Cyclic Codes Research Articles

End Users' Perspectives on the Quality and Design of mHealth Technologies During the COVID-19 Pandemic in the Philippines: Qualitative Study.

The COVID-19 pandemic has expanded the use of mobile health (mHealth) technologies in contact tracing, communicating COVID-19-related information, and monitoring the health conditions of the general population in the Philippines. However, the limited end-user engagement in the features and feedback along the development cycle of mHealth technologies results in risks in adoption. The World Health Organization (WHO) recommends user-centric design and development of mHealth technologies to ensure responsiveness to the needs of the end users. The goal of the study is to understand, using end users' perspectives, the design and quality of mHealth technology implementations in the Philippines during the COVID-19 pandemic, with a focus on the areas identified by stakeholders: (1) utility, (2) technology readiness level, (3) design, (4) information, (5) usability, (6) features, and (7) security and privacy. Using a descriptive qualitative design, we conducted 5 interviews and 3 focus group discussions (FGDs) with a total of 16 participants (6, 37.5%, males and 10, 62.5%, females). Questions were based on the Mobile App Rating Scale (MARS). Using the cyclical coding approach, transcripts were analyzed with NVivo 12. Themes were identified. The qualitative analysis identified 18 themes that were organized under the 7 focus areas: (1) utility: use of mHealth technologies and motivations in using mHealth; (2) technology readiness: mobile technology literacy and user segmentation; (3) design: user interface design, language and content accessibility, and technology design; (4) information: accuracy of information and use of information; (5) usability: design factors, dependency on human processes, and technical issues; (6) features: interoperability and data integration, other feature and design recommendations, and technology features and upgrades; and (7) privacy and security: trust that mHealth can secure data, lack of information, and policies. To highlight, accessibility, privacy and security, a simple interface, and integration are some of the design and quality areas that end users find important and consider in using mHealth tools. Engaging end users in the development and design of mHealth technologies ensures adoption and accessibility, making it a valuable tool in curbing the pandemic. The 6 principles for developers, researchers, and implementers to consider when scaling up or developing a new mHealth solution in a low-resource setting are that it should (1) be driven by value in its implementation, (2) be inclusive, (3) address users' physical and cognitive restrictions, (4) ensure privacy and security, (5) be designed in accordance with digital health systems' standards, and (6) be trusted by end users.

Linear $$\ell $$-intersection pairs of cyclic and quasi-cyclic codes over a finite field $${\mathbb {F}}_q$$

Linear $$\ell $$-intersection pairs of cyclic and quasi-cyclic codes over a finite field $${\mathbb {F}}_q$$

Weight enumerators of some irreducible cyclic codes of odd length

Weight enumerators of some irreducible cyclic codes of odd length

Constructions of cyclic codes and extended primitive cyclic codes with their applications

Linear codes with a few weights have many nice applications including combinatorial design, distributed storage system, secret sharing schemes and so on. In this paper, we construct two families of linear codes with a few weights based on special polynomials over finite fields. The first family of linear codes are extended primitive cyclic codes which are affine-invariant. The second family of linear codes are reducible cyclic codes. The parameters of these codes and their duals are determined. As the first application, we prove that these two families of linear codes hold t-designs, where t=2,3. As the second application, the minimum localities of the codes are also determined and optimal locally recoverable codes are derived.

Several families of binary cyclic codes with good parameters

Binary odd-like duadic codes have parameters [n,(n+1)/2], where n is odd. It is well known that the minimum odd weight of odd-like duadic codes has the lower bound n. The binary quadratic-residue codes and the punctured binary Reed-Muller codes of certain order are two families of binary cyclic codes with parameters [n,(n+1)/2,d≥n]. It is very hard to construct an infinite family of binary cyclic codes with length n and dimension near (n+1)/2 whose minimum distances have a square-root bound. Recently, Tang and Ding constructed an infinite family of binary cyclic codes with parameters [2m−1,2m−1,d] whose minimum distances have lower bounds close to the square-root bound. The objective of this paper is to construct several infinite families of binary cyclic codes of length 2m−1 and dimension near 2m−1 whose minimum distance d much exceeds the square-root bound. For m≥9 being odd, an infinite family of binary cyclic codes with parameters [2m,2m−1,d≥3⋅2m−12−1] is presented. For m≥8 being even, an infinite family of binary cyclic codes with parameters [2m,2m−1−1+m2,d≥3⋅2m−22+1] is presented.

On the Construction of Quantum and LCD Codes from Cyclic Codes over the Finite Commutative Rings

Let Fq be a field of order q, where q is a power of an odd prime p, and α and β are two non-zero elements of Fq. The primary goal of this article is to study the structural properties of cyclic codes over a finite ring R=Fq[u1,u2]/⟨u12−α2,u22−β2,u1u2−u2u1⟩. We decompose the ring R by using orthogonal idempotents Δ1,Δ2,Δ3, and Δ4 as R=Δ1R⊕Δ2R⊕Δ3R⊕Δ4R, and to construct quantum-error-correcting (QEC) codes over R. As an application, we construct some optimal LCD codes.

Lee weight distributions of trace codes over F_q+uF_q from irreducible cyclic codes

The construction of two or few-weight codes from trace codes over the ring Fq + uFq, where u2 = 0, was recently presented in [4]. For such construction, the defining sets for the trace codes are given in terms of cyclotomic classes, and for some of these classes, it is shown that it is possible to obtain the Lee weight distributions of the corresponding trace codes. Motivated by this construction, and by the p-ary semiprimitive irreducible cyclic codes over a prime field Fp, the Lee weight distributions of an infinite family of p-ary three-weight codes from trace codes over the ring Fq + uFq, was recentely found in [11]. In this work, we prove that the Lee weight distribution problem for the trace codes constructed in accordance with either [4] or [11], is equivalent to the weight distribution problem for the irreducible cyclic codes. With this equivalence in mind, and by using the already known weight distributions of an infinite family irreducible cyclic codes (semiprimitive and not semiprimitive), we follow the open problem suggested in the Conclusion of [11] to determine the Lee weight distribution of an infinite family of trace codes over the ring Fq + uFq , that includes the infinite family found in [11].

A Bound on the Minimal Field Size of LRCs, and Cyclic MR Codes That Attain It

We prove a new lower bound on the field size of locally repairable codes (LRCs). Additionally, we construct maximally recoverable (MR) codes which are cyclic. While a known construction for MR codes has the same parameters, it produces non-cyclic codes. Furthermore, we prove both necessary conditions and sufficient conditions that specify when the known non-cyclic MR codes may be permuted to become cyclic, thus proving our construction produces cyclic MR codes with new parameters. Furthermore, using our new bound on the field size, we show that the new cyclic MR codes have optimal field size in certain cases. Other known LRCs are also shown to have optimal field size in certain cases.

Block-Wise Computation of Cyclic Redundancy Code Using Factored Toeplitz Matricesin Lieu of Look-Up Table

Cyclic Redundancy Check (CRC) or Cyclic Redundancy Code is a Cyclic Error Detection Code used to preserve the integrity of data in storage and transmission applications. CRC of a stream of message bits is usually calculated block-wise in parallel with the help of a Look-Up Table (LUT) in software or State Space transformation matrix in hardware. Presented here is a novel method and architecture of parallel computation of Cyclic Redundancy Codes without any Look-Up Table for an arbitrary generating polynomial which is programmable at runtime. The method reduces computational complexity and storage requirements by implicitly factorizing the transformation matrix needed to compute the remainder into two simpler Toeplitz matrices. The resulting hardware architecture is suitable for embedded applications.

New entanglement-assisted quantum MDS codes derived from cyclic codes

Entanglement-assisted quantum error-correcting codes, which can be seen as a generalization of quantum error-correcting codes, can be constructed from arbitrary classical linear codes by relaxing the self-orthogonality properties and using pre-shared entangled states between the sender and the receiver, and can also improve the performance of quantum error-correcting codes. In this paper, we construct some families of entanglement-assisted quantum maximum-distance-separable codes with parameters $[[\frac{{{q^2} - 1}}{a},\frac{{{q^2} - 1}}{a} - 2d+2 + c,d;c]]_q$, where $q$ is a prime power with the form $q = am \pm \ell$, $a = \frac{{\ell^2} - 1}{3}$ is an odd integer, $\ell \equiv 2\ (\bmod\ 6)$ or $\ell \equiv 4\ (\bmod\ 6)$, and $m$ is a positive integer. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature.

Infinite Families of Cyclic and Negacyclic Codes Supporting 3-Designs

Interplay between coding theory and combinatorial <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> -designs has been a hot topic for many years for combinatorialists and coding theorists. Some infinite families of cyclic codes supporting infinite families of 3-designs have been constructed in the past 50 years. However, no infinite family of negacyclic codes supporting an infinite family of 3-designs has been reported in the literature. This is the main motivation of this paper. Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q=p^{m}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> is an odd prime and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m \geq 2$ </tex-math></inline-formula> is an integer. The objective of this paper is to present an infinite family of cyclic codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(q)$ </tex-math></inline-formula> supporting an infinite family of 3-designs and two infinite families of negacyclic codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(q^{2})$ </tex-math></inline-formula> supporting two infinite families of 3-designs. The parameters and the weight distributions of these codes are determined. The subfield subcodes of these negacyclic codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(q)$ </tex-math></inline-formula> are studied. Three infinite families of almost MDS codes are also presented. A constacyclic code over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(4)$ </tex-math></inline-formula> supporting a 4-design and seven open problems are also presented in this paper.

Neural decoders with permutation invariant structure

Neural decoders were introduced as a generalization of the classic Belief Propagation (BP) decoding algorithms. In this work, we propose several neural decoders with different permutation invariant structures for BCH codes and punctured RM codes. Firstly, we propose the cyclically equivariant neural decoder which makes use of the cyclically invariant structure of these two codes. Next, we propose an affine equivariant neural decoder utilizing the affine invariant structure of those two codes. Both these two decoders outperform previous neural decoders when decoding cyclic codes. The affine decoder achieves a smaller decoding error probability than the cyclic decoder, but it usually requires a longer running time. Similar to using the property of the affine invariant property of extended BCH codes and RM codes, we propose the list decoding version of the cyclic decoder that can significantly reduce the frame error rate(FER) for these two codes. For certain high-rate codes, the gap between the list decoder and the Maximum Likelihood decoder is less than 0.1 dB when measured by FER.

Skew Constacyclic Codes over a Non-Chain Ring.

In this paper, we investigate the algebraic structure of the non-local ring Rq=Fq[v]/⟨v2+1⟩ and identify the automorphisms of this ring to study the algebraic structure of the skew constacyclic codes and their duals over this ring. Furthermore, we give a necessary and sufficient condition for the skew constacyclic codes over Rq to be linear complementary dual (LCD). We present some examples of Euclidean LCD codes over Rq and tabulate the parameters of Euclidean LCD codes over finite fields as the Φ-images of these codes over Rq, which are almost maximum distance separable (MDS) and near MDS. Eventually, by making use of Hermitian linear complementary duals of skew constacyclic codes over Rq and the map Φ, we give a class of entanglement-assisted quantum error correcting codes (EAQECCs) with maximal entanglement and tabulate parameters of some EAQECCs with maximal entanglement over finite fields.

Construction of Quantum Codes over the Class of Commutative Rings and Their Applications to DNA Codes

Let k,m be positive integers and F2m be a finite field of order 2m of characteristic 2. The primary goal of this paper is to study the structural properties of cyclic codes over the ring Sk=F2m[v1,v2,…,vk]⟨vi2−αivi,vivj−vjvi⟩, for i,j=1,2,3,…,k, where αi is the non-zero element of F2m. As an application, we obtain better quantum error correcting codes over the ring S1 (for k=1). Moreover, we acquire optimal linear codes with the help of the Gray image of cyclic codes. Finally, we present methods for reversible DNA codes.

Linear Codes and Linear Complementary Pairs of Codes Over a Non-Chain Ring

Let [Formula: see text] be an odd prime number, [Formula: see text] for a positive integer [Formula: see text], let [Formula: see text] be the finite field with [Formula: see text] elements and [Formula: see text] be a primitive element of [Formula: see text]. We first give an orthogonal decomposition of the ring [Formula: see text], where [Formula: see text] and [Formula: see text] for a fixed integer [Formula: see text]. In addition, Galois dual of a linear code over [Formula: see text] is discussed. Meanwhile, constacyclic codes and cyclic codes over the ring [Formula: see text] are investigated as well. Remarkably, we obtain that if linear codes [Formula: see text] and [Formula: see text] are a complementary pair, then the code [Formula: see text] and the dual code [Formula: see text] of [Formula: see text] are equivalent to each other.

Gray images of cyclic codes over [formula omitted] and [formula omitted

In the paper, we firstly study the algebraic structures of ZpZpk-additive cyclic codes and give the generator polynomials and the minimal spanning set of these codes. Secondly, a necessary and sufficient condition for the ZpZp2-additive cyclic code whose Gray image is linear (not necessarily cyclic) over Zp is given. Moreover, as for some special families of cyclic codes over Zp2 and ZpZp2, the linearity of the Gray images is determined.

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.