Additive codes generalize linear codes by relaxing the algebraic requirements on the code's structure. A linear code is closed under scalar multiplication by the full field; an additive code is closed under multiplication by a subfield only. Additive codes can, in principle, achieve parameters (minimum distance, rate) that no linear code matches. Several constructions in the literature claim this advantage.
The paper introduces a deterministic test — requiring only the generator matrix — to distinguish genuinely additive codes from codes that are equivalent to linear ones via coordinate permutations, scalings, and field automorphisms. The test is applied to codes from recent literature.
Some codes pass: they are genuinely additive, with no linear equivalent. But one code fails. A previously published additive complementary dual (ACD) code turns out to be equivalent to a linear Hermitian LCD code. The “additive” code was linear all along, dressed in additive notation. The equivalence, once detected, improves known bounds for linear codes — the additive construction achieved parameters thought to be out of reach for linear codes, but since the code is actually linear, those parameters are in reach after all.
The structural lesson: the boundary between linear and additive is not where the literature placed it. Some codes sit on the additive side by notation but the linear side by structure. Without a test, the classification is based on how the code was constructed, not what it is. The test moves classification from syntax (how you write the code) to semantics (what equivalence class it belongs to). Several additive claims survive; one doesn't.