Multiset Deletion-Correcting Codes: Bounds and Constructions
Avraham Kreindel, Isaac Barouch Essayag, Aryeh Lev Zabokritskiy
TL;DR
This work studies multiset deletion-correcting codes in the space $\mathcal{S}_{n,q}$ under the deletion distance $d(S,T)=n-|S\cap T|$, with a focus on extremal regimes and explicit constructions. It develops tight upper bounds via sphere-packing, projection, incidence-graph arguments, and a recursive puncturing bound for the regime $t=n-k$, and provides exact optimal sizes for the extremal cases $t=n-1$ and $t=n-2$, plus a detailed analysis for $t=n-3$. On the constructive side, it completely resolves the binary case, giving an exact formula $S_2(n,t)=\big\lfloor\frac{n+1}{t+1}\big\rfloor$ and a congruence-based optimal construction, and extends to single-deletion codes for general $q$ with a sum-modulo construction that is asymptotically optimal but not always, and a general cyclic Sidon-type construction with redundancy $\log_q\bigl(t(t+1)^{q-2}+1\bigr)$ and linear-time encoding/decoding. Together, these results highlight a sharp dichotomy between the binary case, which admits exact optimal solutions for all deletion levels, and nonbinary alphabets, where boundary effects and high-dimensional geometry complicate optimality. Open questions include determining $S_q(n,1)$ for $q\ge3$ and refining bounds for arbitrary $t$, as well as exploring nonlinear versus lattice-based code optimality in this multiset deletion model.
Abstract
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 $n$.