For variable-length coding with an almost-sure distortion constraint, Zhang et al. show that for discrete sources the redundancy is upper bounded by $\log n/n$ and lower bounded (in most cases) by $\log n/(2n)$, ignoring lower order terms. For a uniform source with a distortion measure satisfying certain symmetry conditions, we show that $\log n/(2n)$ is achievable and that this cannot be improved even if one relaxes the distortion constraint to be in expectation rather than with probability one.

翻译：对于具有几乎必然失真约束的变长编码，Zhang等人证明对于离散信源，冗余度的上界为$\log n/n$，下界（在大多数情况下）为$\log n/(2n)$（忽略低阶项）。针对满足特定对称条件的失真度量下的均匀信源，我们证明$\log n/(2n)$是可达到的，且即使将失真约束放宽为期望约束而非概率为一的约束，该下界也无法进一步改进。