Classification of all \(\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8\)-additive Hadamard codes of a fixed type

Classify all \(\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8\)-additive Hadamard codes of each abstract type up to equivalence, rather than only the recursively constructed family \(H^{t_1,t_2,t_3}\).

Background

The paper proves complete equivalence classification only within the recursively constructed family Ht1,t2,t3H^{t_1,t_2,t_3}. It explicitly notes that the parameter triple does not determine every Z2Z4Z8\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8-additive Hadamard code of the corresponding abstract type.

A generator matrix of type (4,6,12;2,0,1)(4,6,12;2,0,1) is cited as producing a Gray image with rank 13, distinct from the rank 12 of H2,0,1H^{2,0,1}. Thus, additional codes of the same abstract type may exist and require classification.

References

All the statements of this paper therefore concern the recursively constructed family, together with the constructions already proved in to give permutation equivalent codes, and classifying all the $Z_2Z_4Z_8$-additive Hadamard codes of a given abstract type remains open.

— The kernel-block rank profile and a complete classification of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes  (2609.09969 - Bhunia, 9 Sep 2026) in Section 6, Conclusions and further research

In the same direction, the existence of $Z_4Z_8$-additive Hadamard codes, that is, the case $\alpha_1=0$, $\alpha_2\not=0$, $\alpha_3\not=0$, is still open, whereas the case $\alpha_1\not=0$, $\alpha_2=0$, $\alpha_3\not=0$ cannot occur .

— The kernel-block rank profile and a complete classification of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes  (2609.09969 - Bhunia, 9 Sep 2026) in Section 6, Conclusions and further research