Repetition Error Correcting Sets: Explicit Constructions and Prefixing Methods

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 theore...

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Bibliographic Details
Published in:SIAM journal on discrete mathematics Vol. 23; no. 4; pp. 2120 - 2146
Main Authors: Dolecek, Lara, Anantharam, Venkat
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2010
Subjects:
Algebra
Codes
Combinatorics
Combinatorics. Ordered structures
Error correction & detection
Exact sciences and technology
Mathematics
Noise
Sciences and techniques of general use
11A07
Enumeration
residue systems
Error estimation
Generating function
Redundancy
enumeration problems
synchronization error correcting codes
generating functions
Synchronization
Binary code
Immunity
congruencies
Linear code
Upper bound
Prefix code
Concatenation
Residue
Discrete mathematics
94B50
Error correcting code
05A15
Repetition
ISSN:0895-4801, 1095-7146
Online Access:Get full text
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Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:0895-4801
1095-7146
DOI:10.1137/080730093