This paper deals with the problem of determining whether a PD-set exists for a given linear code $ \mathcal C $ and information set $ I $. A computational approach is proposed and illustrated with two exceptional codes with automorphism groups isomorphic to the sporadic simple groups $ \mathrm{M}_{12} $ and $ \mathrm{M}_{22} $, respectively. In both cases, the existence of a PD–set is proven. In general, the algorithm works well whenever the code $ \mathcal C $ has a very large automorphism group.

中文翻译：

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关于PD集的存在：自同构代码组产生的算法

本文涉及确定给定线性代码$ \ mathcal C $和信息集$ I $是否存在PD集的问题。提出了一种计算方法，并用两个例外代码进行了说明，它们的自同构组分别与零星的简单组$ \ mathrm {M} _ {12} $和$ \ mathrm {M} _ {22} $同构。在这两种情况下，都证明存在PD集合。通常，只要代码$ \ mathcal C $具有很大的自同构组，该算法就可以很好地工作。