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On the maximum size of variable-length non-overlapping codes

Geyang Wang, Qi Wang

Abstract

Non-overlapping codes are a set of codewords such that the prefix of each codeword is not a suffix of any codeword in the set, including itself. If the lengths of the codewords are variable, it is additionally required that every codeword is not contained in any other codeword as a subword. Let $C(n,q)$ be the maximum size of $q$-ary fixed-length non-overlapping codes of length $n$. The upper bound on $C(n,q)$ has been well studied. However, the nontrivial upper bound on the maximum size of variable-length non-overlapping codes of length at most $n$ remains open. In this paper, by establishing a link between variable-length non-overlapping codes and fixed-length ones, we are able to show that the size of a $q$-ary variable-length non-overlapping code is upper bounded by $C(n,q)$. Furthermore, we prove that the average length of the codewords in a $q$-ary variable-length non-overlapping codes is lower bounded by $\lceil \log_q \tilde{C} \rceil$, and is asymptotically no shorter than $n-2$ as $q$ approaches $\infty$, where $\tilde{C}$ denotes the cardinality of $q$-ary variable-length non-overlapping codes of length up to $n$.

On the maximum size of variable-length non-overlapping codes

Abstract

Non-overlapping codes are a set of codewords such that the prefix of each codeword is not a suffix of any codeword in the set, including itself. If the lengths of the codewords are variable, it is additionally required that every codeword is not contained in any other codeword as a subword. Let be the maximum size of -ary fixed-length non-overlapping codes of length . The upper bound on has been well studied. However, the nontrivial upper bound on the maximum size of variable-length non-overlapping codes of length at most remains open. In this paper, by establishing a link between variable-length non-overlapping codes and fixed-length ones, we are able to show that the size of a -ary variable-length non-overlapping code is upper bounded by . Furthermore, we prove that the average length of the codewords in a -ary variable-length non-overlapping codes is lower bounded by , and is asymptotically no shorter than as approaches , where denotes the cardinality of -ary variable-length non-overlapping codes of length up to .
Paper Structure (5 sections, 6 theorems, 16 equations, 2 figures)

This paper contains 5 sections, 6 theorems, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Upper bound on the size of variable-length non-overlapping codes
  4. The minimum average length of variable-length non-overlapping codes
  5. Conclusion

Key Result

Lemma 1

Suppose that $S \subseteq \cup_{i=2}^n \mathbb{Z}_q^i$ is a $q$-ary variable-length non-overlapping code. For two distinct codewords $\boldsymbol{u}, \boldsymbol{v} \in S$, we have $\tilde{\boldsymbol{u}} \cap \tilde{\boldsymbol{v}} = \emptyset$, where $\tilde{\boldsymbol{u}}, \tilde{\boldsymbol{v}}

Figures (2)

  • Figure 1: Four possible cases for $\boldsymbol{w} \in \mathrm{Pre}(\boldsymbol{v}') \cap \mathrm{Suf}(\boldsymbol{u}')$.
  • Figure 2: Four possible cases for $\boldsymbol{w} \in \mathrm{Pre}(\boldsymbol{u}') \cap \mathrm{Suf}(\boldsymbol{v}')$.

Theorems & Definitions (13)

  • Definition 1: Non-overlapping codes
  • Definition 2: Prefix codes
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Corollary 1
  • ...and 3 more