On the Effects of Pseudo-Codewords on Independent Rayleigh Flat-Fading Channels

In this work, we consider the pairwise error probability (PEP) of a linear programming (LP) decoder for a general binary linear code as formulated by Feldman et al. (IEEE Trans. Inform. Theory, Mar. 2005) on an independent (or memoryless) Rayleigh flat-fading channel with coherent detection and perfect channel state information (CSI) at the receiver. Let H be a parity-check matrix of a binary linear code and consider LP decoding based on H. The output of the LP decoder is always a pseudo-codeword. We will show that the PEP of decoding to a pseudo-codeword omega when the all-zero codeword is transmitted on an independent Rayleigh flat-fading channel with coherent detection and perfect CSI at the receiver, behaves asymptotically as K(omega)ldr(E_s/N_O)^-chi(omega), where chi(omega) is the support set of omega, i.e., the set of non-zero coordinates, E_s/N_O is the average signal-to-noise ratio (SNR), and K(omega) is a constant independent of the SNR. Thus, the asymptotic decay rate of the error probability with the average SNR is determined by the size of the smallest non-empty stopping set in the Tanner graph of H.