Hamming weight distributions of generalized quasi-cyclic codes over $ \mathrm{GR}(p^2, m) $

Let $ R = \mathrm{GR}(p^2, m) $ denote the Galois ring of characteristic $ p^2 $ with residue field $ \mathbb{F}_q\cong R/\langle p\rangle $ and cardinality $ q^2 $, where $ q = p^m $. This paper determines the Hamming weight distributions of generalized quasi-cyclic (GQC) codes over $ R $ under the assumption that $ \gcd(m_i, p) = 1 $ for every block length $ m_i $. The Chinese Remainder Theorem decomposes each GQC code into constituents over the extension Galois rings $ \mathrm{GR}(p^2, md) $, and the weight problem thereby reduces to counting the positions at which certain trace sums vanish. We develop character-sum formulas for these trace-zero counts, organizing the analysis according to whether the active constituent parameters are zero, nilpotent, or units. Nilpotent constituents reduce to trace equations over the residue field and are evaluated by finite-field Gauss sums, whereas unit and mixed constituents require an analysis of the principal $ p $-layer based on the Teichmüller decomposition, carry traces, and addition carries. One consequence is that residue data alone do not determine the trace-zero count over $ \mathrm{GR}(p^2, m) $: Two units with the same residue but distinct principal $ p $-parts may yield different Hamming weights. For one- and two-constituent GQC codes, we derive explicit finite character-sum formulas covering every combination of nilpotent and unit constituents. Closed evaluations are obtained in the purely nilpotent cases and in selected semiprimitive cases, and the remaining unit cases are expressed through Galois-ring period sums involving carry terms. Together, these formulas determine the Hamming weight distributions of the corresponding families of GQC codes over $ \mathrm{GR}(p^2, m) $.