Repeat-Free Codes

TL;DR

This paper investigates the capacity of repeat-free sequences, develops efficient encoding algorithms with minimal redundancy, and extends the analysis to sequences with local constraints and multi-dimensional cases.

Contribution

It introduces new algorithms for encoding repeat-free sequences with low redundancy and calculates their capacities under various constraints.

Findings

Capacity of repeat-free sequences is calculated.

Efficient encoding algorithms with two bits of redundancy are proposed.

Capacities are extended to sequences with local constraints and multi-dimensional cases.

Abstract

In this paper we consider the problem of encoding data into \textit{repeat-free} sequences in which sequences are imposed to contain any $k$-tuple at most once (for predefined $k$). First, the capacity of the repeat-free constraint are calculated. Then, an efficient algorithm, which uses two bits of redundancy, is presented to encode length-$n$ sequences for $k=2+2\log (n)$. This algorithm is then improved to support any value of $k$ of the form $k=a\log (n)$, for $1<a$, while its redundancy is $o(n)$. We also calculate the capacity of repeat-free sequences when combined with local constraints which are given by a constrained system, and the capacity of multi-dimensional repeat-free codes.