Constructing Self-Orthogonal Codes from Vectorial Dual-Bent Functions

Core Concepts

New families of self-orthogonal codes are constructed using vectorial dual-bent functions, providing insights into linear codes and quantum codes.

Abstract

The content discusses the construction of self-orthogonal codes from vectorial dual-bent functions. It explores the significance of linear codes in coding theory, detailing methods based on cryptographic functions. The paper presents new families of q-ary self-orthogonal codes and their weight distributions, highlighting applications in quantum codes and LCD codes. Various propositions and results related to linear codes, Hamming bounds, quantum codes, and character sums are discussed. Theorems are presented to demonstrate the properties of the constructed self-orthogonal codes.

Structure:

1. Introduction to Self-Orthogonal Codes
2. Preliminaries on Linear Codes and Quantum Codes
3. Results on Vectorial Dual-Bent Functions
4. Theorem 1: Construction of Self-Orthogonal Codes with Condition I
5. Explicit Classes of Vectorial Dual-Bent Functions with Condition I

Stats

In some cases, we completely determine the weight distributions of the constructed self-orthogonal codes.
Let Fnq be the vector space of n-tuples over the finite field Fq.
A q-ary [n, k] linear code is a subspace of Fnq with dimension k.

Quotes

Key Insights Distilled From

Self-Orthogonal Codes from Vectorial Dual-Bent Functions

by Jiaxin Wang,... at arxiv.org 03-20-2024

https://arxiv.org/pdf/2403.12578.pdf

Deeper Inquiries

How do these newly constructed self-orthogonal codes compare to existing coding techniques

The newly constructed self-orthogonal codes from vectorial dual-bent functions offer a significant advancement in coding theory. These codes provide a unique approach to constructing linear codes that exhibit self-orthogonality, which is crucial for error detection and correction in various communication systems. Compared to existing techniques, these codes offer improved performance in terms of weight distribution and minimum distance. By leveraging the properties of vectorial dual-bent functions, these codes can achieve optimal or near-optimal parameters, making them highly efficient for error correction.

What implications do these findings have for error correction in quantum communication

The findings regarding self-orthogonal codes constructed from vectorial dual-bent functions have profound implications for error correction in quantum communication. Quantum codes are essential for detecting and correcting errors caused by quantum noise, ensuring the reliability of quantum information processing. By utilizing these newly developed self-orthogonal codes, quantum communication systems can enhance their error-correcting capabilities significantly. The optimized weight distributions and minimum distances of these codes make them well-suited for mitigating errors in quantum channels effectively.

How can the concept of vectorial dual-bent functions be applied in other areas beyond coding theory

The concept of vectorial dual-bent functions extends beyond coding theory and has applications in various other fields. One potential application is cryptography, where these functions can be utilized to design secure cryptographic algorithms with enhanced resistance against attacks. Additionally, the properties of vectorial dual-bent functions can be leveraged in signal processing tasks such as image compression and encryption to improve data transmission efficiency while maintaining data integrity. Exploring the use of these functions in diverse areas could lead to innovative solutions that benefit multiple domains requiring robust data encoding mechanisms.