On the Hamming Weight Functions of Linear Codes

TL;DR

This work introduces a general secondary construction for linear codes based on the weight function: from any $[n,k]_q$ code ${\mathcal{C}}$ one builds ${\mathcal{C}}^*(w)$ from codewords of fixed weight $w$, and studies its dimension, weight distribution, and connections to extendability and design theory. The authors develop a robust framework using projectivization and duality, and prove key structural results, including a complete characterization for projective two-weight codes, an upper bound on the minimum weight of two-weight codes, and divisibility properties. They also establish a practical method to generate new optimal codes (e.g., via the Golay example) and derive extendability criteria, offering tools to explore the intrinsic combinatorial and geometric structures of codes. The approach has potential to extend to other families such as BCH, algebraic-geometry, and LDPC codes, enabling broader construction and analysis of code families tied to fixed-weight codewords.

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.

Figures (1)

• Figure 1: Codewords of weight $\lambda$ in $\overline{\mathcal{C}^*(w)}$

Theorems & Definitions (49)

• Proposition 1
• Theorem 2: Assmus-Mattson
• Theorem 3: A Generalization of the Assmus-Mattson Theorem
• Example 1
• Theorem 4
• proof
• Theorem 5
• proof
• Proposition 6
• proof