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

TL;DR

This paper introduces a novel construction of linear codes using vectorial plateaued functions, extending previous frameworks to enhance code parameters, and explores their applications in quantum CSS codes with optimal properties.

Contribution

It extends existing code construction methods to three functions and vector-valued functions, providing new families of optimal, few-weight codes with applications to quantum error correction.

Findings

Codes have few weights and are optimal in duals.

Explicit parameters and weight distributions are derived.

Applications to quantum CSS codes are demonstrated.

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.