Combinatorial Capacity Bounds for the $q$-ary Deletion Channel

Hassan Tavakoli; Thinh Nguyen; Bella Bose

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arXiv · 2607.19559

Combinatorial Capacity Bounds for the $q$-ary Deletion Channel

Hassan Tavakoli·Thinh Nguyen·Bella Bose

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Abstract

We study the $q$-ary deletion channel via the pattern-count scalar $N_n(x,y)$, the number of deletion subsets mapping $x\in\Sigma_q^n$ to $y\in\Sigma_q^k$, 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{\Delta_n(d)}{n}, $ from which the simpler certified bound $ C_{q,n}\ge (1-d)\log_2 q-h_2(d)+\frac{\Delta_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.

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Hassan Tavakoli, Thinh Nguyen, Bella Bose. 2026-07-21. Combinatorial Capacity Bounds for the $q$-ary Deletion Channel. https://arxiv.org/abs/2607.19559

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