Low-complexity encoding of binary quasi-cyclic codes based on Galois Fourier transform

This paper presents a novel low-complexity encoding algorithm for binary quasi-cyclic (QC) codes based on matrix transformation. First, a message vector is encoded into a transformed codeword in the transform domain. Then, the transmitted codeword is obtained from the transformed codeword by the inverse Galois Fourier transform. Moreover, a simple and fast mapping is devised to post-process the transformed codeword such that the transmitted codeword is binary as well. The complexity of our proposed encoding algorithm is less than ek(n-k)log₂ e+ne(log²₂ e+log₂ e)+ n/2 elog³₂ e bit operations for binary codes. This complexity is much lower than its traditional complexity 2e²(n - k)k. In the examples of encoding the binary (4095, 2016) and (15500, 10850) QC codes, the complexities are 12.09% and 9.49% of those of traditional encoding, respectively.