Tunstall code is known as an optimal variable-to-fixed length (VF) lossless source code under the criterion of average coding rate, which is defined as the codeword length divided by the average phrase length. In this paper we define the average coding rate of a VF code as the expectation of the pointwise coding rate defined by the codeword length divided by the phrase length. We call this type of average coding rate the average pointwise coding rate. In this paper, a new VF code is proposed. An incremental parsing tree construction algorithm like the one that builds Tunstall parsing tree is presented. It is proved that this code is optimal under the criterion of the average pointwise coding rate, and that the average pointwise coding rate of this code converges asymptotically to the entropy of the stationary memoryless source emitting the data to be encoded. Moreover, it is proved that the proposed code attains better worst-case coding rate than Tunstall code.

Average coding rate of a multi-shot Tunstall code, which is a variation of variable-to-fixed length (VF) lossless source codes, for stationary memoryless sources is investigated. A multi-shot VF code parses a given source sequence to variable-length blocks and encodes them to fixed-length codewords. If we consider the situation that the parsing count is fixed, overall multi-shot VF code can be treated as a one-shot VF code. For this setting of Tunstall code, the compression performance is evaluated using two criterions. The first one is the average coding rate which is defined as the codeword length divided by the average block length. The second one is the expectation of the pointwise coding rate. It is proved that both of the above average coding rate converge to the entropy of a stationary memoryless source under the assumption that the geometric mean of the leaf counts of the multi-shot Tunstall parsing trees goes to infinity.

This paper clarifies two variable-to-fixed length codes which achieve optimum large deviations performance of empirical compression ratio. One is Lempel-Ziv code with fixed number of phrases, and the other is an arithmetic code with fixed codeword length. It is shown that Lempel-Ziv code is asymptotically optimum in the above sense, for the class of finite-alphabet and finite-state sources, and that the arithmetic code is asymptotically optimum for the class of finite-alphabet unifilar sources.