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The simple interpretation of the minimum distance of a linear code obtainedby De Boer and Pellikaan, and later refined by the second author, is furtherdeveloped through the study of various finitely generated graded modules. Weuse the methods of commutative/homological algebra to find connections betweenthe minimum distance and the $\alpha$-invariant of such modules.

Minimum distance of linear codes and the $\alpha$-invariant