Binary Duadic Codes with Square-Root-Like Lower Bounds on Minimum Distance

Core Concepts

Several families of binary duadic codes with length 2^m - 1 and dimension 2^(m-1) are presented, where the minimum distances have a square-root-like lower bound.

Abstract

The paper presents several families of binary duadic codes with length 2^m - 1 and dimension 2^(m-1), where the minimum distances have a square-root-like lower bound. Key highlights: The binary cyclic codes C[r,m,S] and C[r,m,S'] are constructed, where r is an even integer, m is an odd integer, and S is a subset of Z_r with |S| = r/2. It is shown that C[r,m,S] and C[r,m,S'] form a pair of odd-like duadic codes with parameters [2^m - 1, 2^(m-1), d], where the lower bound on the minimum distance d is close to the square-root bound. The parameters of the dual codes and extended codes of these binary duadic codes are also provided. The extended codes are self-dual and doubly-even. An example is given for the case r = 8, where all possible binary duadic codes are constructed and their parameters are analyzed.

Stats

The paper provides the following key metrics and figures: The length n = 2^m - 1 and dimension k = 2^(m-1) of the binary duadic codes C[r,m,S] and C[r,m,S']. The lower bounds on the minimum distances d of the duadic codes, which are close to the square-root bound √n. The parameters of the dual codes C⊥[r,m,S] and C⊥[r,m,S'], as well as the extended codes C[r,m,S] and C[r,m,S'].

Quotes

"By using the BCH bound on cyclic codes, one of the open problems proposed by Liu et al. about binary cyclic codes (Finite Field Appl 91:102270, 2023) is settled." "The extended codes C[r,m,S] and C[r,m,S'] are self-dual and doubly-even."

Key Insights Distilled From

Binary duadic codes and their related codes with a square-root-like lower bound

by Tingting Wu,... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.18053.pdf

Deeper Inquiries

How can the construction of these binary duadic codes be generalized to other finite fields beyond F_2

The construction of binary duadic codes can be generalized to other finite fields beyond F_2 by adapting the techniques used in the paper to the specific characteristics of the new finite field. For example, in a finite field F_q where q is a prime power, the construction of duadic codes can be based on the properties of the field elements and the corresponding primitive elements. The generator polynomials and defining sets of the duadic codes would need to be defined in a way that aligns with the structure of the finite field. Additionally, the minimum distance bounds and properties of the dual and extended codes would need to be re-evaluated based on the characteristics of the new finite field.

What are the potential applications of these binary duadic codes with good minimum distances in communication systems or data storage

The binary duadic codes constructed in the paper with good minimum distances have various potential applications in communication systems and data storage. In communication systems, these codes can be used for error detection and correction, ensuring reliable transmission of data over noisy channels. The good minimum distances of these codes indicate their ability to detect and correct a higher number of errors, making them valuable in scenarios where data integrity is crucial. In data storage systems, these codes can be utilized for efficient encoding and decoding, improving the reliability and robustness of stored data. Overall, the applications of these binary duadic codes extend to any system where error detection and correction are essential for maintaining data integrity.

Can the techniques used in this paper be extended to construct good cyclic codes over other alphabets, such as ternary or quaternary codes

The techniques used in the paper to construct binary duadic codes can be extended to construct good cyclic codes over other alphabets, such as ternary or quaternary codes. By adapting the construction methods and properties of duadic codes to the specific characteristics of ternary or quaternary alphabets, it is possible to create cyclic codes with desirable minimum distances and other important parameters. The generator polynomials, defining sets, and minimum distance bounds would need to be redefined based on the properties of the new alphabets. Additionally, the concepts of dual codes and extended codes can be applied to ternary or quaternary cyclic codes, providing further insights into their properties and applications in various systems.