Cyclic Codes Research Articles

Optimal quinary cyclic codes with three zeros

Optimal quinary cyclic codes with three zeros

Shift-Sum Decoding of Non-Binary Cyclic Codes

Shift-Sum Decoding of Non-Binary Cyclic Codes

A Class of LCD Codes Through Cyclic Codes Over $ZpR$

In recent time, some mixed types of alphabets have been considered for constructing error correcting codes. These constructions include $\bbbz_{2}\bbbz_{4}-$additive codes, $\bbbz_{2}\bbbz_{2}[u]-$linear codes et cetera. In this paper, we studied a class of codes over a mixed ring $\bbbz_{p}R$ where $R=\bbbz_{p}+v\bbbz_{p}+v^{2}\bbbz_{p}, v^{3}=v.$ We determined an algebraic structure of these codes under certain conditions. We have also constructed a class of LCD cyclic codes over $\bbbz_{p}R$. A necessary and sufficient condition for a cyclic code to be a complementary dual (LCD) code has been obtained.

MDS and MHDR Cyclic Codes over Finite Chain Rings

This work establishes a unique set of generators for a cyclic code over a finite chain ring. Towards this, we first determine the minimal spanning set and rank of the code. Furthermore, sufficient as well as necessary conditions for a cyclic code to be an MDS code and for a cyclic code to be an MHDR code are obtained. Finally, to support our results, some examples of optimal cyclic codes are presented.

Retracted: Using a Solution Construction Algorithm for Cyclic Shift Network Coding under Multicast Network to the Transformation of Musical Performance Styles

Retracted: Using a Solution Construction Algorithm for Cyclic Shift Network Coding under Multicast Network to the Transformation of Musical Performance Styles

Structure and Rank of a Cyclic Code over a Class of Nonchain Rings

The rings Z4+νZ4 have been classified into chain rings and nonchain rings based on the values of ν2∈Z4+νZ4. In this paper, the structure of a cyclic code of arbitrary length over the rings Z4+νZ4 for those values of ν2 for which these are nonchain rings has been established. A unique form of generators for a cyclic code over these rings has also been obtained. Furthermore, the rank and cardinality of a cyclic code over these rings have been established by finding a minimal spanning set for the code.

Z2[u]Z2[u]-Additive codes

In this paper, we study mixed alphabet codes over two alphabets [Formula: see text], and its extension [Formula: see text], [Formula: see text]. We define new Gray maps, using which we study cyclic codes and obtain the properties of their generators. Further, we construct constacyclic codes, and linear codes, and observe the useful properties of their generators. Finally, we present some examples of cyclic codes and binary linear codes.

Some new quantum codes from constacyclic codes

Some new quantum codes from constacyclic codes

Hulls of constacyclic codes over finite non-chain rings and their applications in quantum codes construction

Hulls of constacyclic codes over finite non-chain rings and their applications in quantum codes construction

On cyclic and negacyclic codes with one-dimensional hulls and their applications

Linear codes over finite fields with small dimensional hulls have received much attention due to their applications in cryptology and quantum computing. In this paper, we study cyclic and negacyclic codes with one-dimensional hulls. We determine precisely when cyclic and negacyclic codes over finite fields with one-dimensional hulls exist. We also introduce one-dimensional linear complementary pairs of cyclic and negacyclic codes. As an application, we obtain numerous optimal or near optimal cyclic codes with one-dimensional hulls over different fields and, by using these codes, we present new entanglement-assisted quantum error-correcting codes (EAQECCs). In particular, some of these EAQEC codes are maximal distance separable (MDS). We also obtain one-dimensional linear complementary pairs of cyclic codes, which are either optimal or near optimal.

On the construction of constacyclically permutable codes from constacyclic codes

<abstract><p>In this paper, we propose a way to partition any constacyclic code over a finite field in its equivalence classes according to the algebraic structure of the code. Such a method gives the generalization of cyclically permutable codes (CPCs), which are called constacyclically permutable codes (CCPCs), and it is useful to derive a CCPC from a given constacyclic code. Moreover, we present an enumerative formula for the code size of such a CCPC, with all of the terms being positive integers, and we provide an algebraic method to produce such a CCPC.</p></abstract>

Improved and Novel Quantum Codes through CSS Construction from Cyclic Codes over Commutative Non-Local Ring Fp×(Fp + αFp + α2 Fp)

In this study, by using CSS construction, we present the new quantum codes derived from cyclic codes over the commutative non - local ring R = Fp × (Fp + αFp + α2 Fp), where p is an odd prime, α3 = α. Firstly, we investigate the structure of the ring R and study linear codes by defining Lee weight over R. We define a distance – preserving Gray map from R→Fp 4 and discuss the cyclic codes with their dual code by Euclidean inner products on R. Additionally, we find the quantum Singleton defect(QSD) of the quantum code, which is denoted as n + 2 – k – 2d. Magma Software will be employed to provide relevant illustrations for enhanced comprehension.

The polycyclic codes over the finite field $ \mathbb{F}_q $

<p>This article extended the properties of the idempotent generator of cyclic codes to polycyclic codes over the finite field $ \mathbb{F}_q $. In addition, the check matrix of polycyclic codes was provided over $ \mathbb{F}_q $. Specifically, it has been proven that the constacyclic code is an $ {{{\rm{MDS}}}} $ code over $ \mathbb{F}_q $ if and only if its annihilator dual code is also an $ {{{\rm{MDS}}}} $ code. Finally, we have provided some examples of good codes.</p>

$ (\theta_i, \lambda) $-constacyclic codes and DNA codes over $ \mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4} $

<p>In this paper, three new automorphisms were identified over the ring $ \mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4} $ where $ u^3 = u^2 $. With the help of these automorphisms, the characteristic structures of the generator polynomials for the $ \theta_i $-cyclic codes and $ (\theta_i, \lambda) $-constacyclic codes of odd length on this ring were investigated. Also, for all the units over the ring, $ \mathbb{Z}_{4} $-images of $ \theta_i $-cyclic and $ (\theta_i, \lambda) $-constacyclic codes were reviewed with the associated codes based on determined transformations. Using these observations, new and optimal codes were obtained and presented in the table. In addition, a new transformation was identified that involved DNA base pairs with the elements of $ \mathbb{Z}_{4} $. Moreover, a unit reverse polynomial was created, and in this way a new generation method has been built to construct reversible DNA codes over this ring. Finally, this article was further enhanced with supporting examples of the DNA as a part of the study.</p>

Cyclic and Negacyclic Codes with Optimal and Best Known Minimum Distances

Cyclic and Negacyclic Codes with Optimal and Best Known Minimum Distances

Classification of Certain Cyclic LCD Codes

We show that a necessary and sufficient condition for a cyclic code C of length N over a finite chain ring R (whose maximal ideal has nilpotence 2) to be an LCD code is that C = (f(x)), where f(X) is a self-reciprocal monic divisor of XN − 1 in R[X] and x = X + (XN − 1) in R[X]/(XN − 1). A similar, but slightly different, theorem was proved in 2019 by Z. Liu and J. Wang for general finite chain rings (Theorem 25 in [5]). We provide two proofs, both completely different than the proof of Liu and Wang.

Cyclic codes over non-chain ring $ \mathcal{R}(\alpha_1, \alpha_2, \ldots, \alpha_s) $ and their applications to quantum and DNA codes

<abstract><p>Let $ s \geq 1 $ be a fixed integer. In this paper, we focus on generating cyclic codes over the ring $ \mathcal{R}(\alpha_1, \alpha_2, \ldots, \alpha_s) $, where $ \alpha_i \in \mathbb{F}_q\backslash \{0\} $, $ 1 \leq i \leq s $, by using the Gray map that is defined by the idempotents. Moreover, we describe the process to generate an idempotent by using the formula (2.1). As applications, we obtain both optimal and new quantum codes. Additionally, we solve the DNA reversibility problem by introducing $ \mathbb{F}_q $ reversibility. The aim to introduce the $ \mathbb{F}_q $ reversibility is to describe IUPAC nucleotide codes, and consequently, 5 IUPAC DNA bases are considered instead of 4 DNA bases $ (A, \; T, \; G, \; C) $.</p></abstract>

Enumeration and Generation of Cyclically Permutable Codes From Cyclic Codes

Enumeration and Generation of Cyclically Permutable Codes From Cyclic Codes

Minimal and maximal cyclic codes of length 2p

In this paper, we compute the maximal and minimal codes of length 2p over finite fields Fq with p and q are distinct odd primes and f(p) = p – 1 is the multiplicative order of q modulo 2p. We show that, every cyclic code is a direct sum of minimal cyclic codes.

Threshold Circuit Based on Cyclic Codes

Threshold Circuit Based on Cyclic Codes