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

Virginio Fratianni, Sihem Mesnager

TL;DR

This study analyzes both scalar and vectorial frameworks for 3-dimensional linear codes parameterized by two functions, and establishes a theoretical connection between the vectorial approach and the classical first generic construction of linear codes, enabling the resulting codes to be minimal and self-orthogonal.

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.

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

TL;DR

This study analyzes both scalar and vectorial frameworks for 3-dimensional linear codes parameterized by two functions, and establishes a theoretical connection between the vectorial approach and the classical first generic construction of linear codes, enabling the resulting codes to be minimal and self-orthogonal.

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.
Paper Structure (17 sections, 31 theorems, 127 equations, 1 table)

This paper contains 17 sections, 31 theorems, 127 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coding theory
  4. Vectorial and scalar functions over finite fields
  5. Character theory over finite fields
  6. Subfield codes from generic functions
  7. Subfield codes from specific functions
  8. $f(x)=\text{Tr}_{q^m/q}(x)$, $g(y)=\text{Tr}_{q^m/q}(y^2)$, $h(z)=\text{Norm}_{q^m/q}(z)$
  9. $f(x)=\text{Tr}_{q^m/q}(x)$, $g(y)=\text{Tr}_{q^m/q}(y^2)$, $h(z)$ generic
  10. The vectorial framework
  11. Applications in the context of quantum CSS$_T$ codes
  12. Transversal gates and fault tolerance
  13. CSS$_T$ codes in the binary case
  14. CSS phase codes in odd characteristic
  15. CSS phase codes from plateaued functions
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $\mathcal{C}$ be an $[n, k, d]$ code over $\mathbb{F}_q$. Then

Theorems & Definitions (57)

  • Lemma 2.1: Sphere Packing / Hamming Bound
  • Lemma 2.2: Griesmer Bound
  • Lemma 2.3: ref14, Theorem 2.5
  • Lemma 2.4: ref14
  • Definition 2.5
  • Definition 2.6: Bent, Plateaued and Almost Bent Scalar Functions
  • Definition 2.7: Vectorial Plateaued and Almost Bent Functions
  • Remark
  • Lemma 2.8: Lidl
  • Lemma 2.9: Lidl
  • ...and 47 more