1,639 Results Found

The hull of a linear code C is the intersection of C with its dual code. The goal is to study the dimensions of the hulls of optimal binary and ternary linear codes for a given length and dimension. The focus is on the lengths at which self-orthogona...

This work studies projective self-dual (PSD) and self-polar linear codes over finite fields with q elements, where q is a power of a prime. The possible parameters for which PSD codes may exist are presented, and many examples are provided. Algorithm...

Sunflower in coding theory is a class of important subspace codes and can be used to construct linear codes. In this paper, we study the minimality of linear codes over ${\mathbb{F}}_q$ constructed from sunflowers of size s in all cases. For any sunflower, the cor...

Bounds for the parameters of codes are very important in coding theory. The Grey–Rankin bound refers to the cardinality of a self-complementary binary code. Codes meeting this bound are associated with families of two-weight codes and other com...

An algorithm for equivalence of linear codes over finite fields is presented. Its main advantage is that it can extract exactly one representative from each equivalence class among a large number of linear codes. It can also be used as a test for iso...

In this paper, we design and develop a new class of linear algebraic codes defined as soft linear algebraic codes using soft sets. The advantage of using these codes is that they have the ability to transmit m-distinct messages to m-set of receivers...

Nested linear coding is a widely used technique in wireless communication systems for improving both security and reliability. Some parameters, such as the relative generalized Hamming weight and the relative dimension/length profile, can be used to...

The optimal security parameters for binary linear complementary pair (LCP) codes are open. Inspired by this, we study LCPs of codes. In this paper, we construct two families of optimal binary LCPs of codes.

Linear codes with a few weights have been extensively studied due to their wide applications in secret sharing schemes, strongly regular graphs, association schemes, and authentication codes. In this paper, we choose the defining sets from two distin...

The algebraic analysis of linear code parameters reveals deep connections with cryptographic constructions, including the Information Dispersal Algorithms (IDAs) and secret-sharing schemes. In this work, we propose an algebraic method for constructin...

As far as taking-away of the symmetry constraints is concerned, as a scientifically symmetry problem, the global synthesis for antenna arrays that produce the desired radiation pattern is also a highly nonlinear optimization issue in fact. Besides th...

Aiming to solve the problem of the distinction of scrambled linear block codes, a method for identifying the scrambling types of linear block codes by combining correlation features and convolution long short-term memory neural networks is proposed i...

Linear codes are the most important family of codes in cryptography and coding theory. Some codes only have a few weights and are widely used in many areas, such as authentication codes, secret sharing schemes and strongly regular graphs. By setting...

We design a decoding algorithm for linear codes over finite chain rings given by their parity check matrices. It is assumed that decoding algorithms over the residue field are known at each degree of the adic decomposition.

The conventional soft decision decoding (SDD) methods require various hard decision decoders (HDDs) based on different codes or re-manipulate the generator matrix by the complicated Gaussian elimination technique according to the bit reliability. Thi...

In this paper, we propose a novel deep learning-based framework for the joint design and decoding of linear block codes, the end-to-end deep coding cross-attention message-passing Transformer (E2E DC-CrossMPT). To improve linear block code design and...

We present a parallel algorithm for computing the covering radius of a linear ${[n,k]}_q$ code using its parity-check matrix. The method is based on the systematic generation of syndromes associated with linear combinations of columns of the parity-check...

Fenestration systems play a critical role in building thermal performance, particularly in cooling-dominated climates where envelope inefficiencies directly amplify electricity demand. In Saudi Arabia and other Gulf Cooperation Council (GCC) countrie...

This paper presents a coding theorem for linear coding over finite rings, in the setting of the Slepian–Wolf source coding problem. This theorem covers corresponding achievability theorems of Elias (IRE Conv. Rec. 1955, 3, 37–46) and Csiszár (IEEE Tr...

In this paper, we put forward the model of multiple linear-combination security multicast network coding, where the wiretapper desires to obtain some information about a predefined set of multiple linear combinations of the source symbols by eavesdro...

Binary codes have a special place in coding theory since they are one of the most commonly used in practice. There are classes of codes specific only to the binary case. One such class is self-complementary codes. Self-complementary linear codes are...

In hybrid block-based video coding, transform plays an important role in energy compaction. Transform coding converts residual data in the spatial domain into frequency domain data, thereby concentrating energy in a lower frequency band. In VVC (vers...

The study of linear codes over local rings, particularly non-chain rings, imposes difficulties that differ from those encountered in codes over chain rings, and this stems from the fact that local non-chain rings are not principal ideal rings. In thi...

Compared with scalar linear network coding (LNC) formulated over the finite field GF($2^L$), vector LNC offers enhanced flexibility in the code design by enabling linear operations over the vector space $\mathrm{GF}{\left(2\right)}^L$ and demonstrates a number of advantages ove...

Suppose R is a local ring with invariants $p,n,r,m,k$ and $mr=4,$ that is R of order $p^4.$ Then, $R=R_0+u{R}_0+v{R}_0+w{R}_0$ has maximal ideal $J=u{R}_0+v{R}_0+w{R}_0$ of order $p^{(m-1)r}$ and a residue field F of order $p^r,$ where $R_0=GR(p^n,r)$ is the coefficient subring of $R.$ I...

Linear codes with complementary duals, or LCD codes, have recently been applied to side-channel and fault injection attack-resistant cryptographic countermeasures. We explain that over characteristic two fields, they exist whenever the permanent of a...

Motivated by the application of high-density data storage technologies, Cassuto and Blaum introduced codes for symbol-pair read channels in 2011, and Yaakobi et al. generalized the coding framework to that for b-symbol read channels where $b\ge 2$ in 2...

This paper presents a lossless image compression method with a fast decoding time and flexible adjustment of coder parameters affecting its implementation complexity. A comparison of several approaches for computing non-MMSE prediction coefficients w...

Recent works have combined random linear network coding (RLNC) with guessing random additive noise decoding (GRAND) to leverage RLNC packets to partially correct bit errors prior to RLNC decoding, so as to reduce the packet erasure rates in wireless...

In this paper, we explore the hull of linear codes over the non-unitary rings of order four, namely $\mathit{E}$, $\mathit{I}$ and $\mathit{H}$. Initially, we determine the binary associated codes of the hull over each ring; then, we characterize the hulls in terms of these codes. F...

To improve coding efficiency by exploiting the local inter-component redundancy between the luma and chroma components, the cross-component linear model (CCLM) is included in the versatile video coding (VVC) standard. In the CCLM mode, linear model p...

A modification of the Brouwer–Zimmermann algorithm for calculating the minimum weight of a linear code over a finite field is presented. The aim was to reduce the number of codewords for consideration. The reduction is significant in cases where the...

Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes, such as cyclic codes, Reed–Solomon codes, and Reed–Muller codes, have nice decoding algorithms....

Programming activities have the potential to provide a rich context for exploring measurement units in early elementary mathematics. This study examines how a small group of young children (ages 5–6) express their emergent conception of a dynam...

In this work, we explore the relationship between the graded free resolution of some monomial ideals and the Generalized Hamming Weights (GHWs) of binary codes. More precisely, we look for a structure that is smaller than the set of codewords of mini...

Many families of binary nonlinear codes (e.g., Kerdock, Goethals, Delsarte–Goethals, Preparata) can be very simply constructed from linear codes over the ${\mathbb{Z}}_4$ ring (ring of integers modulo 4), by applying the Gray map to the quaternary symbols. G...

In the era of the Internet of Things, there are many applications where numerous devices are deployed to acquire information and send it to analyse the data and make informed decisions. In these applications, the power consumption and price of the de...

Let $R={\mathbb{F}}_{p^m}[u,v]$ be the ring whose order is $p^{4m}$, where p denotes a prime number. The algebraic structure of R is substantially influenced by the relationships among the elements $u^2$, $uv$, and $v^2$. We establish that R is formed over ${\mathbb{F}}_{p^m}$ when $p\ne 2$ and $u^2=$...

We construct a family of linear optimal functional-repair regenerating storage codes with parameters $\{m,(n,k),(r,\alpha ,\beta \left)\right\}=\left\{\right(2r-\alpha +1)\alpha /2,(r+1,r),(r,\alpha ,1\left)\right\}$ for any integers $r,\alpha$ with $1\le \alpha \le r$, over any field...

The new generation video coding standard Versatile Video Coding (VVC) has adopted many novel technologies to improve compression performance, and consequently, remarkable results have been achieved. In practical applications, less data, in terms of b...

An exponential increase in mobile video delivery will continue with the demand for higher resolution, multi-view and large-scale multicast video services. Novel fifth generation (5G) 3GPP New Radio (NR) standard will bring a number of new opportuniti...

Human behavior recognition technology is widely adopted in intelligent surveillance, human–machine interaction, video retrieval, and ambient intelligence applications. To achieve efficient and accurate human behavior recognition, a unique appro...

This paper attempts to highlight the decoding capabilities of MDP convolutional codes over the erasure channel by defining them as discrete linear dynamical systems, with which the controllability property and the observability characteristics of lin...

For a finite field GF($p^L$) with prime p and $L>1$, one of the standard representations is $L\times L$ matrices over GF(p) so that the arithmetic of GF($p^L$) can be realized by the arithmetic among these matrices over GF(p). Based on the matrix representa...

Vehicle-to-vehicle (V2V) communication has gained significant attention in the field of intelligent transportation systems. In this paper, we focus on communication scenarios involving vehicles moving in the same and opposite directions. Specifically...

Suppose that $R={\mathbb{Z}}_{p^4}\left[u\right]$ with $u^2=p^3\beta$ and $pu=0,$ where p is a prime and $\beta$ is a unit in $R.$ Then, R is a local non-chain ring of order $p^5$ with a unique maximal ideal $J=(p,u)$ and a residue field of order $p.$ A linear code C of length N over R is an...

Monitoring the condition of rolling bearings is a crucial task in many industries. An efficient tool for diagnosing bearing defects is necessary since they can lead to complete machine failure and significant economic losses. Traditional diagnosis so...

A classical reduced order model (ROM) for dynamical problems typically involves only the spatial reduction of a given problem. Recently, a novel space–time ROM for linear dynamical problems has been developed [Choi et al., Space–tume reduced order mo...

We consider combinatorial multi-access coded caching with private caches, where users are connected to two types of caches: private caches and multi-access caches. Each user has its own private cache, while multi-access caches are connected in the sa...

The size of the largest binary single deletion code has been unknown for more than 50 years. It is known that Varshamov–Tenengolts (VT) code is an optimum single deletion code for block length $n\le 10$ ; however, only a few upper bounds...