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

2026-07-21

AI Summary

This study investigates the channel capacity of the $q$-ary deletion channel under finite blocklength constraints. By introducing the pattern-count scalar $N_n(x,y)$ to characterize the input–output relationship and leveraging its summation identity, the authors derive—for the first time—an exact closed-form expression for mutual information under uniform input distribution. Employing combinatorial identities, entropy analysis, and probability normalization techniques, they establish provable upper and lower bounds on the capacity: $(1-d)\log_2 q - h_2(d) \le C_{q,n} \le (1-d)\log_2 q$. The tightness of these bounds is verified for specific cases with $q=2,3$ and blocklengths $n=3,5,10$. This work provides the first rigorous analytical framework for finite-length deletion channel capacity grounded in combinatorial structure.

Abstract

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.

Problem

• deletion channel
• capacity bounds
• q-ary
• combinatorial
• information theory

Innovation

• q-ary deletion channel
• pattern-count scalar
• finite-block capacity bounds
• output entropy
• combinatorial identities