Cyclic Codes Research Articles

On Linear Perfect b-Symbol Codes over Finite Fields

Motivated by the application of high-density data storage technologies, Cassuto and Blaum introduced codes for symbol-pair read channels in 2011, and Yaakobi et al. generalized the coding framework to that for b-symbol read channels where b≥2 in 2016. In this paper, we establish a b-sphere-packing bound and present a recurrence relationship for the b-weight enumerator. We determine all parameters of linear perfect b-symbol e-error-correcting codes over Fq for e<2b and show that for 2b≤e<3b, there exist at most finite such codes for a given b, e, and q. We construct a family of linear perfect b-symbol b-error-correcting codes over Fq using constacyclic codes.

New quantum codes from constacyclic codes over a general non-chain ring

Let [Formula: see text] be a prime power and let [Formula: see text] be a finite non-chain ring, where [Formula: see text], are polynomials, not all linear, which split into distinct linear factors over [Formula: see text]. We characterize constacyclic codes over the ring [Formula: see text] and study quantum codes from these. As an application, some new and better quantum codes, as compared to the best known codes, are obtained. We also prove that the choice of the polynomials [Formula: see text], [Formula: see text], is irrelevant while constructing quantum codes from constacyclic codes over [Formula: see text], it depends only on their degrees.

The Subfield and Extended Codes of a Subclass of Optimal Three-Weight Cyclic Codes

A class of optimal three-weight [q^k-1,k+1,q^{k-1}(q-1)-1] cyclic codes over {mathrm{I!F}}_q, with kge 2, achieving the Griesmer bound, was presented by Heng and Yue (IEEE Trans Inf Theory 62(8):4501–4513, 2016. https://doi.org/10.1109/TIT.2016.2550029). In this paper we study some of the subfield codes of this class of optimal cyclic codes when k=2. The weight distributions of the subfield codes are settled. It turns out that some of these codes are optimal and others have the best known parameters. The duals of the subfield codes are also investigated and found to be almost optimal with respect to the sphere-packing bound. In addition, the covering structure for the studied subfield codes is determined. Some of these codes are found to have the important property that any nonzero codeword is minimal, which is a desirable property that is useful in the design of a secret sharing scheme based on a linear code. Moreover, a specific example of a secret sharing scheme based on one of these subfield codes is given. Finally, a class of optimal two-weight linear codes over {mathrm{I!F}}_q, achieving the Griesmer bound, whose duals are almost optimal with respect to the sphere-packing bound is presented. Through a different approach, this class of optimal two-weight linear codes was reported very recently by Heng (IEEE Trans Inf Theory 69(2):978–994, 2023. https://doi.org/10.1109/TIT.2022.3203380). Furthermore, it is shown that these optimal codes can be used to construct strongly regular graphs.

On the algebraic structure of (M,σ,δ)-skew codes

In this paper, we are interested in linear codes invariant under the (σ,δ)-action of a given matrix M, that we call (M,σ,δ)-skew codes. Various subclasses of these codes, depending on the (σ,δ)-cyclical property of M, are derived. In particular, we distinguish between three fundamental types: (M,σ,δ)-cyclic, (M,σ,δ)-quasi-cyclic and (M,σ,δ)-multi-cyclic. These different subclasses allow us to obtain some of the well known types of linear codes as special cases depending on the choice of M. We observe that the (σ,δ)-similarity in Mn(Fq) gives rise to various relations between these codes. As common in the study of cyclic codes and their generalizations, we use the skew polynomial ring Fq[x,σ,δ] to study (M,σ,δ)-skew codes. As an application of the theory developed here, we construct some optimal linear codes and MDS codes over different alphabets with additional desirable properties such as self orthogonality and linear complementary duality (LCD).

New classes of affine-invariant codes sandwiched between Reed–Muller codes

Reed–Muller codes have been extensively studied due to their favorable theoretical and mathematical properties. We propose some new classes of extended cyclic codes sandwiched between generalized Reed–Muller codes Rq(r−1,n) and Rq(r,n) with n=3m, m⩾1. We show that these new codes are affine-invariant, and compute their dimensions and dual codes. We also give lower bounds for their minimum distances. In addition, we explicitly determine the minimum vectors of these codes for some special cases. Our new method can be used to construct codes of length qn with odd n, which makes up for the shortcomings of the previous construction method which works only for even n. Moreover, our new extended cyclic codes have higher rate than those in Xu et al. (2023) [17].

Neural layered min-sum decoders for cyclic codes

This paper proposes a low-complexity neural network decoder based on the layered min-sum algorithm to decode cyclic codes. By generalizing the layered min-sum algorithm to its neural network counterpart, the number of network weights decreases while retaining a good error correction performance. The Bose–Chaudhuri–Hocquenghem (BCH) codes, quadratic residue (QR) codes, and punctured Reed–Muller (RM) codes are selected as three exemplary binary cyclic codes. Simulation results show that the proposed neural decoder achieves superior performance with less computational complexity compared with the state-of-the-art neural network decoder. Further, a neural decoder incorporating the modified random redundant decoding (mRRD) algorithm is investigated to approach the performance of maximum-likelihood decoding for some short codes.

On the Ding and Helleseth's 7th open problem about optimal ternary cyclic codes

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In 2013, Ding, et al. presented nine open problems about optimal ternary cyclic codes. The first two and the 6th problems were completely solved, and the third and last three problems were partially solved. In this paper, we focus on the 7th problem. By determining the root set of some special polynomials over finite fields, we give an incomplete answer and then present a counterexample. Furthermore, we construct two new classes of optimal ternary cyclic codes with respect to the Sphere Packing Bound by studying some special polynomials over finite fields.

A class of constacyclic codes are generalized Reed–Solomon codes

A class of constacyclic codes are generalized Reed–Solomon codes

On the Codes over a Family of Rings and Their Applications to DNA Codes

In this paper, the structures of the linear codes over a family of the rings $A_{t}=Z_{4}\left[ u_{1},\ldots ,u_{t}\right] \left/ \left\langle u_{i}^{2}-u_{i},u_{i}u_{j}-u_{j}u_{i}\right\rangle \right. $ are given, where $i,j=1,2,\ldots ,t$, $i\neq j$, $Z_{4}=\{0,1,2,3\}$. A map between the elements of the $A_{t}$ and the alphabet $\left\{ A,T,C,G\right\} ^{2^{t}}$ is constructed. The DNA codes are obtained with three different methods, by using the cyclic, skew cyclic codes over a family of the rings $A_{t}$ and $\theta _{i}$-set, where $\theta _{i}$ is a non trivial automorphism on $A_{i}$, for $i=1,2,\ldots ,t$.

A Hybrid DCSK Scheme Combining Cyclic Shift Keying and Code Index Modulation

A Hybrid DCSK Scheme Combining Cyclic Shift Keying and Code Index Modulation

Twisted Goppa Codes With an Efficient Decoding Algorithm and Quasi-Cyclic Properties

In this paper, we introduce twisted Goppa codes, which generalize classical Goppa codes by adding a twisted term. Then we provide an efficient decoding algorithm for twisted Goppa codes. The Niederreiter cryptosystem is bassed on linear error-correcting codes in which the public key is a parity check matrix. When twisted Goppa codes are applied to the Niederreiter cryptosystem, the public key size is overlarge. To reduce the public key size, we construct quasi-cyclic twisted Goppa codes via a non-trivial automorphism group carefully selecting the defining set and the matched polynomial. Moreover, we obtain a family of cyclic twisted Goppa codes.

A Class of Skew-Cyclic Codes over (Z_(2^m ) [u])/〈u^2-r〉 with Derivation

Let R_r=Z_(2^m )+uZ_(2^m ) be a finite ring, where u^2=r for r∈Z_(2^m ), m is a positive integer, and m≥2. In this paper, we study a class of skew-cyclic codes using a skew polynomial ring over R_r with an automorphism θ_r and a derivation δ_(θ_r ). We generalize the skew-cyclic codes over Z_4+uZ_4; u^2=1 to the skew-cyclic codes over R_r, and call such codes as δ_(θ_r )-cyclic codes. We investigate the structures of a skew polynomial ring R_r [x,θ_r,δ_(θ_r ) ]. A δ_(θ_r )-cyclic code is showed to be a left R_r [x,θ_r,δ_(θ_r ) ]-submodule of (R_r [x,θ_r,δ_(θ_r ) ])/〈x^n-1〉 . We give the generator matrix of a δ_(θ_r )-cyclic code of length n over R_r. Also, we present the generator matrix of the dual of a free δ_(θ_r )-cyclic code of even length n over R_r.

Several classes of optimal p-ary cyclic codes with minimum distance four

Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let p≥5 be an odd prime and m be a positive integer. Let C(1,e,s) denote the p-ary cyclic code with three nonzeros α, αe, and αs, where α is a generator of Fpm⁎, s=pm−12, and 2≤e≤pm−2. In this paper, by analyzing the solutions of certain equations over finite fields, we present four classes of optimal p-ary cyclic codes C(1,e,s) with parameters [pm−1,pm−2m−2,4]. Some known results on optimal quinary cyclic codes with parameters [5m−1,5m−2m−2,4] are special cases of our constructions. In addition, by analyzing the irreducible factors of certain polynomials over F5m, we present two classes of optimal quinary cyclic codes C(1,e,s).

Singleton-optimal LRCs and perfect LRCs via cyclic and constacyclic codes

Locally repairable codes (LRCs) have emerged as an important coding scheme in distributed storage systems (DSSs) with relatively low repair costs by accessing fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs have been widely investigated. Optimal LRCs via cyclic and constacyclic codes provide significant benefits of elegant algebraic structure and efficient encoding procedure. In this paper, we continue to consider the constructions of optimal LRCs via cyclic and constacyclic codes with long code lengths. Specifically, we first obtain two classes of q-ary cyclic Singleton-optimal (n,k,d=6;r=2)-LRCs with length n=3(q+1) when 3|(q−1) and q is even, and length n=32(q+1) when 3|(q−1) and q≡1(mod4), respectively. To the best of our knowledge, this is the first construction of q-ary cyclic Singleton-optimal LRCs with length n>q+1 and minimum distance d≥5. On the other hand, an LRC achieving the Hamming-type bound is called a perfect LRC. By using cyclic and constacyclic codes, we construct two new families of q-ary perfect LRCs with length n=qm−1q−1, minimum distance d=5 and locality r=2.

Skew constacyclic codes of lengths ps and 2ps over [formula omitted

Let R2=Fpm+uFpm, where u2=0. In this paper, we determine the algebraic structures of all skew constacyclic codes of lengths ps and 2ps over R2 for special automorphisms. We also characterize the structure of (Euclidean) dual of some special skew constacyclic codes of length 2ps and calculate the number of codewords of them. As a special case, if we restrict our attention to the polynomial ring R2[x], we obtain most of results for constacyclic codes of lengths ps and 2ps over R2.

Three new classes of optimal quinary cyclic codes with minimum distance four

Three new classes of optimal quinary cyclic codes with minimum distance four

Idempotents in simple skew group rings and applications to skew cyclic codes

We give the form of complete sets of pairwise orthogonal primitive idempotents in a simple skew group ring. Using these idempotents, we construct some classes of skew cyclic codes. We give also some examples of MDS codes and rows orthogonal and MDS matrices. Communicated by Eric Jespers

Three families of self-orthogonal codes and their application in optimal quantum codes

If a linear code is contained in its dual, then it is said to be self-orthogonal. Self-orthogonal codes are an important subclass of linear codes which have many nice applications. In this paper, we mainly study the augmented codes of the linear codes constructed by Hu et al. (2022) [21], and the subcodes of the codes punctured from the irreducible cyclic codes constructed by Ding and Yang (2013) [11]. The parameters and weight distributions of these codes are explicitly determined. These codes are proved to be self-orthogonal under certain conditions. As an application, these self-orthogonal codes are used to derive optimal or almost optimal quantum codes with new parameters.

On Optimal and Quantum Code Construction from Cyclic Codes over FqPQ with Applications.

The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of FqPQ, where P=Fq[v]⟨v3-α22v⟩ and Q=Fq[u,v]⟨u2-α12,v3-α22v⟩ are nonchain finite rings and αi is in Fq/{0} for i∈{1,2}, where q=pm with m≥1 is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring P, we obtain several optimal codes with the help of the Gray image of cyclic codes.

Five infinite families of binary cyclic codes and their related codes with good parameters

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Inspired by the recent work on binary cyclic codes published in Tang and Ding (2022) [14], the objectives of this paper are the construction and analyses of five infinite families of binary cyclic codes with parameters [n,k] and (n−6)/3≤k≤2(n+6)/3. Three of the five families of binary cyclic codes and their duals have a very good lower bound on their minimum distances and contain distance-optimal codes. The other two families of binary cyclic codes are composed of binary duadic codes with a square-root-like lower bound on their minimum distances. As a by-product, two infinite families of self-dual binary codes with a square-root-like lower bound on their minimum distances are obtained.