We study the universal attainability of the expurgated error exponent for discrete memoryless channels (DMCs). While the random-coding exponent is known to be universally attainable via maximum mutual information (MMI) decoding for DMCs, it remains open whether the expurgated exponent can be attained universally. We show that this is not the case in general. Specifically, we construct a family of DMCs for which no single sequence of codes can attain the expurgated exponent simultaneously for all channels in the family, even at rate zero. In addition, for the same channel family, we show that MMI decoding fails to achieve the expurgated exponent for any channel in the family.

翻译：我们研究了离散无记忆信道（DMC）中删减误差指数的普遍可达性。尽管已知随机编码指数可通过最大互信息（MMI）译码在DMC中普遍达到，但删减指数是否能够普遍达到仍然悬而未决。我们证明这通常并不成立。具体而言，我们构建了一类DMC信道族，使得不存在任何单一码序列能够在该信道族中的所有信道上同时达到删减指数，即使在零速率下也是如此。此外，针对同一信道族，我们证明了MMI译码无法在该信道族中的任何信道上达到删减指数。