 of all lengths and obtained some results about their gray images, also they obtained the structure of cyclic codes over the ring F2 + uF2 + vF2 + uvF2 of any length n in [9], and in the light of the study in [9] they obtained the structure of (1 + v)-constacycliccodesovertheringF2 + uF2 + vF2 + uvF2of odd lengths n as in [10]. ...
... In [6], Xu Xiaofang and Liu Xiusheng they obtained the structure of the ring Fq + uFq + vFq + uvFq, where q is a power of the prime p and u 2 = v 2 = 0, uv = vu. Also they obtained the structure of cyclic codes over the ring Fq + uFq + vFq + uvFq of all lengths n as a generalization of the work done in [9] on the ring F2 + uF2 + vF2 + uvF2. ...
In this paper, we study the structure of linear and self dual codes of an arbitrary length n overhearing Fq + uFq + vFq + uvFq, where q is a power of the prime p and u2 = v2 = 0, uv = vu, Also we obtain the structure of consta-cyclic codes of length n = q − 1 over the ring Fq + uFq + vFq + uvFq in the light of studying cyclic codes over Fq + uFq + vFq + uvFq in [6]. This study is a generalization and extension of the works in [7], [8], and [10]. GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 57-71
... A polynomial f (x) ∈ R[x] is said to be regular if it is not a zerodivisor. The structure of cyclic codes of length n over R is given in Yildiz and Karadeniz (2011). In this section, we further simplify the structure of cyclic codes of odd length over R and give the generators for their duals. ...
... In this section, we further simplify the structure of cyclic codes of odd length over R and give the generators for their duals. Hence we recall the basic notations and definitions that are given in Yildiz and Karadeniz (2011). The following theorem presents the structure of a cyclic code of length n over R. ...
... The Lee weight of any c ∈ R n is defined as w L (c) = w H (φ(c)). The Gray map φ is a linear isometry from (R n , d L ) to (F 4n 2 , d H ). Therefore, if C is a linear code of length n over R with 2 k codewords and minimum Lee distance d, then φ(C) is a binary [4n, k, d]-linear code (Yildiz and Karadeniz, 2011). ...
... A polynomial f (x) ∈ R[x] is said to be regular if it is not a zerodivisor. The structure of cyclic codes of length n over R is given in Yildiz and Karadeniz (2011). In this section, we further simplify the structure of cyclic codes of odd length over R and give the generators for their duals. ...
... In this section, we further simplify the structure of cyclic codes of odd length over R and give the generators for their duals. Hence we recall the basic notations and definitions that are given in Yildiz and Karadeniz (2011). The following theorem presents the structure of a cyclic code of length n over R. ...
... The Lee weight of any c ∈ R n is defined as w L (c) = w H (φ(c)). The Gray map φ is a linear isometry from (R n , d L ) to (F 4n 2 , d H ). Therefore, if C is a linear code of length n over R with 2 k codewords and minimum Lee distance d, then φ(C) is a binary [4n, k, d]-linear code (Yildiz and Karadeniz, 2011). ...
In this paper, we study the structure of duals of cyclic codes over the ring R = F2 + uF2 + vF2 + uvF2, u2 = v2 = 0, uv = vu. We determine a unique set of generators for these codes. We also determine a minimal spanning set for a class of cyclic codes of odd length over R. A sufficient condition for a cyclic code of odd length over R to contain its dual is presented. We give a necessary and sufficient condition for a cyclic code of odd lengths over R to be reversible complement. Further, we construct DNA codes as images of reversible complement cyclic codes of odd length over R.
... Yildiz and Karadeniz in [20] have considered the ring F 2 [u, v]/ u 2 , v 2 , uv − vu , which is not a chain ring, and studied cyclic codes of odd length over that. They have found some good binary codes as the Gray images of these cyclic codes. ...
... We also find the Hamming distance of these codes for length p l . Note that the rank and the Hamming distance of these cyclic codes for p = 2 have not been discussed in [12,18,20]. Hence getting the rank and the Hamming distance of the cyclic codes over R u 2 ,v 2 ,p is new even for p = 2. ...
... Let w L and w H denote the Lee weight and Hamming weight respectively. Following [20], we define the Lee weight as follows ...
In this paper, we study cyclic codes over the ring . We find a unique set of generators for these codes.
We also study the rank and the Hamming distance of these codes.
... These rings are more difficult to deal with but they enjoy Gray maps that are interesting. Some examples on this direction are codes over the ring Z 2 [u, v]/ u, v [12,13]. Some further study over more generalized rings of this type that include the ring Z 2 [u, v]/ u, v are being carried out pretty recently [6,12,13]. ...
... Some examples on this direction are codes over the ring Z 2 [u, v]/ u, v [12,13]. Some further study over more generalized rings of this type that include the ring Z 2 [u, v]/ u, v are being carried out pretty recently [6,12,13]. ...
... So, there is no straightforward way of expressing the generating matrix. For instance in [10] and in [13] some special definitions or cases are adapted in order to express the linear code by generating sets. Here, we make use of the map φ as above and by considering the image of the code we define its generator matrix and obtain the related results. ...
In this paper, we study the structure of linear codes over the non chain ring
. In order to study the codes, we first study the structure of this ring via a distance preserving Gray map which also induces a relation between codes over this ring and ternary codes. Further, the algebraic structure of cyclic and dual codes is also studied. A MacWilliams type Identity between the Gray weight enumerators of the original code and its dual is established. In all cases examples that illustrate the theorems and lemmas are provided. Also, a BCH-type bound and an example that attains this bound is presented.
... Their structure over finite chain rings is now well known [9]. They have also been studied over other rings such as F 2 + uF 2 , u 2 = 0, [3]; F 2 + uF 2 + vF 2 + uvF 2 , u 2 = v 2 = 0, uv = vu, [12]; and F 2 + vF 2 , v 2 = v, [13]. ...
... x n −1 is not in general a principal ideal ring, as the next result shows. The result is a generalization of [12,Lemma 2.4]. ...
In this paper, we have studied cyclic codes over the ring
, . We have considered cyclic codes of odd
lengths. A sufficient condition for a cyclic code over R to be a
-free module is presented. We have provided the general form of
the generators of a cyclic code over R and determined a formula for the ranks
of such codes. In this paper we have mainly focused on principally generated
cyclic codes of odd length over R. We have determined a necessary condition
and a sufficient condition for cyclic codes of odd lengths over R to be
R-free.
... Future research tasks include exploring whether codes such as Hamming Code [17,20,21,69], Cyclic Code [31,101,133,134], Reed-Muller Code [51,70,87,127], Turbo Code [13,102], BCH Code [2,22,65], and Ternary Code [1] can be extended into hypercodes and superhypercodes. Additionally, investigating their mathematical properties and potential applications will be a significant focus. ...
This paper explores extensions of Binary Code, Gray Code, and Floorplan using the frameworks of hyperstruc-tures and superhyperstructures. Binary codes are subsets of fixed-length binary strings used for data encoding, while Gray codes are sequences where consecutive strings differ by one bit. Floorplans are geometric arrangements of modules within defined boundaries, adhering to constraints like area and aspect ratios. Hyperstructures extend power set concepts into advanced mathematical models, and superhyperstructures further generalize these models through-th power sets, enabling iterative and hierarchical abstractions.
... Future research tasks include exploring whether codes such as Hamming Code [17,20,21,69], Cyclic Code [31,101,133,134], Reed-Muller Code [51,70,87,127], Turbo Code [13,102], BCH Code [2,22,65], and Ternary Code [1] can be extended into hypercodes and superhypercodes. Additionally, investigating their mathematical properties and potential applications will be a significant focus. ...
This paper explores extensions of Binary Code, Gray Code, and Floorplan using the frameworks of hyperstruc-tures and superhyperstructures. Binary codes are subsets of fixed-length binary strings used for data encoding, while Gray codes are sequences where consecutive strings differ by one bit. Floorplans are geometric arrangements of modules within defined boundaries, adhering to constraints like area and aspect ratios. Hyperstructures extend power set concepts into advanced mathematical models, and superhyperstructures further generalize these models through-th power sets, enabling iterative and hierarchical abstractions.
... In [18], Yildiz and Karadeniz examined cyclic codes of odd length over the ring ...
Let p be an odd prime. In this paper, we have determined the Hamming distances for constacyclic codes of length over the finite commutative non-chain ring . Also their symbol-pair distances are completely obtained.
... Yildiz and Karadeniz [34] studied cyclic codes over the non-chain ring R = 2 + u 2 + v 2 + uv 2 , where u 2 = 0, v 2 = 0 and uv = vu with respect to the homogeneous weight. Later on, Yildiz and Kelebek [35] studied homogeneous weight for an infinite family of rings ...
Let R = Z 4 [ u , v ] / ⟨ u 2 - 2 , u v - 2 , v 2 , 2 u , 2 v ⟩ be a ring, where Z 4 is a ring of integers modulo 4. This ring 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 R . Employing these reversible cyclic codes, we obtain reversible cyclic DNA codes of length n , based on the deletion distance over the ring R . We also construct a bijection Γ between the elements of the ring R and S D 16 . As an application of Γ , the reversibility problem which occurs in DNA k -bases has been solved. Moreover, we introduce a Gray map Ψ hom : R n → F 2 8 n with respect to homogeneous weight w hom over the ring 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.
... These rings have been used as alphabets of certain cyclic codes by several authors and some good binary and ternary codes have been obtained as Gray images of these codes (see, for examples, [16,24,31,33]). We divide the units of R u 2 ,v 2 ,2 m in the following five categories: λ 1 = α + γ v + δuv, λ 2 = α + βu + δuv, λ 3 = α + βu + γ v + δuv, λ 4 = α + δ 1 uv,λ 5 = α, where α, β, γ , δ 1 ∈ F * 2 m and δ ∈ F 2 m . We obtain all self-dual λ-constacyclic codes of length 2 s over the ring R u 2 ,v 2 ,2 m for the units λ 1 , λ 2 and λ 3 . ...
In this paper, we classify all self-dual -constacyclic codes of length over the finite commutative local ring corresponding to units of the forms , , , where and . Moreover, the Hamming distance of these -constacyclic codes are completely determined.
... This family of rings is a generalization of such rings such as A k which were studied in [1,2] and rings that were studied in [3][4][5]. More directly it is a generalization of the family of rings R k where were studied extensively in such paper as [10][11][12][13]. ...
We construct an infinite family of commutative rings Rq,Δ and we study codes over these rings as well as the structure of the rings. We define a canonical Gray map from Rq,Δ to vectors over the residue finite field of q elements and use it to relate codes over Rq,Δ to codes over the finite field Fq. Finally, we determine the parameters for when self-dual codes exist and give various constructions for self-dual codes over Rq,Δ.
... The structure of cyclic codes over various rings is investigated by many authors (e.g., [2,3,5,10]). In [13], Yıldız studied cyclic codes of odd length over F 2 + uF 2 + vF 2 + uvF 2 . In [12], the algebraic structure of cyclic codes over the ring Z 4 + uZ 4 , where u 2 = 0 , is determined. ...
Abstract: In this paper, we study cyclic codes over the ring R = Z4+uZ4+u2Z4 , where u3 = 0. We investigate Galois
extensions of this ring and the ideal structure of these extensions. The results are then used to obtain facts about cyclic codes over R. We also determine the general form of the generator of a cyclic code and �nd its minimal spanning sets.
Finally, we obtain many new linear codes over Z4 by considering Gray images of cyclic codes over R.
... In the classical study of the structure of linear cyclic codes over rings, most of the rings are assumed to be commutative. For example, linear cyclic codes over different types of finite commutative chain rings have been studied in [1, 3, 4, 6-8, 10, 12, 14, 15, 18, 20, 22, 23], and those over some finite commutative non-chain rings have been studied in [13,19,21,[24][25][26]. ...
In this study, we consider the finite (not necessary commutative) chain ring , where θ is an automorphism of , and completely explore the structure of left and right cyclic codes of any length N over , that is, left and right ideals of the ring . For a left (right) cyclic code, we determine the structure of its right (left) dual. Using the fact that self-dual codes are bimodules, we discuss on self-dual cyclic codes over . Finally, we study Gray images of cyclic codes over and as some examples, three linear codes over with the parameters of the best known ones, but with different weight distributions, are obtained as the Gray images of cyclic codes over .
... Their structure over finite chain rings is now wellknown [8]. They have also been studied over other rings such as F 2 + uF 2 , u 2 = 0 [6]; F 2 + uF 2 + vF 2 + uvF 2 , u 2 = v 2 = 0, uv = vu [18]; and F 2 + vF 2 , v 2 = v [20]. ...
In this paper, we have studied cyclic codes over the ring [Formula: see text], [Formula: see text]. We have provided the general form of the generators of a cyclic code over [Formula: see text] and obtained a minimal spanning set for such codes and determined their ranks. We have determined a necessary condition and a sufficient condition for cyclic codes over [Formula: see text] to be [Formula: see text]-free. For [Formula: see text], we have shown that [Formula: see text] is a local ring, and the complete ideal structure of [Formula: see text] is determined. Some examples are presented.
... In these studies, the ground rings associated with codes are finite chain rings in general, and linear codes over this class of finite rings have been characterized in several literatures [8−10] . Recently, linear and cyclic codes over the ring F 2 + uF 2 + vF 2 + uvF 2 have been considered by Yildiz and Karadenniz in [11] and [12], and some good binary codes have been obtained as the images under two Gray maps. Because the ring F 2 +uF 2 +vF 2 +uvF 2 is not a finite chain ring, some techniques used in these literatures are different from those in the previous papers. ...
This paper studies (1 + u)-constacyclic codes over the ring F 2 + uF 2 + vF 2 + uvF 2. It is proved that the image of a (1+u)-constacyclic code of length n over F 2+uF 2+vF 2+uvF 2 under a Gray map is a distance invariant binary quasi-cyclic code of index 2 and length 4n. A set of generators of such constacyclic codes for an arbitrary length is determined. Some optimal binary codes are obtained directly from (1 + u)-constacyclic codes over F 2 + uF 2 + vF 2 + uvF 2. © 2012 Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg.
In this paper, we study on linear, cyclic, self-dual and self-orthogonal codes over the ring Rq. We especially focus on extremal self-dual codes over . Our method is easy and constructing self-dual codes over . We define a distance-invariant Gray map from Rq to the finite field . It is proved that the images of self-dual codes of length n over Rq under this Gray map correspond to self-dual codes of length 2n over . Using all of these codes, we construct stabilizer quantum codes over . We obtain some self-dual codes and some optimal quantum codes.
In this paper, we study cyclic codes over the ring R = ℤ4 + uℤ4, u2 = 0. We discuss the Galois ring extensions of R and the ideal structure of these extensions. We have studied cyclic codes of odd lengths over R and also presented 1-generator cyclic codes over R interms nth roots of unity. Some examples are given to illustrate the results.
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