Incremental redundancy with ACK/NACK feedback produces a variable-length stop-feedback (VLSF) code constrained to have $m$ decoding times, with an ACK/NACK feedback to the transmitter at each decoding time. This paper focuses on the numerical evaluation of the maximal achievable rate of random VLSF codes as a function of $m$ for the binary-input additive white Gaussian noise channel, binary symmetric channel, and binary erasure channel (BEC). Leveraging Edgeworth and Petrov expansions, we develop tight approximations to the tail probability of length-$n$ cumulative information density that are accurate for any blocklength $n$. We reduce Yavas et al.'s non-asymptotic achievability bound on VLSF codes with $m$ decoding times to an integer program of minimizing the upper bound on the average blocklength subject to the average error probability, minimum gap, and integer constraints. We develop two distinct methods to solve this program. Numerical evaluations show that Polyanskiy's achievability bound for VLSF codes, which assumes $m = \infty$, can be approached with a small $m$ for all three channels. For BEC, we consider systematic transmission followed by random linear fountain coding. This allows us to obtain a new achievability bound stronger than a previous bound and new VLSF codes whose rate further outperforms Polyanskiy's bound.

翻译：带ACK/NACK反馈的增量冗余方案会受到一种可变长停止反馈（VLSF）码的约束，该码限定在$m$个译码时刻，且每个译码时刻向发射端发送ACK/NACK反馈。本文聚焦于数值评估随机VLSF码在二进制输入加性高斯白噪声信道、二进制对称信道及二进制擦除信道（BEC）下最大可达速率随$m$的变化规律。通过利用Edgeworth展开和Petrov展开，我们推导出对长度为$n$的累积信息密度的尾部概率的紧致近似，该近似对任意分组长度$n$均精确。我们将Yavas等人关于含$m$个译码时刻的VLSF码的非渐近可达性界转化为整数规划问题，即：在满足平均错误概率、最小间隔及整数约束的条件下，最小化平均分组长度的上界。我们提出了两种不同方法以求解该规划问题。数值评估表明：对于假设$m=\infty$的Polyanskiy VLSF码可达性界，在这三种信道中均可通过较小的$m$逼近该界。针对BEC，我们考虑采用系统传输后接随机线性喷泉码的方案。由此获得的新可达性界优于先前界，且所构造的新型VLSF码的速率可进一步超越Polyanskiy界。