We consider discrete message passing (MP) decoding of low-density parity check (LDPC) codes based on information-optimal symmetric look-up table (LUT). A link between discrete message labels and the associated log-likelihood ratio values (defined in terms of density evolution distributions) is established. This link gives rise to an algebraic structure on the message labels and leads to an interpretation of LUT decoding as a form of quantized belief propagation. We then exploit the algebraic structure for low-complexity LUT decoder designs. Our LUT decoding framework is the first to also apply to irregular LDPC codes by taking into account the degree distribution in a joint LUT design. We exploit the relation between LUT decoding and belief propagation to obtain stability conditions and irregular LDPC code designs optimized for LUT decoding. The resulting decoders outperform floating-point precision min-sum decoders at LUT resolutions as low as 3 bit s for regular codes and 4 bits for irregular codes.