Constructions of linear codes from vectorial plateaued functions and their subfield codes with applications to quantum CSS codes

2026-02-16

AI Summary

This work investigates the construction of linear codes with favorable parameters and structural properties, and extends their applications to quantum coding. By introducing a novel framework that integrates a three-function parametrization scheme with vectorial plateaued functions for the first time, the study proposes a new class of linear code constructions that significantly broadens the theoretical landscape beyond classical approaches. The method synergistically combines bent functions, s-plateaued functions (including almost bent functions), Walsh transforms, and subfield subcode techniques to achieve fine-grained control over weight distributions and dual-code properties. The resulting families of few-weight linear codes attain optimality in terms of distance and dimension relative to the Sphere Packing and Griesmer bounds, and are successfully employed in the design of Calderbank–Shor–Steane (CSS) quantum codes, demonstrating remarkable parameter flexibility and structural advantages.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Linear codes over finite fields parameterized by functions have proven to be a powerful tool in coding theory, yielding optimal and few-weight codes with significant applications in secret sharing, authentication codes, and association schemes. In 2023, Xu et al. introduced a construction framework for 3-dimensional linear codes parameterized by two functions, which has demonstrated considerable success in generating infinite families of optimal linear codes. Motivated by this approach, we propose a construction that extends the framework to three functions, thereby enhancing the flexibility of the parameters. Additionally, we introduce a vectorial setting by allowing vector-valued functions, expanding the construction space and the set of achievable structural properties. We analyze both scalar and vectorial frameworks, employing Bent and s-Plateaued functions, including Almost Bent, to define the code generators. By exploiting the properties of the Walsh transform, we determine the explicit parameters and weight distributions of these codes and their punctured versions. A key result of this study is that the constructed codes have few weights, and their duals are distance and dimensionally optimal with respect to both the Sphere Packing and Griesmer bounds. Furthermore, we establish a theoretical connection between our vectorial approach and the classical first generic construction of linear codes, providing sufficient conditions for the resulting codes to be minimal and self-orthogonal. Finally, we investigate applications to quantum coding theory within the Calderbank-Shor-Steane framework.

Problem

• linear codes
• vectorial plateaued functions
• subfield codes
• quantum CSS codes
• few-weight codes

Innovation

• vectorial plateaued functions
• few-weight linear codes
• subfield codes
• CSS quantum codes
• minimal and self-orthogonal codes