On the equivalence between additive and linear codes

2026-03-16

AI Summary

This work addresses a fundamental problem in algebraic coding theory: effectively distinguishing strictly additive codes from those equivalent to linear codes. We propose the first deterministic criterion based solely on the generator matrix to determine whether a quaternary additive code is equivalent to a linear code. This method clarifies the essential structural differences between additive and linear codes and revises the current classification of ACD codes. Specifically, it confirms the strictly additive nature of several quaternary additive codes and demonstrates that a code previously believed to be nonlinear is in fact a linear Hermitian LCD code. Consequently, this result improves the best-known parameter bounds for this class of linear codes.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Applications of cryptography

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.

Problem

• additive codes
• linear codes
• code equivalence
• strict additivity
• generator matrix

Innovation

• additive codes
• linear codes
• equivalence test
• generator matrix
• complementary dual codes