We introduce the Hadamard full propelinear codes that factorize as direct product of groups such that their associated group is $ C_{2t}\times C_2 $. We study the rank, the dimension of the kernel, and the structure of these codes. For several specific parameters we establish some links from circulant Hadamard matrices and the nonexistence of the codes we study. We prove that the dimension of the kernel of these codes is bounded by $ 3 $ if the code is nonlinear. We also get an equivalence between circulant complex Hadamard matrix and a type of Hadamard full propelinear code, and we find a new example of circulant complex Hadamard matrix of order $ 16 $.

中文翻译：

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在Hadamard上具有相关群的全凸代码 \ begin {document} $ C_ {2t} \ times C_2 $ \ end {document}

我们介绍了Hadamard完全线性代码，这些代码分解为组的直接乘积，以使其关联的组为$ C_ {2t} \ times C_2 $。我们研究了等级，内核的维数以及这些代码的结构。对于几个特定参数，我们建立了循环Hadamard矩阵与我们研究代码的不存在之间的某些联系。我们证明，如果代码是非线性的，则这些代码的内核维以$ 3 $为边界。我们还得到了循环复Hadamard矩阵与一类Hadamard完全线性代码之间的等价关系，并且找到了一个阶为$ 16 $的循环复Hadamard矩阵的新示例。