A Simple Tight Bound on Error Probability of Block Codes with Application to Turbo Codes

A simple bound on the probability of decoding error for block codes is derived in closed form. This bound is based on the bounding techniques developed by Gallager. We obtained an upper bound both on the word-error probability and the bit-error probability of block codes. The bound is simple, since it does not require any integration or optimization in its final version. The bound is tight since it works for signal-to-noise ratios (SNRs) very close to the Shannon capacity limit. The bound uses only the weight distribution of the code. The bound for nonrandom codes is tighter than the original Gallager bound and its new versions derived by Sason and Shamai and by Viterbi and Viterbi. It also is tighter than the recent simpler bound by Viterbi and Viterbi and simpler than the bound by Duman and Salehi, which requires two-parameter optimization. For long blocks, it competes well with more complex bounds that involve integration and parameter optimization, such as the tangential sphere bound by Poltyrev, elaborated by Sason and Shamai, and investigated by Viterbi and Viterbi, and the geometry bound by Dolinar, Ekroot, and Pollara. We also obtained a closed-form expression for the minimum SNR threshold that can serve as a tight upper bound on maximum-likelihood capacity of nonrandom codes. We also have shown that this minimum SNR threshold of our bound is the same as for the tangential sphere bound of Poltyrev. We applied this simple bound to turbo-like codes.