On the Hamming Weight Functions of Linear Codes

TL;DR

This paper introduces a novel framework for constructing and analyzing linear codes based on their weight functions, offering new bounds and characterizations for two-weight codes and revealing their combinatorial structures.

Contribution

The paper proposes a new method for secondary construction of linear codes using weight functions, advancing understanding of code properties and bounds.

Findings

Established an upper bound on the minimum weight of two-weight codes

Characterized all two-weight codes that attain this bound

Derived divisibility properties of parameters of two-weight codes

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.