Non-binary adaptive LDPC codes for frequency selective channels: code construction and iterative decoding

Probabilistic shaping and non-binary codes

“…Principles of this extension to the non-binary case are implicit from Chapter 5 of [16] with the use of the z-transform. Nevertheless, we give here the exact expression of the probability of a sum of prime random symbols modulo p. This expression is directly related to Hartmann-Rudolph symbol-by-symbol probabilistic decoding [24] in the special case of a single-parity check code and its generalization to characteristic p [25].…”

Geometric shaping: low-density coding of Gaussian-like constellations

“…q 2 transitions in the syndrome trellis representing a check node instead of p 2 . If a field structure replaces the ring structure, a fast transform of q log q is much appreciable than p log p [18] [19].…”

Low-Density Parity-Check Codes and Their Rateless Relatives

“…However, non-binary LDPC codes suffer from the disadvantage of having an increased number of possible values, which renders the classification of symbols more complex and hence naturally increases the decoding complexity imposed. Non-binary codes have been applied for transmission over non-dispersive Rayleigh fading channels [52], over frequency selective channels [53] and multiple-input multiple-output (MIMO) channels [54]- [57]. The results in [51] were also substantiated by Hu et al [58], who proposed a construction for irregular non-binary LDPC codes defined over GF(q) constructed using the so-called progressive edge growth (PEG) algorithm and demonstrated that the performance of these codes improves upon increasing the Galois field size 2 q .…”