q元删除信道的组合容量界
Combinatorial Capacity Bounds for the $q$-ary Deletion Channel
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中文总结 AI 辅助
研究q元删除信道，通过模式计数标量\(N_n(x,y)\)及相关和恒等式得出有限块容量区间等结果，给出精确均匀输入速率及小\(d\)界，通过数值实验在特定参数下证实了所有界。
中文摘要 AI 辅助
我们通过模式计数标量\(N_n(x,y)\)研究q元删除信道，它是将\(x\in\Sigma_q^n\)映射到\(y\in\Sigma_q^k\)的删除子集数量，可分解转移概率。\(N_n\)上的两个和恒等式证明了随机归一化，并在均匀输入下得出精确的闭式输出熵。这些给出了有限块容量区间\((1 - d)\log_2 q - h_2(d)\;\leq\; C_{q,n}\;\leq\; (1 - d)\log_2 q\)。还给出了精确的均匀输入速率，得出更简单的认证界\(C_{q,n}\geq (1 - d)\log_2 q - h_2(d)+\frac{\Delta_n(d)}{n}\)。小\(d\)界\(C_q(d)\geq\log_2 q + d\log_2 d + O(d)\)对所有\(q\geq 2\)成立。在\(n = 3,5,10\)和\(q = 2,3\)的数值实验证实了所有界。
英文摘要
We study the \(q\)-ary deletion channel via the pattern-count scalar \(N_n(x,y)\), the number of deletion subsets mapping \(x\inΣ_q^n\) to \(y\inΣ_q^k\), which factorizes the transition probability. Two sum identities on \(N_n\) certify stochastic normalization and, under uniform input, yield an exact closed-form output entropy. These give the finite-block capacity sandwich \( (1-d)\log_2 q-h_2(d)\;\le\; C_{q,n}\;\le\;(1-d)\log_2 q. \) The exact uniform-input rate is \( \frac{1}{n}I_U(X;Y) =(1-d)\log_2 q+\frac{1}{n}H_{\mathrm{Bin}}(n,1-d)-h_2(d)+\frac{Δ_n(d)}{n}, \) from which the simpler certified bound \( C_{q,n}\ge (1-d)\log_2 q-h_2(d)+\frac{Δ_n(d)}{n} \) follows. The small-\(d\) bound \(C_q(d)\ge\log_2 q+d\log_2 d+O(d)\) follows for all \(q\ge 2\). Numerical experiments at \(n=3,5,10\) and \(q=2,3\) confirm all bounds.
