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On Hadamard full propelinear codes with associated group C_{2t}\times C_2

“…Remark 2. In the conditions of Proposition 6, if t is even, then C cannot be an HFP(t, Q u )-code because its associated group is C t ×C 2 2 . If the value of t is odd, then HFP(t, 2, 2, 2 u ) HFP(2t, 2, 2 u ) and HFP(t, 4 u , 2) HFP(2t, 4 u ) HFP(4t u , 2).…”

“…Let 4t = 2 s t , where t is odd. For any binary vector x of even length we say that x (1) , x (2) are the projections over the first and the second half part of x, respectively. Let v ∈ K(C) \ u .…”

“…In Subsection 3.2, we prove that if the dimension of the kernel is greater than 1, then either there exists a circulant Hadamard matrix or there exist a code of Subsection 3.1. In Subsection 3.3, we present a conjecture about the nonexistence of a kind of Hadamard full propelinear code that is equivalent to Arasu et al conjecture [2] about the nonexistence of circulant complex Hadamard matrices. Also, we show that the circulant complex Hadamard matrix of order 2t = 16, introduced by Arasu et al in [2], corresponds to a Hadamard full propelinear code with a group structure isomorphic to C 2t × C 4 with rank equal to 11 and dimension of the kernel equal to 2.…”

“…In Subsection 3.3, we present a conjecture about the nonexistence of a kind of Hadamard full propelinear code that is equivalent to Arasu et al conjecture [2] about the nonexistence of circulant complex Hadamard matrices. Also, we show that the circulant complex Hadamard matrix of order 2t = 16, introduced by Arasu et al in [2], corresponds to a Hadamard full propelinear code with a group structure isomorphic to C 2t × C 4 with rank equal to 11 and dimension of the kernel equal to 2. Moreover, we have found another nonequivalent Hadamard full propelinear code with a group structure isomorphic to C 2t × C 4 of length 4t = 32 with rank equal to 13 and dimension of the kernel equal to 1 which corresponds to a circulant complex Hadamard matrix of order 16.…”

“…For example, the cocyclic Hadamard conjecture (de Launey and Horadam) states that there is a cocyclic Hadamard matrix of order 4t for all t ∈ N, that implies Ito's conjecture, but not conversely. The cocycles over the groups C t × C 2 2 , for t odd, were studied by Baliga and Horadam [5]. The solution set includes all Williamson Hadamard matrices, so this family of groups is potentially a uniform source for generation of Hadamard matrices.…”