Multiset Deletion-Correcting Codes: Bounds and Constructions

We study error-correcting codes in the space $\mathcal{S}_{n,q}$ of length-$n$ multisets over a $q$-ary alphabet, motivated by permutation channels in which ordering is completely lost and errors act solely by deletions of symbols, i.e., by reducing symbol multiplicities. Our focus is on the \emph{extremal deletion regime}, where the channel output contains $k=n-t$ symbols. In this regime, we establish tight or near-tight bounds on the maximum code size. In particular, we determine the exact optimal code sizes for $t=n-1$ and for $t=n-2$, develop a refined analysis for $t=n-3$, and derive a general recursive puncturing upper bound for $t=n-k$ via a reduction from parameters $(n,k)$ to $(n-1,k-1)$. On the constructive side, we completely resolve the binary multiset model: for all $t\ge1$ we determine $S_2(n,t)$ exactly and give an explicit optimal congruence-based construction. We then study single-deletion codes beyond the binary case, presenting general $q$-ary constructions and showing, via explicit small-parameter examples, that the natural modular construction need not be optimal for $q\ge3$. Finally, we present an explicit cyclic Sidon-type linear construction for general $(q,t)$ based on a single congruence constraint, with redundancy $\log_q\!\bigl(t(t+1)^{q-2}+1\bigr)$ and encoding and decoding complexity linear in the blocklength $n$.