Repetition Error Correcting Sets: Explicit Constructions and Prefixing Methods

Abstract

In this paper we study the problem of finding maximally sized subsets of binary strings (codes) of equal length that are immune to a given number $r$ of repetitions, in the sense that no two strings in the code can give rise to the same string after $r$ repetitions. We propose explicit number theoretic constructions of such subsets. In the case of $r=1$ repetition, the proposed construction is asymptotically optimal. For $r\geq1$, the proposed construction is within a constant factor of the best known upper bound on the cardinality of a set of strings immune to $r$ repetitions. Inspired by these constructions, we then develop a prefixing method for correcting any prescribed number $r$ of repetition errors in an arbitrary binary linear block code. The proposed method constructs for each string in the given code a carefully chosen prefix such that the resulting strings are all of the same length and such that despite up to any $r$ repetitions in the concatenation of the prefix and the codeword, the original codeword can be recovered. In this construction, the prefix length is made to scale logarithmically with the length of strings in the original code. As a result, the guaranteed immunity to repetition errors is achieved while the added redundancy is asymptotically negligible.