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The encoding complexity of a general (en,ek) quasi-cyclic code isO[(e^2)(n-k)k]. This paper presents a novel low-complexity encoding algorithmfor quasi-cyclic (QC) codes based on matrix transformation. First, a messagevector is encoded into a transformed codeword in the transform domain. Then,the transmitted codeword is obtained from the transformed codeword by theinverse Galois Fourier transform. For binary QC codes, a simple and fastmapping is required to post-process the transformed codeword such that thetransmitted codeword is binary as well. The complexity of our proposed encodingalgorithm is O[e(n-k)k] symbol operations for non-binary codes andO[ek(n-k)(log_2 e)] bit operations for binary codes. These complexities aremuch lower than their traditional counterpart O[(e^2)(n-k)k]. For example, ourcomplexity of encoding a 64-ary (4095,2160) QC code is only 1.59% of that oftraditional encoding, and our complexities of encoding the binary (4095, 2160)and (8176, 7154) QC codes are respectively 9.52% and 1.77% of those oftraditional encoding. We also study the application of our low-complexityencoding algorithm to one of the most important subclasses of QC codes, namelyQC-LDPC codes, especially when their parity-check matrices are rank deficient.

A Low-Complexity Encoding of Quasi-Cyclic Codes Based on Galois Fourier Transform