1Error Exponents for Channel Coding with Application to Signal Constellation Design

Abstract — This paper concerns error exponents and the struc-ture of input distributions maximizing the random coding expo-nent for a stochastic channel model. The following conclusions are obtained under general assumptions on the channel statistics: (i) The optimal distribution has a finite number of mass points, or in the case of a complex channel, the amplitude has finite support. (ii) A new class of algorithms is introduced based on the cutting-plane method to construct an optimal input dis-tribution. The algorithm constructs a sequence of discrete distributions, along with upper and lower bounds on the random coding exponent at each iteration. (iii) In some numerical example considered, the resulting code significantly out-performs traditional signal constellation schemes such as QAM and PSK for all rates below the capacity. Index Terms — Information theory, channel coding, error ex-ponents, fading channels. I