Abstract

This paper describes the design and analysis of low-density parity-check (LDPC) codes over rings and shows how these codes, when mapped onto appropriate signal constellations, can be used to effect bandwidth-efficient modulation. Specifically, LDPC codes are constructed over the integer rings Zm and G2m and mapped onto phase-shift keying (PSK)-type signal sets to yield geometrically uniform signal space codes. This paper identifies and addresses the design issues that affect code performance. Examples of codes over Z8 and G64 mapped onto 8-ary and 64-ary signal sets at a spectral efficiency of 1.5 and 2.0 bits per second per hertz (b/s/Hz) illustrate the approach; simulation of these codes over the additive white Gaussian noise (AWGN) channel demonstrates that this approach is a good alternative to bandwidth-efficient techniques based on binary LDPC codes—e.g., bit-interleaved coded modulation.