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In this paper, we consider how to partition the parity-check matrices (PCMs) to reduce the hardware complexity and computation delay for the row layered decoding of quasi-cyclic low-density parity-check (QC-LDPC) codes. First, we formulate the PCM partitioning as an optimization problem, which targets to minimize the maximum column weight of each layer while maintaining a block cyclic shift property among different layers. As a result, we derive all the feasible solutions for the problem and propose a tight lower bound $ω_{LB}$ on the minimum possible maximum column weight to evaluate a solution. Second, we define a metric called layer distance to measure the data dependency between consecutive layers and further illustrate how to identify the solutions with desired layer distance from those achieving the minimum value of $ω_{LB}=1$, which is preferred to reduce computation delay. Next, we demonstrate that up-to-now, finding an optimal solution for the optimization problem with polynomial time complexity is unachievable. Therefore, both enumerative and greedy partition algorithms are proposed instead. After that, we modify the quasi-cyclic progressive edge-growth (QC-PEG) algorithm to directly construct PCMs that have a straightforward partition scheme to achieve $ω_{LB} $ or the desired layer distance. Simulation results showed that the constructed codes have better error correction performance and smaller average number of iterations than the underlying 5G LDPC code.