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On the equivalence between additive and linear codes

Kanat Abdukhalikov, Duy Ho

Abstract

Additive codes have attracted considerable attention for their potential to outperform linear codes. However, distinguishing strictly additive codes from those that are equivalent to linear codes remains a fundamental challenge. To resolve this ambiguity, we introduce a deterministic test that requires only the generator matrix of the code. We apply this test to verify the strict additivity of several quaternary additive codes recently reported in the literature. Conversely, we demonstrate that a previously known additive complementary dual (ACD) code is equivalent to a linear Hermitian LCD code, thereby improving the best-known bounds for such linear codes.

On the equivalence between additive and linear codes

Abstract

Additive codes have attracted considerable attention for their potential to outperform linear codes. However, distinguishing strictly additive codes from those that are equivalent to linear codes remains a fundamental challenge. To resolve this ambiguity, we introduce a deterministic test that requires only the generator matrix of the code. We apply this test to verify the strict additivity of several quaternary additive codes recently reported in the literature. Conversely, we demonstrate that a previously known additive complementary dual (ACD) code is equivalent to a linear Hermitian LCD code, thereby improving the best-known bounds for such linear codes.
Paper Structure (8 sections, 3 theorems, 25 equations, 3 tables, 1 algorithm)

This paper contains 8 sections, 3 theorems, 25 equations, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Additive codes
  4. Equivalence of additive codes
  5. Inner products and ACD codes
  6. Equivalence test for linearity of codes
  7. Applications of the equivalence test
  8. Conclusion

Key Result

Theorem 3.1

Let $C \subseteq \mathbb{F}_{q^2}^n$ be an additive code of $\mathbb{F}_q$-dimension $k$, and let $G \in \mathbb{F}_q^{k \times 2n}$ be a generator matrix of $C$. Then $C$ is equivalent to an $\mathbb{F}_{q^2}$-linear code if and only if there exist $R \in \textrm{GL}_k(\mathbb{F}_q)$ and a block-di

Theorems & Definitions (12)

  • Definition 2.1: Monomial Equivalence
  • Definition 2.2: $\mathrm{SL}$-Equivalence
  • Remark 1
  • Remark 2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 2 more