Classification of nonlinear Hadamard codes

Classify all nonlinear Hadamard codes up to equivalence, extending beyond the presently classified subclasses such as the recursively constructed \(\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8\)-linear family.

Background

Hadamard codes are binary codes with two times as many codewords as their length and minimum distance equal to half their length. The paper studies one structured subclass obtained as Gray-map images of additive codes over mixed alphabets.

The introduction explicitly places the paper's internal classification in the broader context of the unresolved classification of nonlinear Hadamard codes. The results classify only the recursively constructed $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear family and therefore do not resolve the classification problem for all nonlinear Hadamard codes.