如何用反馈击败球形包装约束
How to beat the sphere-packing bound with feedback
摘要：
球形包装界$E_{sp}(R)$界定了固定长度分组码的可靠性函数。对于对称信道，即使当允许从解码器到编码器的严格因果无噪声反馈时，它仍然是有效的界限。要打破束缚，就必须改变问题。虽然长期以来已知可变长度块码在用预期块长度权衡错误概率时可以做得更好，但是这种对应性表明{ \em固定延迟}设置也为通用信道提供了这样的机会。虽然$E_{sp}(R)$继续限制使用{ \em无}反馈的对称信道的误码和固定端到端延迟之间的折衷，但称为"聚焦边界“的新边界给出了对反馈可以做的事情的限制。如果低速率可靠的流量控制是自由的（即。噪声信道具有严格的正零误差容量），则可以渐近地实现聚焦界。即使当信道不具有零错误容量时，也可以通过合成适当可靠的信道来承载流控制信息来基本上击败球形填充边界。
英文摘要：
The sphere-packing bound $E_{sp}(R)$ bounds the reliability function for fixed-length block-codes. For symmetric channels, it remains a valid bound even when strictly causal noiseless feedback is allowed from the decoder to the encoder. To beat the bound, the problem must be changed. While it has long been known that variable-length block codes can do better when trading-off error probability with expected block-length, this correspondence shows that the {\em fixed-delay} setting also presents such an opportunity for generic channels. While $E_{sp}(R)$ continues to bound the tradeoff between bit error and fixed end-to-end latency for symmetric channels used {\em without} feedback, a new bound called the ``focusing bound'' gives the limits on what can be done with feedback. If low-rate reliable flow-control is free (ie. the noisy channel has strictly positive zero-error capacity), then the focusing bound can be asymptotically achieved. Even when the channel has no zero-error capacity, it is possible to substantially beat the sphere-packing bound by synthesizing an appropriately reliable channel to carry the flow-control information.
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