It is well-known that each left ideal in a matrix rings over a finite field is generated by an idempotent matrix. In this work we compute the number of left ideals in these rings, the number of different idempotents generating each left ideal, and give explicitly a set of idempotent generators of all left ideals of a given rank. We then apply these results to give examples of left group codes that have best possible minimum weight.

中文翻译：

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矩阵环和纠错码的左理想

众所周知，矩阵的每个左理想在有限域上都由幂等矩阵产生。在这项工作中，我们计算这些环中左理想的数量，产生每个左理想的不同幂等数，并明确给出一组给定等级的所有左理想的幂等生成子。然后，我们将这些结果应用于给出左组代码示例，这些示例具有最大可能的最小权重。