Article
Quantum Science & Technology
Yanqing Dai, Xiusheng Liu
Summary: This paper proposes two new methods for constructing QSCs and provides concrete examples. These methods can effectively correct the effects of quantum noise on qubits and misalignment in block synchronization.
QUANTUM INFORMATION PROCESSING
(2022)
Article
Computer Science, Theory & Methods
Jian Gao, Xiangrui Meng, Fang-Wei Fu
Summary: This paper determines the weight distribution of several classes of double cyclic codes over Galois rings using Gauss sums.
DESIGNS CODES AND CRYPTOGRAPHY
(2022)
Article
Mathematics, Applied
Jon-Lark Kim, Dong Eun Ohk
Summary: This paper presents a new type of DNA codes over noncommutative rings E and F, based on quasi self-dual codes. By utilizing quasi self-duality, fixed GC-content constraint weight distributions and reverse-complement constraint minimum distributions of those codes can be described.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2022)
Article
Mathematics, Applied
Xiaoxiao Li, Minjia Shi
Summary: In this paper, we construct several infinite families of codes over the chain ring R = F-2[u]/< u(k)>, and compute the homogeneous weight distributions of these codes when simplicial complexes are generated by a single maximal element. Through the Gray map, it is determined that some codes are minimal while others are distance optimal. These codes have minimal codewords for inclusion of supports, making them suitable for secret sharing schemes.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2022)
Article
Computer Science, Theory & Methods
Varsha Chauhan, Anuradha Sharma, Sandeep Sharma, Monika Yadav
Summary: This paper investigates the Hamming weights and weight distributions of multi-twisted codes, identifying optimal equidistant linear codes with connections to combinatorial designs and other useful minimal linear codes. Illustrative examples and listings of various linear codes are provided for further understanding.
DESIGNS CODES AND CRYPTOGRAPHY
(2021)
Article
Mathematics
Minjia Shi, Michael Kiermaier, Sascha Kurz, Patrick Sole
Summary: This study constructs strongly walk-regular graphs by building coset graphs of the duals of codes. The weight distribution is classified, and necessary conditions on the weight distribution are derived in a special case, leading to the construction of infinite examples. As a byproduct, alternative proof for the nonlinearity of the Kerdock code is also presented.
GRAPHS AND COMBINATORICS
(2022)
Article
Mathematics
S. A. Katre, Kshipra Wadikar
Summary: The paper investigates necessary and sufficient trace conditions for matrices over noncommutative rings to be sums of kth powers, extending results from commutative rings. It deduces Vaserstein's result for sum of squares of matrices and provides nice trace conditions for sum of cubes. Additionally, it gives a sufficient condition for an n x n matrix over a noncommutative ring to be a sum of kth powers.
LINEAR & MULTILINEAR ALGEBRA
(2021)
Article
Mathematics
S. A. Katre, Deepa Krishnamurthi
Summary: In this paper, we investigate the conditions under which a matrix over a non-commutative ring can be expressed as a sum of powers. We prove that a matrix is the sum of pth powers if and only if its trace can be written as a sum of pth powers and commutators modulo pR. We also provide necessary and sufficient conditions for a matrix to be written as a sum of fourth powers when n is greater than or equal to 2.
LINEAR & MULTILINEAR ALGEBRA
(2022)
Article
Mathematics, Applied
Adel Alahmadi, Asmaa Melaibari, Patrick Sole
Summary: This paper presents two methods for constructing linear codes over the rings E and I using the adjacency matrices of three-class association schemes. The constructions yield QSD or Type IV codes under certain conditions, and many codes with minimum distance exceeding 4 are presented. The form of the generator matrices of the codes with these constructions prompted some new results on free codes over E and I.
Article
Mathematics, Applied
Shitao Li, Minjia Shi
Summary: In this paper, we construct two infinite families of new two-weight codes over Z(2m) by their generator matrices, which generalize the previous results. We also construct some optimal codes and prove that all codes in one of the families are self-orthogonal. Finally, we determine the linearity of the Gray images of the codes constructed for Lee metric.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)
Article
Computer Science, Theory & Methods
Xiangrui Meng, Jian Gao, Fang-Wei Fu, Fanghui Ma
Summary: In this study, the Hamming weight distribution of Q2DC codes is determined using the trace representation and Gauss sums, and some classes of Q2DC codes satisfying certain conditions are constructed.
DESIGNS CODES AND CRYPTOGRAPHY
(2023)
Article
Computer Science, Theory & Methods
Hongming Ru, Chunming Tang, Yanfeng Qi, Yuxiao Deng
Summary: This paper explores the construction of linear codes with two or three weights using three different defining sets, and determines the weight distributions of these codes, leading to optimal codes that meet certain bounds.
ADVANCES IN MATHEMATICS OF COMMUNICATIONS
(2021)
Article
Mathematics, Applied
Minjia Shi, Xiaoxiao Li
Summary: In this paper, we construct new families of codes using simplicial complexes and compute their Lee weight distributions. Through the Gray map, we demonstrate that some of these codes are minimal and distance optimal.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Mathematics, Applied
Abdulaziz Deajim, Mohamed Bouye, Kenza Guenda
Summary: This article explores the properties of matrix-product codes given a commutative ring R and matrix A, presenting various sufficient and necessary conditions for the formation of complementary dual (LCD) codes. Applications of LCD matrix-product codes from torsion codes over finite chain rings are considered, along with the demonstration of the existence of asymptotically good sequences of such codes over these rings.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2021)
Article
Mathematics, Applied
Ridhima Thakral, Sucheta Dutt, Ranjeet Sehmi
Summary: This paper investigates non-trivial linear complementary pairs of lambda-constacyclic codes over finite commutative chain rings, and derives an expression for the total number of such pairs. Additionally, a complete characterization of non-trivial pairs is obtained for finite chain rings with nilpotency index 2, using the algebraic structure of lambda-constacyclic codes.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2023)