Let $\mathcal {R}$ be a commutative Frobenius local ring. A result that the injective hull of an LCD code $\mathcal {C}$ over $\mathcal {R}$ is free of dimension $\ell (\mathcal {C})$ , where $\ell (\mathcal {C})$ is the minimum over the cardinalities of the generating sets of $\mathcal {C}$ , is proved in this correspondence. Applying this result, a concise proof for the main result in a recent paper by Sanjit Bhowmick et al. is derived. Furthermore, the LCD $\lambda $ -constacyclic codes with $\lambda $ being a unit, $\pi (\lambda ^{2})=1$ and $\lambda ^{2} \neq 1$ , where $\pi $ is the natural projection of $\mathcal {R}$ to its residue field, are characterized, as another application of our result.

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关于Frobenius环的LCD代码的注意事项

让 $ \ mathcal {R} $ 是可交换的Frobenius本地环。结果是LCD代码的内注式 $ \数学{C} $ 超过 $ \ mathcal {R} $ 没有维度 $ \ ell（\ mathcal {C}）$ ， 在哪里 $ \ ell（\ mathcal {C}）$ 是发电机组基数的最小值 $ \数学{C} $ 证明了这一点。应用此结果，是Sanjit Bhowmick最近发表的论文中主要结果的简要证明等。派生。此外，LCD $ \ lambda $ -等距代码 $ \ lambda $ 作为一个单位， $ \ pi（\ lambda ^ {2}）= 1 $ 和 $ \ lambda ^ {2} \ neq 1 $ ， 在哪里 $ \ pi $ 是...的自然投影 $ \ mathcal {R} $ 对其残渣场进行表征，作为我们结果的另一个应用。