Simple Asymptotic Bounds on Statistical Decoder Error Rate
Statistical decoders are one of the most robust decoders for positional modulations like FSK and PPM. As we show in this work they are applicable to any (unknown) channels that have non-zero distance between received signals. This makes it possible to use statistical decoders in NOMA random access communication systems with bad Channel State Information. In this work we consider the problem of data transmission over unknown memoryless channels. To the author's knowledge this problem was not studied in literature till now. We propose repetition Kautz-Singleton codes and statistical decoders as a solution to this problem. To estimate the performance of the proposed solution we propose a lower and an upper asymptotic bounds on error rate for statistical decoder. These bounds are evaluated for Kolmogorov-Smirnov goodness-of-fit criteria and compared to a computer simulation. The lower bound seems to be close to the simulation result. The upper bound is not that close to the simulation result but it still holds. To the author's knowledge this is the first technique to derive bounds on error rate for any distance-based statistical decoder. These bounds for the Kolmogorov-Smirnov goodness-of-fit criterion also show that the error rate should be inversely proportional to the square root of code distance. Other goodness-of-fit criteria might yield asymptotically better results.