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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| Vydáno v: | SIAM journal on discrete mathematics Ročník 23; číslo 4; s. 2120 - 2146 |
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| Jazyk: | angličtina |
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Philadelphia, PA
Society for Industrial and Applied Mathematics
01.01.2010
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| ISSN: | 0895-4801, 1095-7146 |
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| 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. |
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| AbstractList | 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. |
| Author | Anantharam, Venkat Dolecek, Lara |
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| Cites_doi | 10.1215/ijm/1255631587 10.1073/pnas.35.8.481 10.1090/S0002-9904-1949-09219-4 10.1215/ijm/1255631807 10.1090/S0025-5718-96-00669-2 |
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| Keywords | 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 |
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| References | R3 Levenshtein V. I. (R5) 1966; 10 R12 Rosser J. B. (R8) 1962; 6 Levenshtein V. I. (R4) 1965; 1 R7 Gilbert E. N. (R2) 1961; 5 Varshamov R. R. (R11) 1965; 26 |
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| SubjectTerms | Algebra Codes Combinatorics Combinatorics. Ordered structures Error correction & detection Exact sciences and technology Mathematics Noise Sciences and techniques of general use |
| Title | Repetition Error Correcting Sets: Explicit Constructions and Prefixing Methods |
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