Classically, the quantization of the soft information supplied to a finite-precision decoder is chosen to optimize a certain criterion which does not depend on the characteristics of the existing code. This work studies code-aware quantizers, for finite-precision min-sum decoders, which optimize the noise threshold of the existing family of Low-Density Parity-Check (LDPC) codes. We propose a code-aware quantizer with lower complexity than that obtained by optimizing all decision levels and approaching its performance, for few quantization bits. We show that code-aware quantizers outperform code-independent quantizers in terms of noise threshold for both regular and irregular LDPC codes. To overcome the error floor behavior of LDPC codes, we propose the design of the quantizer for a target error probability at the decoder output. The results show that the quantizer optimized to get a zero error probability could lead to a very bad performance for practical range of signal to noise ratios. Finally, we propose to design jointly irregular LDPC codes and code-aware quantizers for finite-precision min-sum decoders. We show that they achieve significant decoding gains with respect to LDPC codes designed for infinite-precision belief propagation decoding, but decoded by finite-precision min-sum. © 2016 IEEE.