Cyclic Codes Research Articles

Mixed skew cyclic codes over rings

This paper explores different types of skew cyclic codes by generating special subclasses with additional desirable properties. Specifically, we are interested in skew cyclic codes over mixed rings. We study some algebraic and structural properties of these codes and their constructions. We study skew cyclic codes over the mixed alphabet ring $\mathbb{F}_q(\mathbb{F}_q+v\mathbb{F}_q)$ under a mixed automorphism $(\theta,\tilde{\theta})$ and we give the structure of these codes for an arbitrary length via the non-commutative ring $\mathbb{F}_q[x,\theta](\mathbb{F}_q+v\mathbb{F}_q)[x,\tilde{\theta}]$. A condition for the existence of linear complementary dual (LCD) codes (which play an important role in practical applications such as armoring implementations against side-channel attacks and fault injection attacks) are explored specifically for skew cyclic codes.

On generating sets of 2D cyclic codes of length 2ks and corrections to some previous results

In this paper, we first study when two-dimensional cyclic codes of length [Formula: see text] have generating sets of a certain form. Then we deduce that several of the main results of the papers [Z. Rajabi and K. Khashyarmanesh, Repeated-root two-dimensional constacyclic codes of length [Formula: see text], Finite Fields Appl. 50 (2018) 122–137] and [Z. Sepasdar and K. Khashyarmanesh, Characterizations of some two-dimensional cyclic codes correspond to the ideals of [Formula: see text], Finite Fields Appl. 41 (2016) 97–112] are not accurate. We also present counterexamples for these results and find other conditions needed for them to hold.

"None of us asked to be in this community." Understanding nephrotic syndrome through TikTok: patient and caregiver perspectives.

TikTok, a popular social media platform, is increasingly used for health information dissemination; however, the accuracy and quality of medical content remain uncertain, including in the context of nephrotic syndrome (NS). This study aims to identify prominent patient and caregiver experiences with NS on TikTok and demonstrate how they may vary. A convenience sample of TikTok videos containing the hashtag "nephrotic syndrome" posted between July 1, 2020, and February 29, 2024, was analyzed. Videos underwent cyclical and inductive coding, followed by content and discourse analysis to identify common themes and narratives. One hundred twenty-three videos were included in the analysis. 62.6% of videos (N = 77) consisted of caregivers sharing their experiences of their child's disease. Three prominent topics included: (1) navigating healthcare and managing illness, where users shared their disease journeys; (2) emotional and physical wellbeing, where caregivers focused on physical disease signs while patients highlighted the mental health toll of the illness; and (3) education, awareness, and support systems, where users shared feelings of social isolation post-diagnosis. The discourse analysis revealed language portraying patients as "warriors," reflecting the resiliency promoted by TikTok support systems. We uncovered a hidden disease burden associated with NS that affects everyday life, reinforcing the importance of the journey and stress patients and caregivers experience outside of the clinician's office. Our findings also highlight that patient priorities may differ from those reported by caregivers, particularly in pediatrics. TikTok may also be an outlet for feelings of isolation and community-building within NS.

Three families of large cyclic subspace codes

Three families of large cyclic subspace codes

Equivalence of constacyclic codes with shift constants of different orders

Equivalence of constacyclic codes with shift constants of different orders

Quantum Codes as an Application of Constacyclic Codes

The main focus of this paper is to analyze the algebraic structure of constacyclic codes over the ring R=Fp+w1Fp+w2Fp+w22Fp+w1w2Fp+w1w22Fp, where w12−α2=0, w1w2=w2w1, w23−β2w2=0, and α,β∈Fp∖{0}, for a prime p. We begin by introducing a Gray map defined over R, which is associated with an invertible matrix. We demonstrate its advantages over the canonical Gray map through some examples. Finally, we create new and improved quantum codes from constacyclic codes over R using Calderbank–Shore–Steane (CSS) construction.

On generalized Sidon spaces

Sidon spaces have been introduced by Bachoc, Serra and Zémor as the q-analogue of Sidon sets, classical combinatorial objects introduced by Simon Szidon. In 2018 Roth, Raviv and Tamo introduced the notion of r-Sidon spaces, as an extension of Sidon spaces, which may be seen as the q-analogue of Br-sets, a generalization of classical Sidon sets. Thanks to their work, the interest on Sidon spaces has increased quickly because of their connection with cyclic subspace codes they pointed out. This class of codes turned out to be of interest since they can be used in random linear network coding. In this work we focus on a particular class of them, the one-orbit cyclic subspace codes, through the investigation of some properties of Sidon spaces and r-Sidon spaces, providing some upper and lower bounds on the possible dimension of their r-span and showing explicit constructions in the case in which the upper bound is achieved. Moreover, we provide further constructions of r-Sidon spaces, arising from algebraic and combinatorial objects, and we show examples of Br-sets constructed by means of them.

On Negacyclic Codes of Length 8 p s over F p m + u F p m

In this paper, we provide a correction regarding the structure of negacyclic codes of length 8 p s over R = F p m + u F p m when p m ≡ 3 ( mod 8 ) as classified in [1]. Among other results, we determine the number of codewords and the dual of each negacyclic code.

Some Results for Double Cyclic Codes over Fq+vFq+v2Fq.

Let Fq be a finite field with an odd characteristic. In this paper, we present a new result about double cyclic codes over a finite non-chain ring. Specifically, we study the double cyclic code over Fq+vFq+v2Fq with v3=v, which is isomorphic to Fq×Fq×Fq. This study mainly involves generator polynomials and generator matrices. The generating polynomial of the dual code is also obtained. We show the relationship between the generating polynomials of the double cyclic codes and those of their dual codes. Finally, as an application of these results, we construct some optimal codes over F3.

Constacyclic codes over F q S T and their applications

Let q = p r , where p is an odd prime and r is a positive integer. Consider a ring F q S T , where S = F q + u F q + v F q + uv F q with u 2 = 1 , v 2 = 1 , uv = vu and T = F q + u F q + v F q + w F q + uv F q + uw F q + vw F q + uvw F q with u 2 = 1 , v 2 = 1 , w 2 = 1 , uv = vu , uw = wu , vw = wv . In this paper, we examine the algebraic structure of constacyclic codes over the ring F q S T of block length ( α , β , γ ) . Further, a construction of quantum error-correcting codes (QECCs) from constacyclic codes over F q S T is also given. Moreover, we derive a number of new QECCs.

A note on the hull and linear complementary pair of cyclic codes

A note on the hull and linear complementary pair of cyclic codes

A class of constacyclic codes of length 2pˢ over Fp^m[u,v] / ⟨u², v², uv − vu⟩

Let p be an odd prime number. This paper investigates λ-constacyclic codes of length 2ps over the ring Rᵤ,ᵥ = Fpm[u,v] / ⟨u², v², uv − vu⟩, where λ = ψ + ρu + ηv + τuv, with ψ, ρ, η, τ ∈ Fpm, ψ ∈ F*pm, and ρ, η not both zero. When λ = μ² for some μ ∈ Rᵤ,ᵥ, the structures of all λ-constacyclic codes of length 2ps over Rᵤ,ᵥ are determined and can be expressed as the sum of a (−μ)-constacyclic code and a μ-constacyclic code of length ps. When λ is not a square, we classify these codes into four distinct types, provide the number of codewords for each type, and give the details of their dual codes.

New entanglement-assisted MDS quantum constacyclic codes

Entanglement-assisted quantum error-correcting codes can be seen as a new-type of quantum error-correcting codes and can be constructed from arbitrary linear codes which should not satisfy the dual-containing condition by utilizing shared entangled states between the sender and the receiver in advance. In this paper, we construct several new classes of entanglement-assisted quantum maximum-distance-separable codes from constacyclic codes and cyclic codes by exploiting small pre-shared entangled states, respectively. These codes are new in the sense that they are not covered by the codes available in the literature.

Trace dual of additive cyclic codes over finite fields

Trace dual of additive cyclic codes over finite fields

ON QUANTUM CODES CONSTRUCTION FROM CONSTACYCLIC CODES OVER THE RING I_q[u,v] / <u^2-a^2, v^2-a^2, uv-vu>

This paper focuses on studying the properties of constacyclic codes, quantum error-correcting codes. The code is studied over a specific mathematical structure called the ring $\mathfrak{S}$, which is defined as $\mathfrak{S}=\mathfrak{I}_q[\mathfrak{u},\mathfrak{v}]/\langle \mathfrak{u}^2-\alpha^2,~ \mathfrak{v}^2-\alpha^2,~\mathfrak{u}\mathfrak{v}-\mathfrak{v}\mathfrak{u} \rangle$, where $\mathfrak{I}_q$ is a finite field of $q$ elements, $\alpha$ be the nonzero elements of the field $\mathfrak{I}_q$ and $q$ is a power of an odd prime $p$ such that $q=p^m, ~\textup{for}~ m \ge 1$. The paper also introduces a Gray map and use it to decompose constacyclic codes over the ring $\mathfrak{S}$ into a direct sum of constacyclic codes over $\mathfrak{I}_q$. We construct new and better quantum error-correcting codes over the ring $\mathfrak{S}$ (cf.; Table 1 and Table 2). Moreover, we also obtain best known linear codes as well as best dimension linear codes (cf.; Table 4).

General Structure and Properties of Cyclic Codes Over \(GF_2\)

This article provides a comprehensive analysis of cyclic codes in GF(2), focusing on their general structure and key properties. Cyclic codes are characterized by their generator polynomials, which define their structure and play a crucial role in encoding and decoding processes. The cyclical shift property, inherent in cyclic codes, facilitates efficient implementation using shift register circuits, making them practical for real-world applications. The discussion highlights the algebraic properties that distinguish cyclic codes from other linear block codes, emphasizing their ability to detect and correct errors.

Additive cyclic codes over 𝔽q3

Additive cyclic codes over 𝔽q3

Dynamic Injection and Permutation Coding for Enhanced Data Transmission.

In this paper, we propose a novel approach to enhance spectral efficiency in communication systems by dynamically adjusting the mapping between cyclic permutation coding (CPC) and its injected form. By monitoring channel conditions such as interference levels and impulsive noise strength, the system optimises the coding scheme to maximise data transmission reliability and efficiency. The CPC method employed in this work maps information bits onto non-binary symbols in a cyclic manner, aiming to improve the Hamming distance between mapped symbols. To address challenges such as low data rates inherent in permutation coding, injection techniques are introduced by removing δ column(s) from the CPC codebook. Comparative analyses demonstrate that the proposed dynamic adaptation scheme outperforms conventional permutation coding and injection schemes. Additionally, we present a generalised mathematical expression to describe the relationship between the spectral efficiencies of both coding schemes. This dynamic approach ensures efficient and reliable communication in environments with varying levels of interference and impulsive noise, highlighting its potential applicability to systems like power line communications.

A new construction of cyclic subspace codes

A new construction of cyclic subspace codes

On Some Aspects Of Degenerated Cyclic Codes

On Some Aspects Of Degenerated Cyclic Codes