On the Fixed-Length-Burst Levenshtein Ball with Unit Radius

Consider a length-$n$ sequence $\bm{x}$ over a $q$-ary alphabet. The \emph{fixed-length Levenshtein ball} $\mathcal{L}_t(\bm{x})$ of radius $t$ encompasses all length-$n$ $q$-ary sequences that can be derived from $\bm{x}$ by performing $t$ deletions followed by $t$ insertions. Analyzing the size and structure of these balls presents significant challenges in combinatorial coding theory. Recent studies have successfully characterized fixed-length Levenshtein balls in the context of a single deletion and a single insertion. These works have derived explicit formulas for various key metrics, including the exact size of the balls, extremal bounds (minimum and maximum sizes), as well as expected sizes and their concentration properties. However, the general case involving an arbitrary number of $t$ deletions and $t$ insertions $(t>1)$ remains largely uninvestigated. This work systematically examines fixed-length Levenshtein balls with multiple deletions and insertions, focusing specifically on \emph{fixed-length burst Levenshtein balls}, where deletions occur consecutively, as do insertions. We provide comprehensive solutions for explicit cardinality formulas, extremal bounds (minimum and maximum sizes), expected size, and concentration properties surrounding the expected value.