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

Abstract

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 $3\le k\le 60$, and derive a polynomial-time finite-graph test. For growing alphabets, the normalized correction loss lies between $\ln \ddot{I}†$ 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 $r$.

44 pages, 6 tables. Accompanying reproducibility package: https://doi.org/10.5281/zenodo.21398315