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A Weak Structural Form of Commutative Equivalence in Finite Codes

Dean Kraizberg

Abstract

We investigate the structural relationship between prefix-free codes over the binary alphabet and a class of unlabeled rooted trees, which we call \emph{symmetric} trees. We establish a canonical correspondence between prefix-free codes and symmetric trees, preserving not only the lengths of codewords but also some additional commutative structure. Using this correspondence, we provide a result related to the commutative equivalence conjecture. We show that for every code, there exists a prefix-free code such that, for each fixed word length, the sums of powers of two determined by the occurrences of a distinguished symbol are equal.

A Weak Structural Form of Commutative Equivalence in Finite Codes

Abstract

We investigate the structural relationship between prefix-free codes over the binary alphabet and a class of unlabeled rooted trees, which we call \emph{symmetric} trees. We establish a canonical correspondence between prefix-free codes and symmetric trees, preserving not only the lengths of codewords but also some additional commutative structure. Using this correspondence, we provide a result related to the commutative equivalence conjecture. We show that for every code, there exists a prefix-free code such that, for each fixed word length, the sums of powers of two determined by the occurrences of a distinguished symbol are equal.

Paper Structure

This paper contains 7 sections, 5 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some Background on Codes
  3. Symmetric Trees and Main Results
  4. Proofs of Main Results
  5. Proof of Theorem \ref{['theorem:correspondence_symmetric_pfc']}
  6. Proof of Theorem \ref{['theorem:sum_of_powers_equivalence']}
  7. Further Discussion

Key Result

Theorem 1.3

Let $C$ be a code over an alphabet of size $k$. Then Conversely, let $A$ be a finite multiset of positive integers satisfying Then there exists a prefix-free code $C$ such that the multiset of codeword lengths of $C$ is precisely $A$, that is, $\{ \lvert c \rvert : c \in C \} = A$ as multisets.

Figures (1)

  • Figure 1: Region in which we seek a word that is neither a prefix of nor has a prefix in the existing code $T_0$.

Theorems & Definitions (16)

  • Definition 1.2
  • Theorem 1.3: Kraft--McMillan
  • Definition 1.4
  • Conjecture 1.5: Schützenberger
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • Definition 2.1
  • ...and 6 more