On the Pairwise Error Probability of Linear Programming Decoding on Independent Rayleigh Flat-Fading Channels

In this paper, 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. Inf. 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 pseudocode-word. We will show that the PEP of decoding to a pseudocodeword w when the all-zero codeword is transmitted on the above-mentioned channel, behaves asymptotically as K(omega) ldr (E_s/N₀)^{-|chi(omega)|}, where chi(omega) is the support set of omega, i.e., the set of nonzero coordinates, E_s/N₀ is the average signal-to-noise ratio (SNR), and K(omega) is a constant independent of the SNR. Note that the support set chi(omega) of omega is a stopping set. Thus, the asymptotic decay rate of the error probability with the average SNR is determined by the size of the smallest nonempty stopping set in the Tanner graph of H. As an example, we analyze the well-known (155,64) Tanner code and present performance curves on the independent Rayleigh flat-fading channel.