仿射Grassmann码及其相关码的对偶
Duals of Affine Grassmann Codes and their Relatives
摘要：
仿射格拉斯曼码是广义Reed-Muller码的一种变体，与格拉斯曼码密切相关。这些代码是在最近的工作中介绍的[2]。在这里，我们考虑，更一般地，仿射格拉斯曼码的一个给定的水平。我们明确地确定任何级别的仿射格拉斯曼码的对偶，并计算其最小距离。进一步改进了[2]中关于仿射Grassmann码的自同构群的结果。最后，我们证明了仿射格拉斯曼码及其子码具有由其最小重量码字生成的线性码的性质。这为广义Reed-Muller码的相应结果提供了一个干净的模拟。
英文摘要：
Affine Grassmann codes are a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work [2]. Here we consider, more generally, affine Grassmann codes of a given level. We explicitly determine the dual of an affine Grassmann code of any level and compute its minimum distance. Further, we ameliorate the results of [2] concerning the automorphism group of affine Grassmann codes. Finally, we prove that affine Grassmann codes and their duals have the property that they are linear codes generated by their minimum-weight codewords. This provides a clean analogue of a corresponding result for generalized Reed-Muller codes.
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