Cyclic Codes Research Articles

A generalization of the pascal matrix and an application to coding theory

Suppose that m , k , t are integers with m , k ≥ 1 and 0 ≤ t and A is the k × k matrix with the ( i , j ) -entry ( t + ( j − 1 ) m i − 1 ) . When t = 0 and m = 1, this is the upper triangular Pascal matrix. Here, first we study the properties of this matrix, in particular, we find its determinant and its LDU decomposition and also study its inverse. Then by using this matrix we present a generalization of the Mattson-Solomon transform and a polynomial formulation for its inverse to the case that the sequence length and the characteristic of the base field are not coprime. At the end, we use this generalized Mattson-Solomon transform to present a lower bound on the length of repeated root cyclic codes, which can be seen as a generalization of the BCH bound.

Hulls of double cyclic codes

In this paper, we characterize hulls of double cyclic codes over Z2. Firstly, we give the generators of hulls of double cyclic codes under the separable case and the non-separable case with coprime odd block lengths. Then, using the concept of a generating function in combinatorics, the solution of dimensions of hulls of double cyclic codes over Z2 is given. Furthermore, the enumeration of double cyclic codes over Z2 with hulls of a fixed dimension is determined. As an application, some good entanglement-assisted quantum error-correcting codes (EAQECCs) are obtained by the hulls of non-separable double cyclic codes.

New Optimal Linear Codes With Hierarchical Locality

Locally repairable codes with hierarchical locality (H-LRCs) are designed to correct different numbers of erasures, which play a crucial role in large-scale distributed storage systems. In this paper, we construct three classes of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -ary optimal H-LRCs by employing matrix product codes, concatenated codes and cyclic codes, respectively. The first two constructions are based on the idea of constructing new codes from old, which produces several new classes of optimal H-LRCs whose lengths can reach up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q^{2}+q$ </tex-math></inline-formula> or unbounded. The final construction generates a class of new optimal cyclic H-LRCs whose lengths divide <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q-1$ </tex-math></inline-formula> . Compared with the previously known ones, our constructions are new in the sense that their parameters are not covered by the codes available in the literature.

On Symbol-Pair Distance of a Class of Constacyclic Codes of Length 3ps over Fpm+uFpm

Let p≠3 be any prime. In this paper, we compute symbol-pair distance of all γ-constacyclic codes of length 3ps over the finite commutative chain ring R=Fpm+uFpm, where γ is a unit of R which is not a cube in Fpm. We give the necessary and sufficient condition for a symbol-pair γ-constacyclic code to be an MDS symbol-pair code. Using that, we provide all MDS symbol-pair γ-constacyclic codes of length 3ps over R. Some examples of the symbol-pair distance of γ-constacyclic codes of length 3ps over R are provided.

Double Constacyclic Codes Over Two Finite Commutative Chain Rings

Many kinds of codes which possess two cycle structures over two special finite commutative chain rings, such as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ </tex-math></inline-formula> -additive cyclic codes and quasi-cyclic codes of fractional index etc., were proved asymptotically good. In this paper we extend the study in two directions: we consider any two finite commutative chain rings with a surjective homomorphism from one to the other, and consider double constacyclic structures. We construct an extensive kind of double constacyclic codes over two finite commutative chain rings. And, developing a probabilistic method suitable for quasi-cyclic codes over fields, we prove that the double constacyclic codes over two finite commutative chain rings are asymptotically good.

The b-symbol weight distributions of all semiprimitive irreducible cyclic codes

Up to a new invariant mu (b), the complete b-symbol weight distribution of a particular kind of two-weight irreducible cyclic codes, was recently obtained by Zhu et al. (Des Codes Cryptogr 90(5):1113–1125, 2022). The purpose of this paper is to simplify and generalize the results of Zhu et al., and obtain the b-symbol weight distributions of all one-weight and two-weight semiprimitive irreducible cyclic codes.

AMDS symbol-pair codes from repeated-root cyclic codes

Symbol-pair codes are proposed to guard against pair-errors in symbol-pair read channels. The minimum symbol-pair distance is of significance in determining the error-correcting capability of a symbol-pair code. One of the central themes in symbol-pair coding theory is the constructions of symbol-pair codes with the largest possible minimum symbol-pair distance. Maximum distance separable (MDS) and almost maximum distance separable (AMDS) symbol-pair codes are optimal and sub-optimal regarding the Singleton bound, respectively. In this paper, six new classes of AMDS symbol-pair codes are explicitly constructed through repeated-root cyclic codes. Remarkably, one class of such codes has unbounded lengths and the minimum symbol-pair distance of another class can reach 13.

Optimized Higher-Order Polarization Weight Incremental Selective Decoding and Forwarding in Cooperative Satellite Sensor Networks

An optimized higher-order polarization weight (HPW) incremental selective decoding and forwarding (ISDF) scheme is proposed for cooperative transmissions in satellite sensor networks (SSNs). First, an HPW method is designed with subchannel power allocation in encoding to enhance the antiinterference. Second, a cooperative decision in the destination is used to avoid error propagation for better outage probability (OP) and bit error rate (BER) performance. It switches between direct and cooperative transmissions alternatively by channel state information (CSI). Finally, the HPW coding information, cyclic redundancy codes (CRCs), and four special frame structures of polar subcodes are jointly fast successive cancellation list (FSCL) decoded to improve efficiency and BER performance. Finally, we apply the optimized polarization encoding and decoding to the relay system to form a new ISDF relay system for better coding efficiency, BER, and OP performance. Simulation results show that the proposed scheme obtains approximately 0.8 and 1 dB performance gains at OP of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${{10}^{-{2}}}$ </tex-math></inline-formula> and BER of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${{10}^{-{6}}}$ </tex-math></inline-formula> , respectively, when compared with those of the current ISDF scheme. Therefore, the cooperative scheme possesses efficient decoding and cooperation, which endows it a promising candidate relay scheme in SSNs.

QUANTUM CODES OBTAINED THROUGH -CONSTACYCLIC CODES OVER

The structural properties and construction of quantum codes over using Constacyclic codes over the finite commutative non-chain ring = where are the focus of htis paperand is field having elements with characteristic p where p is an prime such that . A Gray map is defined between and . Decomposing constacyclic codes into cyclic and negacyclic codes over yields the parameters of quantum codes over . Some examples of quantum codes of arbitrary length are also obtained as an application.

Quantum codes from constacyclic codes over S_{k}

Let S_{k}={mathbb{F}}_{q}[u_{1},u_{2},ldots ,u_{k}]/langle u^{3}_{i}=u_{i},u_{i}u_{j}=u_{j}u_{i}=0 rangle , where 1leq i,jleq k, q=p^{m}, p is an odd prime. First, we define two new Gray maps phi _{k} and varphi _{k}, and study their Gray images. Further, we determine the structure of constacyclic codes and their dual codes, and give a necessary and sufficient conditions of constacyclic codes to contain their duals. Finally, we obtain some new quantum codes over mathbb{F}_{q} by using CSS construction, and compare the constructed codes better than the existing literature.

EAQEC codes from two distinct constacyclic codes

EAQEC codes from two distinct constacyclic codes

Construction of Cyclic Redundancy Check Codes for SDDC Decoding in DRAM Systems

Single device data correction (SDDC) is a main reliability, availability, and serviceability feature of DRAM systems in servers due to the significant hard-failure rate associated with DRAM devices. To correct errors in one DRAM device, error pattern is determined by even parity bits and error location is determined by the error pattern and cyclic redundancy check (CRC) bits in SDDC decoding. In this brief, a SDDC decoding scheme is proposed, which improves the error-correction performance by uniquely determining the error location. For that purpose, requirements for binary CRC generator polynomials to uniquely determine the error location are derived. Based on these requirements, a systematic method for constructing CRC generator polynomials is proposed, which guarantees 100% error-correction rate. Finally, it is confirmed that the proposed SDDC decoding scheme has lower decoding complexity compared with various ECC schemes and also shows 100% SDDC decoding success through simulation.

A Tight Upper Bound on the Number of Non-Zero Weights of a Cyclic Code

Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}$ </tex-math></inline-formula> be a simple-root cyclic code and let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {G}$ </tex-math></inline-formula> be the subgroup of the automorphism group of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}$ </tex-math></inline-formula> generated by the cyclic shift of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}$ </tex-math></inline-formula> and the scalar multiplications of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}$ </tex-math></inline-formula> . In this paper, we find an explicit formula for the number of orbits of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {G}$ </tex-math></inline-formula> on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}\setminus \{\mathbf {0}\}$ </tex-math></inline-formula> . Consequently, an explicit upper bound on the number of non-zero weights of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}$ </tex-math></inline-formula> is immediately derived and a necessary and sufficient condition for codes meeting the bound is exhibited. Several reducible and irreducible cyclic codes meeting the bound are presented, revealing that our bound is tight. In particular, we find that some infinite families of irreducible cyclic codes constructed in (Ding, 2009) meet our bound; we then conclude that such known codes enjoy an additional property that any two codewords with the same weight belong to the same <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {G}$ </tex-math></inline-formula> -orbit, a fact that may not have been known before. Our main result improves and generalizes some of the results in (Shi et al., 2019).

DNA Code from Cyclic and Skew Cyclic Codes over F4[v]/⟨v3⟩

The main motivation of this work is to study and obtain some reversible and DNA codes of length n with better parameters. Here, we first investigate the structure of cyclic and skew cyclic codes over the chain ring R:=F4[v]/⟨v3⟩. We show an association between the codons and the elements of R using a Gray map. Under this Gray map, we study reversible and DNA codes of length n. Finally, several new DNA codes are obtained that have improved parameters than previously known codes. We also determine the Hamming and the Edit distances of these codes.

Tailored XZZX codes for biased noise

Quantum error correction (QEC) for generic errors is challenging due to the demanding threshold and resource requirements. Interestingly, when physical noise is biased, we can tailor our QEC schemes to the noise to improve performance. Here we study a family of codes having XZZX-type stabilizer generators, including a set of cyclic codes generalized from the five-qubit code and a set of topological codes that we call generalized toric codes (GTCs). We show that these XZZX codes are highly qubit efficient if tailored to biased noise. To characterize the code performance, we use the notion of effective distance, which generalizes code distance to the case of biased noise and constitutes a proxy for the logical failure rate. We find that the XZZX codes can achieve a favorable resource scaling by this metric under biased noise. We also show that the XZZX codes have remarkably high thresholds that reach what is achievable by random codes, and furthermore they can be efficiently decoded using matching decoders. Finally, by adding only one flag qubit, the XZZX codes can realize fault-tolerant QEC while preserving their large effective distance. In combination, our results show that tailored XZZX codes give a resource-efficient scheme for fault-tolerant QEC against biased noise.

Constacyclic Codes Over 𝑭𝒒[𝒖] /〈𝒖𝟑 = 𝟎〉 and Their Application of Constructing Quantum Codes

Let 𝑅 = 𝐹𝑞+u𝐹𝑞+𝑢^2𝐹𝑞, 𝑢^3=0 be a finite chain ring. In this paper, we give the structure of constacyclic codes over 𝑅 and obtain self-orthogonal codes over 𝐹𝑞 by using the Gray map from 𝑅𝑛 to 𝐹𝑞^(3𝑛). As an application, we present a construction of quantum codes from the codes obtained from this class.

(β,γ)-Skew QC Codes with Derivation over a Semi-Local Ring

In this article, we consider a semi-local ring S=Fq+uFq, where u2=u, q=ps and p is a prime number. We define a multiplication yb=β(b)y+γ(b), where β is an automorphism and γ is a β-derivation on S so that S[y;β,γ] becomes a non-commutative ring which is known as skew polynomial ring. We give the characterization of S[y;β,γ] and obtain the most striking results that are better than previous findings. We also determine the structural properties of 1-generator skew cyclic and skew-quasi cyclic codes. Further, We demonstrate remarkable results of the above-mentioned codes over S. Finally, we find the duality of skew cyclic and skew-quasi cyclic codes using a symmetric inner product. These codes are further generalized to double skew cyclic and skew quasi cyclic codes and a table of optimal codes is calculated by MAGMA software.

When do quasi-cyclic codes have $\mathbb F_{q^l}$-linear image?

A length $ml$, index $l$ quasi-cyclic code can be viewed as a cyclic code of length $m$ over the field $\mathbb F_{q^l}$ via a basis of the extension $\mathbb F_{q^l}/\mathbb F_{q}$. This cyclic code is an additive cyclic code. In [C. Güneri, F. Özdemir, P. Solé, On the additive cyclic structure of quasi-cyclic codes, Discrete. Math., 341 (2018), 2735-2741], authors characterize the $(l,m)$ values for one-generator quasi-cyclic codes for which it is impossible to have an $\mathbb F_{q^l}$-linear image for any choice of the polynomial basis of $\mathbb F_{q^l}/\mathbb F_{q}$. But this characterization for some $(l,m)$ values is very intricate. In this paper, by the use of this characterization, we give a more simple characterization.

Multi-orbit cyclic subspace codes and linear sets

Cyclic subspace codes gained a lot of attention especially because they may be used in random network coding for correction of errors and erasures. Roth, Raviv and Tamo in 2018 established a connection between cyclic subspace codes (with certain parameters) and Sidon spaces. These latter objects were introduced by Bachoc, Serra and Zémor in 2017 in relation with the linear analogue of Vosper's Theorem. This connection allowed Roth, Raviv and Tamo to construct large classes of cyclic subspace codes with one or more orbits. In this paper we will investigate cyclic subspace codes associated to a set of Sidon spaces, that is cyclic subspace codes with more than one orbit. Moreover, we will also use the geometry of linear sets to provide some bounds on the parameters of a cyclic subspace code. Conversely, cyclic subspace codes are used to construct families of linear sets which extend a class of linear sets recently introduced by Napolitano, Santonastaso, Polverino and the author. This yields large classes of linear sets with a special pattern of intersection with the hyperplanes, defining rank metric and Hamming metric codes with only three distinct weights.

Cyclic codes of length 5ps over Fpm + uFpm and their duals

For an odd prime p ? 5, the structures of cyclic codes of length 5ps over R = Fpm + uFpm (u2 = 0) are completely determined. Cyclic codes of length 5ps over R are considered in 3 cases, namely, p ? 1 (mod 5), p ? 4 (mod 5), p ? 2 or 3 (mod 5). When p ? 1 (mod 5), a cyclic code of length 5ps over R can be expressed as a direct sum of a cyclic code and ?ps i -constacyclic codes of length ps over R, where ?ps i = ?i(pm?1)ps/10, i = 1,3,7,9. When p ? 4 (mod 5), it is equivalent to pm ? 1 (mod 5) when m is even and pm ? 4 (mod 5) when m is odd. If pm ? 1 (mod 5) when m is even, then a cyclic code of length 5ps over R can be obtained as a direct sum of a cyclic code and ?ps i -constacyclic codes of length ps over R, where ?psi = ?i(pm?1)ps/10, i = 1,3,7,9. If pm ?/ 4 (mod 5) when m is odd, then a cyclic code of length 5ps over R can be expressed as a direct sum of a cyclic code of length ps over R and an ?1 and ?2-constacyclic code of length 2ps over R, for some ?1, ?2 ? Fpm\{0}. If p ? 2 or 3 (mod 5) such that pm ?/ 1 (mod 5), then a cyclic code of length 5ps over R can be expressed as C1 ? C2, where C1 is an ideal of R[x]/?xps?1? and C2 is an ideal of R[x]/(x4+x3+x2+x+1)ps ?. We also investigate all ideals of R[x]/?(x4+x3+x2+x+1)ps ? to study detail structure of a cyclic code of length 5ps over R. In addition, dual codes of all cyclic codes of length 5ps over R are also given. Furthermore, we give the number of codewords in each of those cyclic codes of length 5ps over R. As cyclic and negacyclic codes of length 5ps over R are in a one-by-one equivalent via the ring isomorphism x ? ?x, all our results for cyclic codes hold true accordingly to negacyclic codes.