AI 中文总结
本文研究LCD码的射影性，证明所有双边LCD码为射影，并将Frobenius环的刻画推广至非交换情形，应用于群码的LCD判定。
AI 中文摘要
有限交换链环上的LCD码已知是自由的。然而，如果环不是不可分解的，则总是存在一个射影但非自由的LCD理想。我们证明所有双边LCD码都是射影的。我们将有限交换Frobenius环通过码及其对偶的大小条件的刻画推广到非交换情形。该结果被应用于证明群码是LCD码当且仅当它们由一个中心幂等元生成。
英文摘要
LCD codes over finite commutative chain rings are known to be free. However, if the ring is not indecomposable, there always exists an LCD ideal that is projective but not free. We prove that all two-sided LCD codes are projective. We generalise the characterisation of finite commutative Frobenius rings by means of the size condition of a code and its dual to the noncommutative setting. This result is applied to prove that group codes are LCD if and only if they are generated by a central idempotent.
