Position recovery on a circle based on coding theory

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Abstract

We treat the problem of designing periodical binary sequences for the purpose of being able to recover the position of a specified bit in the sequence in respect to a defined starting position. This can be useful for example for position recovery on a rotating device. Known sequences which can serve this purpose are the de-Bruijn sequences of periodicity 2n or maximal sequences of periodicity 2n-1. Both such sequences have the property that any consecutive n bits in them form a different pattern, and therefore define uniquely their position with respect to a defined starting point. Although each bit in a maximal length sequence of periodicity 2n-1 starts a different n-tuple, the complexity of recovering the position of a given n-tuple, based on its specific pattern, is that of performing a log operation over GF(2n), which is exponential. Maximal length sequences are characterized by a structure under which location indices that belong to the same cyclotomic coset are assigned the same value. A new class of binary sequences of periodicity 2n-1, characterized by the same structure (location indices that belong to the same cyclotomic coset are assigned the same value) is presented in this paper. It is shown that in these sequences the recovery of a location index of any specified bit is of complexity that is linear with n. There is however an additional complexity defined as ‘space complexity’, which is related to the fact that the bits whose values determine the location of a specified bit are not all located in the direct vicinity of the bit whose location is to recovered. The resemblance between the space complexity encountered here, and that encountered in a standard binary search is discussed.

Original languageEnglish
Title of host publicationAlgebraic Coding - 1st French-Soviet Workshop, Proceedings
EditorsGerard Cohen, Antoine Lobstein, Gilles Zemor, Simon Litsyn
PublisherSpringer Verlag
Pages87-101
Number of pages15
ISBN (Print)9783540551300
DOIs
StatePublished - 1 Jan 1992
Event1st French-Soviet Workshop on Algebraic Coding, 1991 - Paris, France
Duration: 22 Jul 199124 Jul 1991

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume573 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference1st French-Soviet Workshop on Algebraic Coding, 1991
Country/TerritoryFrance
CityParis
Period22/07/9124/07/91

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