Narrow Results

In this paper, we first generalize the class of linear codes by Ding and Ding [A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Trans. Inform. Theory61 (2015) 5835–5842, https://ieeexplore.ieee.org/document/7226827]. Then we mainly study the augmented codes of this generalized class of linear codes. For one thing, we use Gaussian sums to determine the parameters and weight distributions of the augmented codes in some cases. It is shown that the augmented codes are self-orthogonal and have only a few nonzero weights. For another thing, the locality of the augmented codes is proved to be 2, which indicates that the augmented codes are useful in distributed storage. Besides, the augmented codes are projective as the minimum distance of their duals is proved to be 3. In particular, we obtain several (almost) optimal linear codes and locally recoverable codes.

In this paper, we propose a new class of error-detecting codes based on quasigroups using automorphisms of the finite field ${𝔽}_{p^n}$ and the additive group $({𝔽}_p,+)$. We demonstrate that these codes are effective in detecting burst errors caused by hardware damage or noisy transmission. We also explore the codes ability to detect various types of double errors, including transposition errors, twin errors, and phonetic errors. Our analysis extends beyond bit-level errors to include block-level errors. Finally, we provide a comparative analysis and demonstrate the real-world applications of our code.

In this paper, we study $n$-MacWilliams right $R$-modules. It is shown that $(1)$ the MacWilliams extension property for $n=1$ implies that $M_R$ is an automorphism-invariant module (the converse holds if $M_R$ is finitely cogenerated), $(2)$ for $n=2$, the MacWilliams extension property implies that $M_R$ is quasi-injective which yields that if the matrix ring $M_n(R)$ over a right finitely cogenerated ring $R$ is right 1-MacWilliams, then $R$ is also right 1-MacWilliams. We also show that $(3)$ a right automorphism-invariant ring containing no infinite orthogonal sets of idempotents with essential right socle is right pseudo-Frobenius and $(4)$ if $R$ is a right $1$-MacWilliams ring and it has ACC on right annihilators, then $R$ is quasi-Frobenius which generalize answers the question of when is a MacWilliams ring quasi-Frobenius and when is a quasi-Frobenius ring a right artinian, right automorphism-invariant?

Let $p$ be an odd prime number, $q=p^m$ for a positive integer $m$, let ${𝔽}_q$ be the finite field with $q$ elements and $\omega$ be a primitive element of ${𝔽}_q$. We first give an orthogonal decomposition of the ring $R={𝔽}_q+\nu {𝔽}_q$, where ${\nu }^2=a^3,$ and $a={\omega }^{2l}$ for a fixed integer $l$. In addition, Galois dual of a linear code over $R$ is discussed. Meanwhile, constacyclic codes and cyclic codes over the ring $R$ are investigated as well. Remarkably, we obtain that if linear codes ${𝒞}$ and ${𝒟}$ are a complementary pair, then the code ${𝒞}$ and the dual code ${{𝒟}}^{{\perp }_E}$ of ${𝒟}$ are equivalent to each other.

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.

Let ${\mathbb{Z}}_4$ be the ring of integers modulo 4. We study the $\mathrm{\Lambda }$-constacyclic and $(\theta ,\mathrm{\Lambda })$-cyclic codes over the non-chain ring $R={\mathbb{Z}}_4[u,v]/⟨u^2=1,v^2=0,uv=vu=0⟩$ for a unit $\mathrm{\Lambda }=1+2u+2v$ in $R$. We define several Gray maps and find that the respective Gray images of a quasi-cyclic code over ${\mathbb{Z}}_4$ are cyclic, quasi-cyclic or permutation equivalent to this code. For an odd positive integer $n$, we determine the generator polynomials of cyclic and $\mathrm{\Lambda }$-constacyclic codes of length $n$ over $R$. Further, we prove that a $(\theta ,\mathrm{\Lambda })$-cyclic code of length $n$ is a $\mathrm{\Lambda }$-constacyclic code if $n$ is odd, and a $\mathrm{\Lambda }$-quasi-twisted code if $n$ is even. A few examples are also incorporated, in which two parameters are new and one is best known to date.

Recently, a number of researches studied relationship between codes and BCK-algebras, dealing with the category codes in a BCK-algebra have not been considered in earlier works. This paper investigates a code constructed by a BCK-algebra and also a BCK-algebra constructed based on code. The suggested rendered algorithm constructs the code based on BCK-algebra, the fixed dimension of this code is guaranteed by a BCK-algebra with some condition. We introduce the category of linear codes and denoted by $\mathrm{\text{LC}}$. In addition, a subcategory of BCK-algebra is described and denoted by ${\mathrm{\text{E}}}_{\mathrm{\text{BCK}}}$. Finally, we prove that $\mathrm{\text{LC}}$ is equivalent to ${\mathrm{\text{E}}}_{\mathrm{\text{BCK}}}$.

In this paper we will analyze one linear code from the theoretical point of view. Namely, the code definition is based on linear quasigroups. In the previous work we classified the quasigroups of order 4 according to their probability of undetected errors. Now, in this paper we will conclude whether the linear quasigroups that are in the same class in this classification obtain equal number of surely detected incorrectly transmitted bits. Also, we will classify the linear quasigroups of order 4 according to the number of errors that the code surely detects when they are used for coding. At the end we will make conclusion which quasigroups of order 4 are overall best for coding having in mind both important parameters for every code for error detection: the number of errors that the code surely detects and the probability of undetected errors.

Separating codes have been studied due to their applications to digital finger printing, the state assignments, automata theory and to construct hash functions. In this paper, we study the necessary and sufficient conditions for a code to be a $(3,1)$ and $(3,2)$-separating systems for q-ary level and also satisfy its intersecting properties. We also construct a bound for $(3,1)$ separating system.

In this paper, we study mixed alphabet codes over two alphabets $Z_2$, and its extension $Z_2[u]=Z_2+u{Z}_2$, $u^2=0$. We define new Gray maps, using which we study cyclic codes and obtain the properties of their generators. Further, we construct constacyclic codes, and linear codes, and observe the useful properties of their generators. Finally, we present some examples of cyclic codes and binary linear codes.

A binary $[n,k]$-linear code ${𝒞}$ is a $k$-dimensional subspace of ${𝔽}_2^n$. For $x\in {𝔽}_2^n$, the set $x+{𝒞}$ is a coset of ${𝒞}$. In this work, we study a partial ordering on the set of cosets of a binary linear code ${𝒞}$ of length $n$: for $x,y\in {𝔽}_2^n$, $x\preccurlyeq y$ provided that $supp(x)\subseteq supp(y)$, and we construct its associated Hasse diagram. We give general and particular examples that help to understand the concept and we obtain general properties of this graph.

Codes over integer rings equipped with the Lee metric are gaining attention among researchers recently. In this paper, we investigate some specific linear codes over integer rings and characterize their dimensions and Lee weights using greatest common divisors (GCD). Furthermore, we construct equidistant Lee codes over ${\mathbb{Z}}_{2^b},{\mathbb{Z}}_{2p}\text{and}{\mathbb{Z}}_p$ which have applications in error detection and error correction.

We consider the problem of computing the weight distribution of a linear code of dimension k over a composite finite field ${\mathbb{F}}_q$ where q = 2^m. Due to the trace function we reduce the arithmetic operations over the composite field to those over a prime field. This allows us to apply a transform of Walsh-Hadamard type. The codes are represented by their characteristic vector with respect to a given generator matrix and a generator matrix of the k-dimensional simplex code $S_{q:k}$. The developed algorithm has complexity O(kmq^k−1).

The equivalence test is a main part in any classification problem. It helps to prove bounds for the main parameters of the considered combinatorial structures and to study their properties. In this paper, we present algorithms for equivalence of linear codes, based on their relation to multisets of points in a projective geometry.

We present a generalization of Walsh-Hadamard transform that is suitable for applications in Coding Theory, especially for computation of the weight distribution and the covering radius of a linear code over a finite field. The transform used in our research, is a modification of the Vilenkin-Chrestenson transform. Instead of using all the vectors in the considered space, we take a maximal set of nonproportional vectors, which reduces the computational complexity.

This work explains the role of Möller algorithm and Gröbner technology in the description of linear codes. We survey several results of the authors about FGLM techniques applied to linear codes as well as some results concerning the structure of the code.

Linear codes over finite chain rings correspond to multisets of points in finite projective Hjelmslev geometries. This survey contains an introduction to both subjects, working out the links between coding theory and geometry over this class of rings.

We study algebraic coding theory, especially the theory of zeta functions for linear codes. This is a survey article of zeta functions for linear codes, with a brief introduction to coding theory.