Extremal Deletion-Ball Intersections under Run-Count and Lower-Order Deletion-Ball Intersection Constraints

2026-06-24

AI Summary

This study addresses extremal problems for q-ary sequences under constraints on the intersection of deletion balls and bounds on the number of runs, with applications to sequence reconstruction and reconstruction code design. By analyzing pairs of sequences whose low-order deletion ball intersections are restricted and whose run counts are bounded, the work derives a tight upper bound on their higher-order deletion ball intersections and establishes a radius lifting theorem that extends reconstruction guarantees from radius \(s\) to a larger radius \(t\). The paper innovatively provides, for the first time in finite settings, a tight upper bound incorporating low-order deletion correction constraints. Combining run-structure characterization with combinatorial analysis, it yields an asymptotically achievable bound and develops a duality theory applicable to both deletion and insertion scenarios. Key contributions include improved upper bounds for the binary case, achievability of the leading term of the intersection under run constraints, and the radius lifting theorem along with its counterpart for insertion balls.

Abstract

Motivated by sequence reconstruction and reconstruction codes, we study extremal intersections of deletion balls over a fixed $q$-ary alphabet. Let $Σ_q^n$ be the set of sequences of length $n$ over $Σ_q$, and let $D_t(x)$ denote the set of all sequences obtained from $x\inΣ_q^n$ by deleting exactly $t$ symbols. Our first result gives a finite upper bound under a lower-order deletion-correction constraint. We prove that if $x,y\inΣ_q^n$ satisfy $D_{s-1}(x)\cap D_{s-1}(y)=\varnothing$, then \[ |D_t(x)\cap D_t(y)| \le \binom{2s}{s}\binom{n-s}{t-s}. \] For binary alphabets, this strengthens a recent asymptotic upper bound of Pham, Goyal, and Kiah (2025, JCTA). We then investigate deletion-ball intersections under simultaneous constraints on run counts and lower-order deletion-ball intersections. For fixed $0<γ\le1$, integers $1\le s\le t$, and $m\ge1$, we show that if $x,y\inΣ_q^n$ have at most $γn$ runs and satisfy $|D_s(x)\cap D_s(y)|\le m$, then \[ |D_t(x)\cap D_t(y)|\le \frac{mγ^{t-s}}{(t-s)!}n^{t-s}+O_{s,t,m}(n^{t-s-1}). \] Moreover, the leading term can be attainable whenever $m$ is realized by a fixed finite-length seed pair. As a consequence, we obtain a direct lifting theorem for deletion reconstruction codes, transferring reconstruction properties from radius $s$ to larger radii $t$. Finally, we establish a parallel insertion theory and derive corresponding results for insertion-ball intersections and insertion reconstruction codes.

Problem

• deletion-ball intersection
• sequence reconstruction
• run-count constraint
• reconstruction codes
• extremal combinatorics

Innovation

• deletion-ball intersection
• reconstruction codes
• run-count constraint
• lifting theorem
• insertion-ball intersection