Linear Codes Research Articles

Nonlinear numerical simulation of insertion loss for structures having a compressed softened porous layer sandwiched by double walls under acoustic excitation

We perform nonlinear numerical simulation of insertion loss for structures having compressed porous layers sandwiched by double walls under acoustic excitation. The double walls are consisted of a steel base plate and a plastic cover plate. We compare the insertion loss including Urethane foam with felt. These porous layers are compressed by a heavy frame around the cover plate. As increasing weight of the frame, Urethane foam has soft-hardening characteristics in its restoring force. If we set appropriate weight of the frame to get 33% compression, lower rigidity is obtained for Urethane foam. This leads to softer porous layer for the double walls. And vibration and sound decrease from the base plate to the cover plate. For felt, the hardening characteristics appear. This leads to the higher rigidity and the more transmission. In the numerical calculation using LS-Dyna as a nonlinear FEM code, geometric nonlinearity is considered. For compressed porous materials, Storaker model is adopted in consideration of material nonlinearity with hysteresis. Further, under compression, we also investigate the reason of the softening mechanism for the Urethane foam using another model of microscopic structures. We reveal that the softening mechanism is due to geometric nonlinearity by using multiple Kelvin cells.

The second-order football-pool problem and the optimal rate of generalized-covering codes

The goal of the classic football-pool problem is to determine how many lottery tickets are to be bought in order to guarantee at least n−r correct guesses out of a sequence of n games played. We study a generalized (second-order) version of this problem, in which any of these n games consists of two sub-games. The second-order version of the football-pool problem is formulated using the notion of generalized-covering radius, recently proposed as a fundamental property of linear codes. We consider an extension of this property to general (not necessarily linear) codes, and provide an asymptotic solution to our problem by finding the optimal rate function of second-order covering codes given a fixed normalized covering radius. We also prove that the fraction of second-order covering codes among codes of sufficiently large rate tends to 1 as the code length tends to ∞.

Singleton-type bounds for list-decoding and list-recovery, and related results

List-decoding and list-recovery are important generalizations of unique decoding and receive considerable attention over the years. We study the optimal trade-off among the list-decoding (resp. list-recovery) radius, the list size, and the code rate, when the list size is constant and the alphabet size is large (both compared with the code length). We prove a new Singleton-type bound for list-decoding, which, for a wide range of parameters, is asymptotically tight up to a 1+o(1) factor. We also prove a Singleton-type bound for list-recovery, which is the first such bound in the literature. We apply these results to obtain near optimal lower bounds on the list size for list-decodable and list-recoverable codes with rates approaching capacity.Moreover, we show that under some indivisibility condition of the parameters and over a sufficiently large alphabet, the largest list-decodable nonlinear codes can have much more codewords than the largest list-decodable linear codes. Such a large gap is not known to exist in unique decoding. We prove this by a novel connection between list-decoding and the notion of sparse hypergraphs in extremal combinatorics.Lastly, we show that list-decodability or recoverability implies in some sense good unique decodability.

On generalized quasi-cyclic codes over [formula omitted

Based on good algebraic structures and practicabilities, generalized quasi-cyclic (GQC) codes play important roles in coding theory. In this paper, we study some results on GQC codes over Z4 including normalized generating sets, minimal generating sets and normalized generating sets of their dual codes. As an application, new Z4-linear codes and good binary nonlinear codes are constructed from GQC codes over Z4.

Near-MDS codes from maximal arcs in PG(2,q)

The singleton defect of an [n,k,d] linear code C is defined as s(C)=n−k+1−d. Codes with S(C)=0 are called maximum distance separable (MDS) codes, and codes with S(C)=S(C⊥)=1 are called near maximum distance separable (NMDS) codes. Both MDS codes and NMDS codes have good representations in finite projective geometry. MDS codes over Fq with length n and n-arcs in PG(k−1,q) are equivalent objects. When k=3, NMDS codes of length n are equivalent to (n,3)-arcs in PG(2,q). In this paper, we deal with the NMDS codes with dimension 3. By adding some suitable projective points in maximal arcs of PG(2,q), we can obtain two classes of (q+5,3)-arcs (or equivalently [q+5,3,q+2] NMDS codes) for any prime power q. We also determine the exact weight distribution and the locality of such NMDS codes and their duals. It turns out that the resultant NMDS codes and their duals are both distance-optimal and dimension-optimal locally recoverable codes.

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).