arXiv subjects

Aryeh Lev Zabokritskiy

Publications and source records attributed to Aryeh Lev Zabokritskiy.

23 records · Page 2Linked to original sources

arXiv2026

Extinction Depth and q-ary Error-Correcting Codes for the Limited Permutation Channel

In the radius-one limited permutation channel, errors consist of disjoint adjacent transpositions. A correcting code must separate distinct codewords: their error balls may not contain a common received word. Hamming distance does not ensure this, because disjoint swaps can make words differing in many positions confusable. For block-concatenation codes, earlier work tested each possible collision only through the longer initial block. This sufficient condition is not necessary: we exhibit a valid ternary block set that it does not certify. We introduce extinction depth, which tracks unresolved first-block pairs through later block extensions, and prove that their extinction at one common finite horizon certifies correction at every length. The criterion gives explicit $q=3$ and $q=5$ block sets with rates above $0.6777475$ and $0.6694926$. No block-set-independent depth bound exists: we give an exact linear family, verify a quadratic formula for every $3\leq k\leq 60$, and derive a polynomial-time finite-graph test. For growing alphabets, the normalized correction loss lies between $\lnφ$ and $\ln(1.82560995)$, while explicit finite-length covers improve finite-alphabet upper bounds. We develop a parallel directed-extinction theory for detection, including an all-length block criterion, a set unresolved by the earlier test, and exact corridor depth $4k$. Stable type lifting yields optimal first-order loss $\sqrt{2}$, and weak-zigzag codes improve the $q=3,4$ lower bounds. Finally, window restriction, cancellation, and a bounded pending-input frontier extend the criterion and its finite-state verification to every fixed displacement radius $r$.

cs.IT↗

arXiv2026

Multiset Deletion Codes: Cyclic Constructions, Bounds, and Exact Results

We study deletion-correcting codes in the space of length-$n$ multisets over a $q$-ary alphabet. We present an explicit cyclic Sidon-type construction for arbitrary alphabet size $q$ and deletion radius $t$, defined by a single congruence modulo $t(t+1)^{q-2}+1$. The construction has redundancy at most $\log_q(t(t+1)^{q-2}+1)$ and admits linear-time online decoding for fixed $q$ and $t$ after finite preprocessing. We prove that its syndrome classes are asymptotically balanced and compare several general upper bounds. For a single deletion, we show that the natural sum-modulo construction is asymptotically optimal for every fixed $q$. We also obtain exact results for $q=3$ and $q=4$, including uniqueness results for optimal codes in the relevant parameter ranges, and formulate conjectures for prime alphabets.

cs.IT↗

arXiv2026

Perfect Matchings with Prescribed Differences Beyond Hall: The Two-Hole Problem

The Balister--Győri--Schelp (BGS) conjecture asks whether every zero-sum list of $2^{s-1}$ nonzero vectors in $\mathbb{F}_2^s$ is the prescribed-difference profile of a perfect matching. The conjecture remains open in general, whereas the classical Hall hyperplane case is solved when all prescribed differences cross between two affine copies of a hyperplane. We isolate the smallest mixed case beyond Hall: exactly two prescribed differences are internal. Although only two requests have changed type, the complete Hall permutation is replaced by a prescribed-difference bijection between two punctured copies of the hyperplane, with two unknown deleted vertices on each side. We call this the two-hole problem. We develop a new combinatorial method for prescribed-difference matchings, based on counting and the character structure of the binary vector space. Unlike the known Hall-type methods, which construct a matching through a sequence of local algorithmic choices, our approach proves existence through a global noncancellation phenomenon. This loss of algorithmic structure is compensated by a different advantage: the method can retain global boundary information that local exchanges do not control. As a first application, it gives a new proof of the binary Hall theorem, and it then yields a complete solution of the two-hole problem with no multiplicity assumption. We also give direct constructive proofs for symmetric even-multiplicity two-hole and four-hole families. More broadly, the new technique provides a framework for studying further subfamilies of the BGS problem by measuring how far their matching structure departs from the Hall case.

math.CO↗

arXiv2026

A $q$-ary Local Criterion for the Radius-One Limited Permutation Channel and Almost-Optimal Binary Block-Concatenation Codes

The radius-one limited permutation channel $\operatorname{LPC}_{\infty}(1)$ maps a transmitted word to any word obtained by an arbitrary set of pairwise disjoint adjacent transpositions. This is the $r=1$ case of the $\ell_\infty$-limited permutation channel of Langberg et al., and is also the zero-error version of simultaneous adjacent-swap errors. We study zero-error block-concatenation codes for this channel. Our first contribution is a $q$-ary two-stage local criterion for certifying free block-concatenation codes. The criterion replaces the global all-length confusability problem by finitely many local checks between blocks: a same-length truncated-ball test and a second-stage prefix test for unequal lengths. In the binary case, it yields explicit block-concatenation codes of rates $0.649872$, $0.652018$, and $0.653618$. The best construction improves the previous string-concatenation rate $0.642805$ and comes within $0.013049$ of the known upper bound $2/3$. Although the criterion is only sufficient, we prove that it is rate-complete: for every alphabet size $q$, the supremum of $q$-ary block rates certified by the criterion is exactly the $q$-ary zero-error capacity $C_0^{(q)}$. Thus it imposes no asymptotic rate loss. We also give an exact product-automaton verifier which decides, for a fixed prefix-free binary block set, whether the induced finite-length codes are correcting for all lengths. Finally, motivated by feedback settings, we study error detection. We prove a $q$-ary pairing upper bound and give a $q$-ary local detecting criterion. In the binary case, we construct a detecting block-concatenation code of rate $0.756707$, compared with the upper bound $\frac12\log_2 3\approx0.792481$.

cs.IT↗

arXiv2026

Polynomial Constructions and Deletion-Ball Geometry for Multiset Deletion Codes

We study error-correcting codes in the space $\mathcal{S}_{n,q}$ of length-$n$ multisets over a $q$-ary alphabet under the deletion metric, motivated by permutation channels in which ordering is completely lost and errors act only on symbol multiplicities. We develop two complementary directions. First, we present polynomial Sidon-type constructions over finite fields, in both projective and affine forms, yielding multiset $t$-deletion-correcting codes in the regime $t<q$ with redundancy $t+O(1)$, independent of the blocklength $n$. Second, we develop a geometric analysis of deletion balls in $\mathcal{S}_{n,q}$. Using difference-vector representations together with a diagonal reduction of the relevant generating functions, we derive exact generating-function expressions for individual deletion-ball sizes, exact formulas for the number of ordered pairs of multisets at a fixed distance $m$, and consequently for the average ball size. We prove that radius-$r$ deletion balls are minimized at extreme multisets and maximized at the most balanced multisets, giving a formal global characterization of extremal centers in $\mathcal{S}_{n,q}$. We further relate the maximal-ball value to the ideal difference set $S_{q-1}(r,r)$ through boundary truncation, obtaining explicit closed forms for $q=2$ and $q=3$. These geometric results lead to volume-based bounds on code size, including sphere-packing upper bounds, a boundary-aware analysis of code--anticode arguments, and Gilbert--Varshamov-type lower bounds governed by exact average ball sizes. For fixed $q$ and $t$, the resulting average-ball lower bound matches the interior-difference-set scale asymptotically.

cs.IT↗