Threshold of overflow probability in terms of smooth max-entropy for variable-length compression allowing errors

This paper deals with one-shot fixed-to-variable length source coding allowing error probability. We adopt the criterion of the overflow probability and treat prefix and non-prefix codes. The infimum of the threshold of the overflow probability is investigated under the condition that the error probability and the overflow probability are bounded above by positive constants. We show this threshold is evaluated in terms of the smooth max-entropy and clarify the difference of the thresholds between prefix and non-prefix codes. Further, we elucidate the benefit of allowing a nonzero error compared with lossless coding.