Free and projective LCD codes

2026-09-14

AI Summary

研究解决了非可分解环上LCD码的自由性和射影性问题，通过推广Frobenius环特征到非交换环境，并证明群码是LCD当且仅当由中心幂等元生成。

Abstract

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.

Problem

• LCD codes
• finite commutative chain rings
• projective
• noncommutative
• group codes

Innovation

• projective LCD codes
• noncommutative setting
• central idempotent