The Kraft sum as a monotone function on the refinement-ordered set of uniquely decipherable codes
Stephan Foldes
TL;DR
This paper investigates the properties of the Kraft sum as a monotone function within the refinement order of uniquely decipherable codes, providing characterizations and structural insights into these codes.
Contribution
It introduces a new perspective on UD codes by analyzing the Kraft sum as a monotone function and characterizes codes with equal Kraft sums through specific equalities.
Findings
Kraft sum is monotone on the refinement-ordered set of UD codes
Chains of UD codes with the same Kraft sum are of a simple sequence type
Characterization of UD codes via equalities involving Kraft sums
Abstract
The set of all uniquely decipherable (UD) codes is partially ordered by refinement, meaning that all strings in the cruder code can be represented as concatenations of strings taken from the finer code. The Kraft sum is a monotone (increasing) function on this poset. In the refinement order, chains of UD codes having the same Kraft sum are necessarily of the simple sequence type. A characterization of UD codes in terms of equalities involving Kraft sums is also given.
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