Linear Codes Research Articles

DNA Code Design Based on the Cosets of Codes over Z4

DNA code design is a challenging problem, and it has received great attention in the literature due to its applications in DNA data storage, DNA origami, and DNA computing. The primary focus of this paper is in constructing new DNA codes using the cosets of linear codes over the ring Z4. The Hamming distance constraint, GC-content constraint, and homopolymers constraint are all considered. In this study, we consider the cosets of Simplex alpha code, Kerdock code, Preparata code, and Hadamard code. New DNA codes of lengths four, eight, sixteen, and thirty-two are constructed using a combination of an algebraic coding approach and a variable neighborhood search approach. In addition, good lower bounds for DNA codes that satisfy important constraints have been successfully established using Magma software V2.24-4 and Python 3.10 programming in our comprehensive methodology.

Experimental and Numerical Investigations on the Dynamic Response of Blades with Dual Friction Dampers

Friction dampers are widely employed to reduce blade resonance vibration amplitude in turbomachinery. In this paper, a study was performed on the forced response of two blades with dual friction dampers. Numerical simulation and experimental testing were conducted. Firstly, the dynamics of the blade and dual friction damper system assembly are modeled. A nonlinear code based on the multi-harmonic balance method was developed to calculate the resonance response. In this analysis, both the blade and the damper are modeled with the finite element and the matrices reduced with the component mode synthesis method, while the contact forces are modeled with a one-dimensional variable normal load array element. Secondly, a test rig made of two blades and dual friction dampers, the material of which was steel, was established to measure the nonlinear frequency response function curves of the blade system. The results indicate that when a dual friction damper is applied, superior vibration reduction characteristics are demonstrated, with the system exhibiting an average 21% reduction in the response amplitude levels and an increase of 3% in the frequency shifting range compared to a single damper. Dampers positioned at relatively higher locations contribute significantly to the vibration reduction process. In the end, the numerical predictions match very well with the experimental ones.

The Homogeneous Gray image of linear codes over the Galois ring GR(4, m)

The Homogeneous Gray image of linear codes over the Galois ring GR(4, m)

High capacity, secure audio watermarking technique integrating spread spectrum and linear predictive coding

High capacity, secure audio watermarking technique integrating spread spectrum and linear predictive coding

Complexity of The Systems of Linear Restrictions over a Finite Field

This paper continues the results obtained in [1]. In the previous paper, we formulated the problem of the unknown vector recovering from linear dependencies with this vector, which act as constraints on it. The next step, after finding out some algebraic and combinatorial properties, is to give basic estimates of complexity for the main problem as well as for related problems. Such related problems can be obtained by fixing some parameters of the main problem or applying constraints on the number of restrictions in the system. Such an analysis makes possible to arrange the problem of recovering an unknown vector based on partial information into the general computational complexity framework in order to approach existing theoretical results to its solution. The obtained theoretical results can be used in algebraic cryptanalysis of stream ciphers and cryptosystems based on linear codes.

A class of t-weight codes and its applications

In this paper, we constructed a class of [Formula: see text]-weight linear codes over [Formula: see text] under the homogeneous weight metric by their generator matrices, where [Formula: see text] and [Formula: see text] The Gray images of some class of these codes over [Formula: see text] are [Formula: see text]-ary nonlinear codes, which have the same weight distributions as that of the two-weight [Formula: see text]-ary linear codes of type SU1 in the sense of [R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc. 18(2) (1986) 97–122]. Also, we obtained the minimum distance of the dual codes of the constructed codes. Further, we discussed some optimal linear codes over [Formula: see text] with respect to Plotkin-type bound from the constructed codes when [Formula: see text] Furthermore, we investigated the applications in strongly regular graphs and secret sharing schemes.

Some Quaternary Additive Codes Outperform Linear Counterparts

The additive codes may have better parameters than linear codes. However, it is still a challenging problem to efficiently construct additive codes that outperform linear codes, especially those with greater distances than linear codes of the same lengths and dimensions. This paper focuses on constructing additive codes that outperform linear codes based on quasi-cyclic codes and combinatorial methods. Firstly, we propose a lower bound on the symplectic distance of 1-generator quasi-cyclic codes of index even. Secondly, we get many binary quasi-cyclic codes with large symplectic distances utilizing computer-supported combination and search methods, all of which correspond to good quaternary additive codes. Notably, some additive codes have greater distances than best-known quaternary linear codes in Grassl’s code table (bounds on the minimum distance of quaternary linear codes http://www.codetables.de) for the same lengths and dimensions. Moreover, employing a combinatorial approach, we partially determine the parameters of optimal quaternary additive 3.5-dimensional codes with lengths from 28 to 254. Finally, as an extension, we also construct some good additive complementary dual codes with larger distances than the best-known quaternary linear complementary dual codes in the literature.

Run Length Limited Error Control Codes Derived from Reed Solomon Codes

Reed Solomon codes were standardized for numerous wireless communication systems. Most practical Reed Solomon codes belong to non-binary linear block codes defined over finite fields with characteristic two. Each linear block code contains one codeword composed of all zeros. The concatenation of this and also other codewords can lead to long, theoretically even infinite, runs of equal symbols. Such long runs do not support synchronization in wireless receivers and therefore are unwanted. In this paper it is shown that extended and some appropriately shortened Reed Solomon codes constructed over finite fields with characteristic two can be transformed into Run Length Limited Reed Solomon codes. The presented method, if applicable, allows for doing it without inserting additional redundancy. Another advantage is that after the transformation, if some round conditions are fulfilled, the decoding does not have to be rebuilt.

A Simple Family of Non-Linear Analog Codes

A Simple Family of Non-Linear Analog Codes

Designing an ultra-efficient Hamming code generator circuit for a secure nano-telecommunication network

Communication links forming secure telecommunications networks rely on various technologies such as message switching, circuit switching, or packet switching to transmit messages and data. Hamming codes, a family of linear error-correcting codes, are commonly used in communication networks to detect and correct one-bit and two-bit errors. However, reducing power consumption, occupied area, and latency in secure telecommunication networks remains a challenge for future information and communication technology. To address these challenges, emerging technologies like quantum dots offer potential solutions. Quantum-dot cellular automata (QCA) stands as a promising frontier in nanotechnology for enhancing secure telecommunications networks. It opens up the possibility of crafting high-performance, energy-efficient digital circuits. This research harnesses the potential of QCA and introduces groundbreaking innovations: a 3-8 decoder employing a single-layer layout and a 3-input XOR gate with a multi-layer configuration. These components are utilized in the design of an electronic circuit for Hamming codes, incorporating the QCA-based approach. It is important to note that practical implementation in real-world scenarios presents challenges due to the nature of QCA technology. As a result, the evaluation and validation of the proposed designs heavily rely on simulations using QCADesigner. While experimental validation in real-world scenarios is limited, the simulations provide insights into the functionality and feasibility of the suggested designs. By leveraging QCA, the proposed Hamming code circuit significantly enhances cell count, occupied area, and clock latency. The suggested design can be adapted to fit different generating matrices in Hamming codes without requiring drastic modifications to the underlying architecture.

Minimal linear codes derived from weakly regular bent and plateaued functions

The construction of (minimal) linear codes from functions has received much attention in the literature. In this paper, we derive several minimal codes from the sets of pre-images of weakly regular plateaued and bent functions over the odd characteristic finite fields. Based on the recent construction method, we obtain six-weight and seven-weight minimal codes from these functions. The Hamming weights in proposed codes are calculated from the character sums of the sets based on the Walsh spectrum of the employed functions, while the weight distributions of the codes are determined from both the sizes of the employed sets and the Walsh distributions of the employed functions.

Linear codes invariant under cyclic endomorphisms

In this paper, we study linear codes invariant under a cyclic endomorphism [Formula: see text] called [Formula: see text]-cyclic codes. Since every cyclic endomorphism can be represented by a cyclic matrix with respect to a given basis, all of these matrices are similar. For simplicity we restrict ourselves to linear codes invariant under the right multiplication by a cyclic matrix [Formula: see text] that we call [Formula: see text]-cyclic codes, and when [Formula: see text] is the companion matrix [Formula: see text] of a given nonzero polynomial [Formula: see text] we call them [Formula: see text]-cyclic codes. The similarity relation between matrices helps us find connections between [Formula: see text]-cyclic codes and [Formula: see text]-cyclic codes, where [Formula: see text] is the minimal polynomial of [Formula: see text] The class of [Formula: see text]-cyclic codes contains cyclic codes and their various generalizations such as constacyclic codes, right and left polycyclic codes, monomial codes, and others. As common in the study of cyclic codes and their generalizations, we make use of the one-to-one correspondence between [Formula: see text]-cyclic codes and ideals of the polynomial ring [Formula: see text] where [Formula: see text] is the minimal polynomial of [Formula: see text] This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials among others. Next, we study the duality of these codes, where we show that the [Formula: see text]-dual of an [Formula: see text]-cyclic code is an [Formula: see text]-cyclic code, where [Formula: see text] is the adjoint matrix of [Formula: see text] with respect to [Formula: see text] and we explore some important results on the duality of these codes. Finally, we give examples as applications of some of the results and we construct some optimal codes.

Linear codes and cyclic codes over finite rings and their generalizations: a survey

Linear codes and cyclic codes over finite rings and their generalizations: a survey

Linear codes of larger lengths with Galois hulls of arbitrary dimensions and related entanglement-assisted quantum error-correcting codes

Galois hulls are generalizations of Euclidean and Hermitian hulls, which have attracted interest because of their important applications in determining the complexity of some algorithms about linear codes and in constructing entanglement-assisted quantum error-correcting codes (EAQECCs). In this paper, the whole contribution is two-folded. On one hand, we propose explicit methods to construct linear codes of larger lengths with Galois hulls of arbitrary dimensions from given Galois self-orthogonal codes. Conditions required in our approach are proven to be relatively weak. On the other hand, we apply these results to construct EAQECCs. Two bounds for EAQECCs constructed from linear codes with prescribed dimensional Galois hull are given and EAQECCs with rates greater than or equal to 12 and positive net rates can be obtained. We also present many interesting examples to explain visually how these two aspects work.

New Results and Bounds on codes over GF(17)

Determining the best possible values of the parameters of a linear code is one of the most fundamental and challenging problems in coding theory. There exist databases of best-known linear codes (BKLC) over small finite fields. In this work, we establish a database of BKLCs over the field GF(17) together with upper bounds on the minimum distances for lengths up to 150 and dimensions up to 6. In the process, we have found many new linear codes over GF(17).

Optimization Objective Function Corona Discharge Acoustic Using Fuzzy c-Means (FcM )

In many electrical networks in Indonesia, insulation failure due to high voltage phenomena like Corona Discharge (CD) still happens. This is a result of our inability to perform early Corona Discharge (CD) identification. This study’s objective is to optimalize the sound properties of Corona Discharge (CD) as a first step throught the early identification of insulation failure in the form of clustering 20 kV cubicle. Based on observations on the needle-rod electrode 3 cm apart, the smallest breakdown was obtained at 34.3 kV. So that the classification of CD sound by 3 clusters starting 20 kV cubicle voltage until before the failure occurs on 33 kV. The temperature in the cubical is between 27.5℃ - 35.3℃ and humidity ranges from 70% - 95%. It was stated in the study that the FcM method was the most widely used and successful method. In this case, FcM can obtain more flexible results that classify data into clusters easily. This research will be carried out using the Fuzzy c-Means (FcM) method. Feature extraction with linear predictive coding (LPC) method, then optimization by using the Fuzzy c-Means (FcM) method which is expected to be used as an initial step for early detection of insulation failure.

New good large (n,r)-arcs in PG(2,29) and PG(2,31)

An $(n, r)$-arc is a set of $n$ points of a projective plane such that some $r$, but no $r+1$ of them, are collinear. The maximumsize of an $(n, r)$-arc in $\PG(2,q)$ is denoted by $m_r(2,q)$. In this article a $(477, 18)$-arc, a $(596,22)$-arc, a $(697,25)$-arc in PG(2,29) and a $(598, 21)$-arc, a $(664, 23)$-arc, a $(699, 24)$-arc, a $(769, 26)$-arc, a $(838,28)$-arc in PG(2,31) are presented. The constructed arcs improve the respective lower bounds on $m_r(2,29)$ and $m_r(2,31)$ in \cite{MB2019}. As a consequence there exist eight new three-dimensional linear codes over the respective finite fields.\\

Partial permutation decoding and PD-sets for [formula omitted]-linear generalized Hadamard codes

It is known that Zps-linear codes, which are the Gray map image of Zps-additive codes (linear codes over Zps), are systematic and a systematic encoding has been found. This makes Zps-linear codes suitable to apply the permutation decoding method. This technique is also based on the existence of r-PD-sets, which are subsets of the permutation automorphism group of the code. In this paper, we study the permutation automorphism group of Zps-linear generalized Hadamard codes of type (n;t1,…,ts) and show how to construct r-PD-sets of size r+1, for all r up to an upper bound, in order to be able to perform a partial permutation decoding for these codes.

Optimal quaternary linear codes with one-dimensional Hermitian hull and related EAQECCs

Optimal quaternary linear codes with one-dimensional Hermitian hull and related EAQECCs

Column cyclic rank metric codes and linear complementary dual rank metric codes

We introduce and study column cyclic rank metric (CCRM) codes over finite fields. We present natural questions for the existence and characterization of CCRM codes. We completely solve these questions only in the first nontrivial case: The case of the minimum rank distance [Formula: see text] and the number of rows [Formula: see text]. We also consider linear complementary dual rank metric (LCDRM) codes both in the setting of CCRM codes and in the general setting of rank metric codes. We also ask the natural questions for the existence and characterization of CCRM codes in the subclass of LCDRM codes. We also completely solve these questions for this subclass of LCDRM codes only in the first nontrivial case that [Formula: see text] and [Formula: see text]. Moreover we extend a characterization of linear complementary dual (LCD) codes of Massey to arbitrary rank metric codes in the general setting.