利用2 × 2分块矩阵构造群环构造新的68自对偶码
New Self-Dual Codes of length 68 from a 2 by 2 block matrix Construction and Group Rings
摘要：
许多用于构造不同长度的极值二元自对偶码的生成矩阵具有G=（I|其中I是n乘n单位矩阵，A是完全由第一行确定的n乘n矩阵。在这项工作中，我们定义了一个生成矩阵，其中A是一个分块矩阵，其中的块来自群环，并且A不完全由第一行中出现的元素决定。利用我们在F_2+uF_2上的构造和码的扩张方法，我们构造了新的长度为68的极值二元自对偶码。此外，通过对所得码采用广义近邻法，我们构造出了W_{68,2}中具有稀有参数γ = 7，8和9的许多新的二元自对偶[68，34，12]-码.特别地，我们发现了92个新的二元自对偶[68，34，12]-码。
英文摘要：
Many generator matrices for constructing extremal binary self-dual codes of different lengths have the form G=(I|A), where I is the n by n identity matrix and A is the n by n matrix fully determined by the first row. In this work, we define a generator matrix in which A is a block matrix, where the blocks come from group rings and also, A is not fully determined by the elements appearing in the first row. By applying our construction over F_2+uF_2 and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to construct many new binary self-dual [68,34,12]-codes with the rare parameters gamma=7,8 and 9 in W_{68,2}. In particular, we find 92 new binary self-dual [68,34,12]-codes.
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