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原文摘要

We determine sharp class-size estimates for multiset codes over $q$ symbols that correct up to $t$ deletions. For fixed $q,t\ge2$, we prove that the syndrome classes defined by Nathanson's geometric-sum labels modulo $M=1+t+\cdots+t^{q-1}$ differ from their average size by $O(n^{q/p-1})$, where $n$ is the blocklength and $p$ is the least prime divisor of $q$. This exponent is sharp, and every prescribed class has size asymptotic to the average. For prime $q$, all classes have equal size when $n\equiv-1,\ldots,-(q-1)\pmod M$. We also present our radix construction, whose labels are zero and the first $q-1$ powers of $t+1$, starting at $1$, modulo $t(t+1)^{q-2}+1$. We determine its sharp class-size deviation exponent. For even $q\ge4$, its syndrome classes have a smaller deviation exponent, although their average size is smaller. Over four symbols and every $t\ge2$, we show that the radix labels still correct up to $t$ deletions modulo a reduced modulus, which gives bounded deviations and eventually larger classes than either unmodified construction. For a fixed number $k\ge2$ of surviving symbols, we prove that some maximum code contains all constant words, reducing the remaining optimization to multisets with multiplicities below $k$. Finally, for one deletion at length four over $q\ge4$ symbols, we show that the best bound from fractional weights on deletion outputs, rounded down, equals the maximum code size exactly when $q\equiv2,4\pmod6$.