On the equivalence between additive and linear codes

TL;DR

This paper introduces a deterministic test to distinguish strictly additive codes from linear codes using only the generator matrix, and applies it to clarify the nature of several quaternary additive codes.

Contribution

The paper presents a new test for code equivalence and demonstrates its effectiveness on existing additive codes, clarifying their relationship to linear codes.

Findings

Verified strict additivity of several quaternary additive codes

Demonstrated equivalence of an additive ACD code to a linear Hermitian LCD code

Improved bounds for certain 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.