Position recovery on a circle based on coding theory
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.
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