Non-Binary LDPC Codes Defined Over the General Linear Group: Finite Length Design and Practical Implementation Issues

Abstract

Abstract-Non-binary LDPC codes are now recognized as a potential competitor to binary coded solutions, especially when the codeword length is small or moderate. More and more works are reported with good performance/complexity tradeoffs, which make non-binary solutions interesting for practical applications, such as 4G-wireless systems or DVB-like systems.In this paper, we show that proposing non-binary LDPC codes build on finite fields is actually a limitation, both from performance and implementation aspects. By considering nonbinary codes on the general linear group, we show in particular that one can obtain a slight performance improvement compared to Galois field codes, with reasonable additional cost in the hardware implementation. The performance gain is quite small, but comes at a slight extra decoding cost, and is obtained by proper generalization of the code optimization techniques that are standard for non-binary LDPC codes on fields.

Design of a GF(64)-LDPC Decoder Based on the EMS Algorithm

“…different code rates and frame lengths). The speedup factor between software simulation 9 and hardware emulation was greater than 100 for all cases. The performance results obtained with the hardware emulator platform were compared to the EMS and BP simulation results.…”

“…is a more general framework than finite Galois fields GF(q) [9]. This leads to hybrid [10] and split or cluster NB-LDPC codes [11], increasing the degree of freedom in terms of code construction while keeping the same decoding complexity.…”

Analysis of Error Floors for Non-binary LDPC Codes over General Linear Group through <i>q</i>-Ary Memoryless Symmetric Channels

“…In the simulation results of this section, we do not employ the methods which reduce the decoding error rates in waterfall regions [4], [13]. Hence, there are no gains in the waterfall regions for non-binary LDPC codes over general linear groups.…”

“…The general linear group of degree m 3 over F 2 m 4 is the set of m 3 × m 3 invertible matrices over F 2 m 4 with the operation of ordinary matrix multiplication and matrix inversion, and denoted by GL(m 3 It is known that the decoding complexity of non-binary LDPC codes over the general linear group GL(m, F 2 ) is larger than that of non-binary LDPC codes over finite field F 2 m for m ≥ 2. On the other hand, the decoding error rates in the waterfall region for optimized non-binary LDPC codes over the general linear group GL(m, F 2 ) is lower than those for optimized non-binary LDPC codes over the finite field F 2 m [4].…”

Analysis of stopping constellation distribution for irregular non-binary LDPC code ensemble

“…Each entry of parity check matrix for non-binary LDPC code defined over the general linear group is an element of the general linear group. It is known that LDPC codes defined over general linear groups outperform LDPC codes defined over finite fields in terms of the decoding performance [4].…”