Block Markov Superposition Transmission of Fourier Transform Pair Codes

This paper proposes a new class of systematic linear codes called Fourier transform pair (FTP) codes. Given any element $\beta$ of order $n$ in a finite field, a codeword in the FTP code consists of an information vector of length $n$ followed by its Fourier transform vector defined with $\beta$ as the parity-check vector. The FTP code has length $2n$, dimension $n$, and minimum distance at least $2\sqrt{n}$. Distinguishingly, an FTP code defined with any element of order three is a maximum distance separable (MDS) code. To improve the performance of FTP codes, we turn to the block Markov superposition transmission (BMST) systems. Simulation results show that BMST-FTP codes have good performances in the water-fall region (within 0.5 $\mathbf{dB}$ away from the corresponding Shannon limits) and reach the corresponding genie-aided (GA) bounds in the error-floor region. Compared with BMST of other short codes, the BMST-FTP codes require shorter encoding memory to achieve comparable performance, potentially leading to low decoding complexity and low latency.