Tangential Sphere Bounds on the Ensemble Performance of ML Decoded Gallager Codes via Their Exact Ensemble Distance Spectrum

An efficient numerical approach to the exact ensemble distance spectrum of Gallager codes has been developed by evaluating powers of polynomials. With the exact ensemble distance spectrum of Gallager codes, tangential sphere upper bounds on their maximum likelihood (ML) decoding performance over binary input AWGN channels are investigated. Numerical results indicate improved bounds have been obtained, better than Sason and Shamai's results (which are based on Gallager's upper bound on the ensemble distance spectrum), especially in the error floor region. Furthermore, some critical properties of Gallager codes, including typical minimum distance and the performance tradeoff in the waterfall and error floor regions, have been considered.