The generalized Bézout theorem of the title describes a Sylvester-like matrix whose rank estimates the number of points common to \(r \geq 2\) plane curves. Then, from the abstract: ``A new approach to determine a lower bound for the minimum distance for algebraic-geometric codes defined from a class of plane curves is introduced, based on the generalized Bézout theorem. Examples of more efficient linear codes are constructed using the generalized Bézout theorem and the new approach. For \(d=4\) the linear codes constructed by the new construction are better than or equal to the known linear codes. For \(d\geq 5\) the new codes are better than the known A.G. codes defined from whole spaces. The Klein codes \([22,16,5]\) and \([22,15,6]\) over \(\text{GF} (2^3)\) and the improved Hermitian code \([64,56,6]\) over \(\text{GF} (2^4)\) are also constructed''.