Optimal Generalized Quasi-Polycyclic Codes over Fq+uFq

This paper develops a structural and constructive theory of right generalized quasi-polycyclic (GQPC) codes over the finite chain ring R=Fq+uFq with u2=0, extending the existing field-based GQPC framework to a ring-theoretic setting. Right GQPC codes over R are modeled as R[x]-submodules of direct products of polycyclic ambient algebras R[x]/⟨xei−αi(x)⟩, induced by vectors αi∈Rei, thereby unifying right quasi-polycyclic and generalized quasi-cyclic codes over R. Under explicit and verifiable factorization conditions on the defining polynomials, we establish a Chinese Remainder Theorem decomposition that reduces right GQPC codes to collections of shorter codes over finite chain-ring extensions of R. This decomposition yields a characterization of ρ-generator right GQPC codes and leads to a canonical normalized generating set with an upper-triangular structure. As a consequence, we obtain an explicit rank formula in terms of the diagonal generator polynomials, together with an effective normalization algorithm. To demonstrate the coding-theoretic impact of the framework, we combine these structural results with a distance-compatible Gray map Φ:R→Fq2 and construct new q-ary linear codes from 2-generator right GQPC codes of index 2 over R. For q=9 and q=3, the resulting Gray images attain optimal or near-optimal parameters with respect to the best-known bounds, confirming that right GQPC codes over Fq+uFq constitute a robust and effective ring-based source of high-quality linear codes.