Let $C$ be a finite maximal prefix code of maximum codeword length $k\ge2$. We prove that, whenever the literal decoder of $C$ is synchronizing, it has a reset word of length at most $(k-1)(4k-5)$. The bound is independent of the alphabet and the number of decoder states. Its main ingredient is a short-detection theorem: a linear observation of the decoder, with terminal weight constant on states of depth $k-1$, is either identically zero or nonzero on a word of length at most $2k-3$. The proof uses a first-return decomposition in noncommuting variables and cancellation of a highest-degree coefficient. For codes whose codeword lengths are $k-1$ and $k$, we prove the sharper bound $2(k-1)^2-1$ and characterize equality. The binary code with the single short word $a^{k-1}$ attains this bound for every $k\ge2$; an age representation and a potential on circular arcs give the matching lower bound. Both upper bounds yield polynomial-time constructions in the explicit code size. The mathematical results and the algorithms are formalized in Lean, with polynomial work bounds in a specified binary cost model.
Every finite nonempty incomplete uniquely decipherable code with maximum word length $k$ has an uncompletable word of length at most $4k^2-3k$. The bound is independent of the number of codewords and their total length. Deleting a complete codeword cycle gives a finite path-counting identity; Kraft equality then suppli...
We prove that some pairs of binary words of every sufficiently large prescribed length $n$ require $\Omega(\log n\log\log n)$ states to separate by a deterministic finite automaton whose letters act as permutations. We obtain this bound by constructing distinct equal-length positive binary words that form an identity o...
A length-$n$ binary $k$-deletion code is a set of binary strings such that if we delete any $k$ bits of a string, leaving a length-$(n-k)$ binary string, we can uniquely recover the codeword. In this paper, we consider $t$-list decodable deletion codes, where after $k$ bits of a codeword are deleted, we can identify a...
We study binary quasi-cyclic LDPC codes whose parity-check matrices are full arrays of single circulant permutation matrices (CPMs), referred to here as CPM-LDPC codes. Their minimum distance is at most $(J+1)!$, where $J$ is the column weight. For every fixed pair of column and row weights $2\le J<L$, we show that thi...
Let $A_q(n,d)$ denote the largest size of a code of length $n$ and minimum Hamming distance $d$ over an alphabet of $q$ symbols. For fixed $q\ge2$ and $d\ge4$ we construct codes with redundancy $n-\log_q A_q(n,d)\le(d-3)\log_q n+O(\sqrt{\log n})$ for all large $n$. No algebraic structure is imposed on the alphabet, and...
It is well known that the length $n$ of a binary Type II code is divisible by $8$. The classical proof relies on Gleason's theorem on weight enumerators. Motivated by a question of Vera Pless, we give elementary proofs of this fact that avoid invariant theory. Our main tool is a congruence modulo $8$ for the determinan...
J. Baek, Jon-Lark Kim· 0 citations
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