🤖 AI Summary
This work investigates the generalized Hamming weights (GHWs) of nested matrix product codes, establishing tight lower bounds and exact values. For nested constructions comprising two or three constituent codes, we derive the first tight GHW lower bounds and introduce a recursive analytical framework leveraging the nested subspace structure of component codes. In particular, when both constituent codes are Reed–Solomon codes, we obtain an explicit closed-form formula for the GHWs. The analysis is further extended to non-nested two-code matrix product codes. Our methodology integrates algebraic coding theory, subspace lattice analysis, and matrix product code construction. The results yield exact GHWs or optimal lower bounds for several classes of matrix product codes, generalize classical GHW characterizations of Reed–Muller and Hermitian codes, and provide a new theoretical tool for evaluating information-theoretic security and error-correction capability of linear codes.