Cross-Bifix-Free Codes Within a Constant Factor of Optimality

TL;DR

This paper introduces a new, nearly optimal construction of cross-bifix-free codes for any length and alphabet size, with improved bounds and connections to Fibonacci sequences, relevant for distributed sequence synchronization.

Contribution

It generalizes previous constructions to longer codes and larger alphabets, providing nearly optimal code sizes and new Fibonacci-based bounds.

Findings

Codes are nearly optimal in size.

Construction applies to any length and alphabet size.

New Fibonacci sequence results aid in size estimation.

Abstract

A cross-bifix-free code is a set of words in which no prefix of any length of any word is the suffix of any word in the set. Cross-bifix-free codes arise in the study of distributed sequences for frame synchronization. We provide a new construction of cross-bifix-free codes which generalizes the construction in Bajic (2007) to longer code lengths and to any alphabet size. The codes are shown to be nearly optimal in size. We also establish new results on Fibonacci sequences, that are used in estimating the size of the cross-bifix-free codes.