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

In this paper, we compare the decoding error rates in the error floors for non-binary low-density parity-check (LDPC) codes over general linear groups with those for non-binary LDPC codes over finite fields transmitted through the q-ary memoryless symmetric channels under belief propagation decoding. To analyze non-binary LDPC codes defined over both the general linear group $\mathrm{GL}(m,\mathbb{F}_2)$ and the finite field $\mathbb{F}_{2^m}$, we investigate non-binary LDPC codes defined over $\mathrm{GL}(m_3,\mathbb{F}_{2^{m_4}})$. We propose a method to lower the error floors for non-binary LDPC codes. In this analysis, we see that the non-binary LDPC codes constructed by our proposed method defined over general linear group have the same decoding performance in the error floors as those defined over finite field. The non-binary LDPC codes defined over general linear group have more choices of the labels on the edges which satisfy the condition for the optimization.