The information-outage probability of finite-length codes over AWGN channels

The performance of random error control codes approaches the Shannon capacity limit as the code length goes to infinity. When the code length is finite, then the code will be unable to achieve arbitrarily low error probability and a nonzero codeword error rate is inevitable. Information-theoretic bounds on codeword error rate may be found as a function of length through traditional methods such as sphere packing. Alternatively, the behavior of finite-length codes can be characterized in terms of an information-outage probability. The information- outage probability is the probability that the mutual information, which is a random variable, is less than the rate. In this paper, a Gaussian approximation is proposed that accurately models the information-outage probability for moderately small codes. The information-outage probability is related to several previously derived bounds, including Shannon's sphere-packing and random coding bounds, as well as a bound on maximal error probability known as Feinstein's lemma. It is shown that the information- outage probability is a useful predictor of achievable error rate.