Related Experiment Video
Updated: Jan 16, 2026
05:39
Generating Strictly Controlled Stimuli for Figure Recognition Experiments
Published on: March 18, 2019
5.5K
Symplectic QSD, LCD, and ACD Codes over a Non-Commutative Non-Unitary Ring of Order Nine
Sarra Manseri1, Patrick Solé2, Adel Alahmadi3
1Department of Mathematics, Central China Normal University, Wuhan 430079, China.
Entropy (Basel, Switzerland)
|September 27, 2025
Summary
New codes, including quasi self-dual (QSD) and linear complementary dual (LCD) codes, are introduced for symplectic inner products over a specific ring. Connections to existing codes and characterizations of right-symplectic additive complementary dual (ACD) codes are established.
Area of Science:
- Coding theory
- Abstract algebra
- Finite fields
Background:
- The study of error-correcting codes is crucial in digital communication and data storage.
- Exploring codes over non-commutative rings offers a generalization beyond traditional fields.
- Symplectic structures are important in various areas of mathematics and physics.
Purpose of the Study:
- To introduce and define novel code families: quasi self-dual (QSD), linear complementary dual (LCD), and additive complementary dual (ACD) codes.
- To investigate these codes specifically for the symplectic inner product.
- To explore these codes over a non-commutative non-unitary ring of order 9.
Main Methods:
- Definition of QSD, LCD, and ACD codes in the context of the symplectic inner product.
- Utilizing properties of the non-commutative non-unitary ring of order 9.
- Establishing theoretical connections and performing characterizations of the defined codes.
Main Results:
- Introduction of QSD, LCD, and ACD codes over the specified ring.
- Demonstration of connections between these new codes and existing symplectic-self-orthogonal and LCD ternary codes.
- Characterization of right-symplectic ACD codes over the ring.
Conclusions:
- The introduced QSD, LCD, and ACD codes provide new structures for coding theory over rings.
- The established connections highlight the interplay between different types of codes and algebraic structures.
- The characterization of right-symplectic ACD codes advances the understanding of these codes in non-commutative settings.
Related Concept Videos
01:28Synthetic Disvision of Polynomials
149
Synthetic division is an efficient algorithmic approach for dividing a polynomial by a linear binomial of the form x - c, where c is a real number. This method is helpful due to its streamlined process, which avoids the more cumbersome steps involved in the traditional long division of polynomials. It simplifies computation and serves as a practical tool for evaluating polynomials and identifying their factors.To perform synthetic division, one begins by listing the coefficients of the...
149
01:29Quadratic Equations in the Complex Number System
395
A quadratic equation in the form ax2+bx+c=0 can have solutions that vary in nature depending on the value of the discriminant, b2−4ac. In this expression, a is the coefficient of the quadratic term x2, b is the coefficient of the linear term x, and c is the constant term. When the discriminant is negative, the equation has no real number solutions. However, by introducing complex numbers through the imaginary unit i, defined by i=-1, these equations can still be solved.The square root of...
395
01:28Cartesian Vector Notation
1.4K
Cartesian vector notation is a valuable tool in mechanical engineering for representing vectors in three-dimensional space, performing vector operations such as determining the gradient, divergence, and curl, and expressing physical quantities such as the displacement, velocity, acceleration, and force. By using Cartesian vector notation, engineers can more easily analyze and solve problems in various areas of mechanical engineering, including dynamics, kinematics, and fluid mechanics. This...
1.4K
01:34Woodward–Hoffmann Selection Rules and Microscopic Reversibility
3.8K
Electrocyclic reactions, cycloadditions, and sigmatropic rearrangements are concerted pericyclic reactions that proceed via a cyclic transition state. These reactions are stereospecific and regioselective. The stereochemistry of the products depends on the symmetry characteristics of the interacting orbitals and the reaction conditions. Accordingly, pericyclic reactions are classified as either symmetry-allowed or symmetry-forbidden. Woodward and Hoffmann presented the selection criteria for...
3.8K
01:16SFG Algebra
316
In Signal Flow Graph (SFG) algebra, the value a node represents is determined by the sum of all signals entering that node. This summed value is then transmitted through every branch leaving the node, making the SFG a powerful tool for visualizing and analyzing control systems.
Each node in an SFG corresponds to a variable, and the interactions between nodes are represented by branches with associated gains. When multiple branches lead into a node, the value at that node is the sum of the...
Each node in an SFG corresponds to a variable, and the interactions between nodes are represented by branches with associated gains. When multiple branches lead into a node, the value at that node is the sum of the...
316
01:29Complex Zeros
240
Complex zeros are the solutions to polynomial equations that include imaginary numbers, specifically, numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit defined by i2=-1. These zeros satisfy the equation P(x) = 0, where P(x) is a polynomial with real or complex coefficients. Since the complex number system includes all real numbers, it provides a complete framework for analyzing all possible roots of a polynomial.Every polynomial of degree n≥1 can be...
240