On the equivalence between additive and linear codes
This work addresses a fundamental challenge in coding theory for researchers, but it is incremental as it builds on existing methods to clarify code equivalences.
The authors tackled the problem of distinguishing strictly additive codes from those equivalent to linear codes by introducing a deterministic test based on the generator matrix. They applied this test to verify the strict additivity of quaternary additive codes and showed that a known additive complementary dual code is equivalent to a linear Hermitian LCD code, improving best-known bounds.
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.