Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Hadamard Z₂Z₄Q₈-codes are Hadamard binary codes coming from a subgroup of the direct product of Z₂, Z₄ and Q ₈ groups, where Q ₈ is the quaternionic group.  characterize Hadamard Z₂Z₄Q₈-codes as a quotient of a semidirect product of Z₂Z₄-linear codes and  show that all these codes can be represented in a standard form, from a set of generators. On the other hand,  show that there exist Hadamard Z₂Z₄Q₈-codes with any given pair of allowable parameters for the rank and dimension of the kernel.
['Pere Montolio', 'Josep Rifà']
I am writing a research paper on a idea Hadamard \mathbb{Z}_{2}\mathbb{Z}_{4}Q_{8}-Codes: Rank and Kernel
[{"from": "human", "value": "I am writing a research paper on a idea Hadamard \\mathbb{Z}_{2}\\mathbb{Z}_{4}Q_{8}-Codes: Rank and Kernel"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Hadamard Z\u2082Z\u2084Q\u2088-codes are Hadamard binary codes coming from a subgroup of the direct product of Z\u2082, Z\u2084 and Q \u2088 groups, where Q \u2088 is the quaternionic group.  characterize Hadamard Z\u2082Z\u2084Q\u2088-codes as a quotient of a semidirect product of Z\u2082Z\u2084-linear codes and  show that all these codes can be represented in a standard form, from a set of generators. On the other hand,  show that there exist Hadamard Z\u2082Z\u2084Q\u2088-codes with any given pair of allowable parameters for the rank and dimension of the kernel."}]