When Variable-Length Codes Meet the Field of Error Detection
Résumé
Given a finite alphabet $A$ and a binary relation $\tau\subseteq A^*\times A^*$, a set $X$ is $\tau$-{\it independent} if $ \tau(X)\cap X=\emptyset$.
Given a quasi-metric $d$ over $A^*$ (in the meaning of \cite{W31}) and $k\ge 1$, we associate the relation $\tau_{d,k}$ defined by $(x,y)\in\tau_{d,k}$ if, and only if, $d(x,y)\le k$ \cite{CP02}.
In the spirit of \cite{JK97,N21}, the error detection-correction capability of variable-length codes can be expressed in term of conditions over $\tau_{d,k}$. With respect to the prefix metric, the factor one, and every quasi-metric associated to (anti-)automorphisms of the free monoid, we examine whether those conditions are decidable for a given regular code.
Mots clés
Anti-reflexive
automaton
automorphism
anti-automorphism
Bernoulli measure
binary relation
channel
code
codeword
complete
distance
error correction
error detection
embedding
factor
free monoid
homomorphism
independent
input word
Kraft inequality
maximal
measure
metric
monoid
output word
prefix
quasi-metric
regular
subsequences
suffix
synchronization constraint
transducer
variable-length code
word
Anti-reflexive
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