We develop an algebraic theory of supports for \(R\)-linear codes of fixed length, where \(R\) is a finite commutative unitary ring. A support naturally induces a notion of generalized weights and allows one to associate a monomial ideal to a code. Our main result states that, under suitable assumptions, the generalized weights of a code can be obtained from the graded Betti numbers of its associated monomial ideal. In the case of \(\mathbb{F}_q\)-linear codes endowed with the Hamming metric, the ideal coincides with the Stanley-Reisner ideal of the matroid associated to the code via its parity-check matrix. In this special setting, we recover the known result that the generalized weights of an \(\mathbb{F}_q\)-linear code can be obtained from the graded Betti numbers of the ideal of the matroid associated to the code. We also study subcodes and codewords of minimal support in a code, proving that a large class of \(R\)-linear codes is generated by its codewords of minimal support.

Mathematics Subject Classifications: 94B05, 13D02, 13F10

Keywords: Linear codes, codes over rings, supports, generalized weights, monomial ideal of a code, graded Betti numbers, matroid