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

TL;DR

This paper introduces a low-complexity encoding algorithm for quasi-cyclic codes using Galois Fourier transform, significantly reducing computational complexity compared to traditional methods.

Contribution

The paper proposes a novel encoding method based on matrix transformation and Galois Fourier transform, lowering complexity for binary and non-binary QC codes.

Findings

Complexity reduced to 1.59% for a 64-ary QC code

Binary QC code encoding complexity decreased to 9.52% and 1.77%

Applicable to QC-LDPC codes with rank-deficient matrices

Abstract

The encoding complexity of a general (en,ek) quasi-cyclic code is O[(e^2)(n-k)k]. This paper presents a novel low-complexity encoding algorithm for 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. For binary QC codes, a simple and fast mapping is required to post-process the transformed codeword such that the transmitted codeword is binary as well. The complexity of our proposed encoding algorithm is O[e(n-k)k] symbol operations for non-binary codes and O[ek(n-k)(log_2 e)] bit operations for binary codes. These complexities are much lower than their traditional counterpart O[(e^2)(n-k)k]. For example, our complexity of encoding a 64-ary (4095,2160) QC code is only 1.59% of that of traditional encoding, and our complexities of encoding the binary (4095, 2160) and (8176, 7154) QC codes are respectively 9.52% and 1.77% of those of traditional encoding. We also study the application of our low-complexity encoding algorithm to one of the most important subclasses of QC codes, namely QC-LDPC codes, especially when their parity-check matrices are rank deficient.