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

This paper presents two novel low-complexity encoding algorithms for quasi-cyclic (QC) codes based on Galois Fourier transform. The key idea behind them is making use of the block diagonal structure of the transformed generator matrix. The first one, named encoding by Galois Fourier transform, is equivalent to the fast implementations of the traditional encoding by Galois Fourier transform. The second one, named encoding in the transform domain (ETD), requires much less computational complexity for encoding binary QC codes. It skips the first step of the first algorithm and applies post-processing to save a large number of Galois field multiplications. Its application to QC-LDPC codes is also studied in this paper. Particularly, the hardware cost of the ETD for RS-based LDPC codes can be greatly reduced by short linear-feedback shift registers.