发表机构
Southern University of Science and Technology（南方科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究循环码的置换自同构群，建立循环码与分圆结合方案的联系，证明 q-仿射群对某些码长不适用，并用算术条件刻画这些码长，其密度为零。
AI 中文摘要
Berger-Charpin 猜想预测，循环码的置换自同构群通常为 $q$-仿射群，该群由指标集上的移位和 Frobenius 乘子生成。文献中并未精确说明“通常”一词的含义，本系列论文的目标是研究 Berger-Charpin 猜想或其变体何时成立。在第一部分中，我们建立了循环码理论与分圆结合方案（等价地，循环群上的 Schur 环）理论之间的自然联系。利用这一联系，我们证明了 $q$-仿射群并非某些码长所应预期的正确群。我们通过一个算术条件刻画了这些码长，并证明它们具有零密度。
英文摘要
The Berger-Charpin conjecture predicts that the permutation automorphism group of a cyclic code is generally the $q$-affine group, which is generated by the shift and the Frobenius multiplier on the index set. The word {\em generally} is not made precise in the literature, and the goal of this series of papers is to study when the Berger-Charpin conjecture or its variants hold. In this first part, we establish a natural connection between the theory of cyclic codes and the theory of cyclotomic association schemes (equivalently, Schur rings over cyclic groups). Using it, we show that the $q$-affine group is not the correct group to expect for certain code lengths. We characterize these lengths by an arithmetic condition, and show that they have density zero.
