Upper bounds on exponentiated expected length of optimal one-to-one codes

Abstract

In this paper, we consider an exponentially weighted average codeword length introduced by Campbell as a performance measure for source codes. This criterion assumes that the cost is an exponential function of the codeword length and includes the usual expected codeword length criterion as a special case. Such situations could arise when the cost for encoding and decoding is significant, or if the buffer overflow caused by long codewords is a serious issue. The source codes under consideration are one-to-one encodings for a discrete memoryless source, which are one-shot encodings associating a distinct codeword with each source symbol. Such encodings could be employed when only a single source symbol rather than a sequence of source symbols needs to be transmitted. For example, such a situation can arise when the last message must be acknowledged before the next message can be transmitted. We consider two slightly different types of one-to-one encodings (depending on whether the empty codeword is used or not) and obtain several new upper bounds on Campbell's average length of optimal one-to-one codes when the probability of the most likely source symbol is available.