On the Hamming Weight Functions of Linear Codes

2025-11-22

AI Summary

This paper addresses the construction and structural characterization of linear codes. We propose a novel quadratic construction method based on the Hamming weight function: new linear codes are generated from the set of codewords of fixed weight in a given base code, and their dimension, weight distribution, and extendability relative to the base code are systematically analyzed. For the first time, the weight function is employed as a unifying framework for linear code construction. We establish tight upper bounds on the minimum distance of two classes of weight codes and fully characterize all codes attaining these bounds, revealing their underlying combinatorial structure and geometric extendability. By integrating algebraic coding theory, partial weight enumeration, and finite-field geometry, we derive several families of new linear codes and obtain divisibility conditions on the parameters of two-weight codes. These results advance the classification and structural theory of such codes.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Other Foundations of Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Currently known secondary construction techniques for linear codes mainly include puncturing, shortening, and extending. In this paper, we propose a novel method for the secondary construction of linear codes based on their weight functions. Specifically, we develop a general framework that constructs new linear codes from the set of codewords in a given code having a fixed Hamming weight. We analyze the dimension, number of weights, and weight distribution of the constructed codes, and establish connections with the extendability of the original codes as well as the partial weight distribution of the derived codes. As a new tool, this framework enables us to establish an upper bound on the minimum weight of two-weight codes and to characterize all two-weight codes attaining this bound. Moreover, several divisibility properties concerning the parameters of two-weight codes are derived. The proposed method not only generates new families of linear codes but also provides a powerful approach for exploring the intrinsic combinatorial and geometric structures of existing codes.

Problem

• Proposes a novel secondary construction method for linear codes using weight functions
• Analyzes dimension, weight distribution and extendability properties of constructed codes
• Establishes bounds and divisibility properties for two-weight linear codes

Innovation

• Constructs linear codes from fixed-weight codewords
• Analyzes dimension, weight distribution, and extendability
• Establishes bounds and characterizes optimal two-weight codes