In this paper, we give a new method for constructing LCD codes. We employ group rings and a well known map that sends group ring elements to a subring of the $n \times n$ matrices to obtain LCD codes. Our construction method guarantees that our LCD codes are also group codes, namely, the codes are ideals in a group ring. We show that with a certain condition on the group ring element $v,$ one can construct non-trivial group LCD codes. Moreover, we also show that by adding more constraints on the group ring element $v,$ one can construct group LCD codes that are reversible. We present many examples of binary group LCD codes of which some are optimal and group reversible LCD codes with different parameters.
翻译:在本文中,我们给出了一种构建液晶编码的新方法。 我们使用组环和一张广为人知的地图,将组环元素发送到美元/乘以n$n$的基质下方,以获得液晶编码。 我们的构建方法保证我们的液晶编码也是组代码, 即, 代码是集团环中的理想。 我们显示,如果对组环元素有某种条件, $v, 美元可以构建非三联组液晶编码。 此外, 我们还表明,通过对组环元素增加更多的限制, $v, 美元, 一个人可以构建可逆的团体液晶编码。 我们提出了许多二联式LCD代码的例子, 其中一些是最佳的, 并且可组合可逆的液晶编码具有不同参数。