Collective Encoding and Iterative Soft-Decision Decoding of Cyclic Codes of Prime Lengths in Galois-Fourier Transform Domain

This paper presents a novel coding scheme for collective encoding and collective iterative soft-decision decoding of cyclic codes of prime lengths in Galois-Fourier transform (GFT) domain. The encoding of a cyclic code is performed on a collection of codewords which are mapped through composition, Hadamard permutations, and GFT into a codeword in an LDPC code with a binary parity-check matrix for transmission. Using this matrix, binary iterative decoding is applied to jointly decode a collection of codewords in the cyclic code. The joint-decoding allows reliability information sharing among the received vectors corresponding to the transmitted codewords in the collection during the iterative decoding process. This joint-decoding and information sharing can achieve a joint-decoding gain over the maximum likelihood decoding of individual codewords. The binary iterative decoding can be performed efficiently and reduces the decoding complexity significantly.