Efficiently encodable non-binary generalized LDPC codes

This writeup examines the construction and performance of non-binary generalized low-density parity-check (GLDPC) codes. Recently, binary GLDPC coding techniques have been shown to improve upon conventional low-density parity-check (LDPC) codes by increasing the minimum codeword length, reducing error floors, and improving performance at lower coding rates. GLDPC codes replace the parity-check constraints of a standard LDPC code with stronger codes, e.g. short linear block codes. At the same time, non-binary LDPC codes have been shown to outperform their binary LDPC counterpart for shorter codeword lengths and for space-time coding applications in multiple-input multiple-output (MIMO) systems. Therefore, in contrast to previous works which have examined binary GLDPC codes, this paper explores non-binary GLDPC codes to combine all of the aforementioned benefits of non-binary LDPC and binary GLDPC codes. Additionally, non-binary GLDPC codes can attain higher coding rates than the counterpart binary GLDPC codes. We first propose a class of GLDPC codes with low encoding complexity, where the number of encoding operations scales linearly with the codeword length. An example is provided to illustrate the performance and complexity benefits of GLDPC codes over non-binary LDPC codes at lower coding rates. In particular, a GF(4) GLDPC code is shown to outperform a strong GF(256) LDPC code while requiring approximately 9x less operations per information bit. It is also demonstrated that with the appropriate parameters, the non-binary GLDPC codes outperform non-binary LDPC codes with respect to both the error floor and waterfall performance. Furthermore, it is shown that performance remains strong for the same coding rate as the constellation changes; this result contrasts previous works which have demonstrated that for non-binary LDPC codes, optimal parameters such as the column weight need to be adjusted for different constellations.