We describe eight composite constructions from group rings where the orders of the groups are 4 and 8, which are then applied to find self-dual codes of length 16 over $ \mathbb{F}_4 $. These codes have binary images with parameters $ [32,16,8] $ or $ [32,16,6] $. These are lifted to codes over $ \mathbb{F}_4+u\mathbb{F}_4 $, to obtain codes with Gray images of extremal self-dual binary codes of length 64. Finally, we use a building-up method over $ \mathbb{F}_2+u\mathbb{F}_2 $ to obtain new extremal binary self-dual codes of length 68. We construct 11 new codes via the building-up method and 2 new codes by considering possible neighbors.

中文翻译：

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由群环和长度为68的新极值自对偶二进制码的复合结构

我们从组环描述了八种复合构造，其中组的顺序为4和8，然后将其应用于查找$ \ mathbb {F} _4 $上长度为16的自对偶代码。这些代码具有参数为$ [32,16,8] $或$ [32,16,6] $的二进制图像。将它们提升为$ \ mathbb {F} _4 + u \ mathbb {F} _4 $上的代码，以获得带有长度为64的极值自对二进制二进制代码的Gray图像的代码。最后，我们在$ \ mathbb {F} _2 + u \ mathbb {F} _2 $获得新的长度为68的极值二进制自对偶代码。我们通过构造方法构造了11个新代码，并考虑了可能的邻居构造了2个新代码。