Generalized low-density parity-check (GLDPC) codes, where single parity-check constraints on the code bits are replaced with generalized constraints (an arbitrary linear code), are a promising class of codes for low-latency communication. We have constructed quasi-cyclic GLDPC codes, where the proportion of generalized constraints is determined by an asymptotic analysis. We have analyzed the complexity and performance of the message passing decoder with various update rules (including standard full-precision sum-product and min-sum algorithms) and quantization schemes for a GLDPC code over the additive white Gaussian noise (AWGN) channel and determined a constraint-to-variable update rule based on the specific codewords of the component codes. This data set includes the simulated GLDPC code constructions and the block error rate performance, which is shown to outperform a variety of state- of-the-art code and decoder designs with suitable lengths and rates for the 5G ultra-reliable low-latency communication regime over an AWGN channel with quadrature PSK modulation.