Umlaut information

The sphere-packing bound quantifies the error exponent for noisy channel coding for rates above a critical value. Here, we study the intriguing single-letter form of the zero-rate limit of the sphere-packing bound, which we call the umlaut information, inspired by the lautum information introduced by Palomar and Verdú. Unlike the latter quantity, the umlaut information is additive for parallel uses of channels. Furthermore, it allows to bridge the zero-rate error exponent of non-signalling-assisted coding with the zero-rate error exponent of list decoding in the large list limit, for which we bound the convergence rate. An extended version of this paper is available at [1].