Optimal Families of Perfect Polyphase Sequences From the Array Structure of Fermat-Quotient Sequences

We show that a p-ary polyphase sequence of period p 2 from the Fermat quotients is perfect. That is, its periodic autocorrelation is zero for all non-trivial phase shifts. We call this Fermat-quotient sequence. We propose a collection of optimal families of perfect polyphase sequences using the Ferm...

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Bibliographic Details
Published in:IEEE transactions on information theory 2016-02, Vol.62 (2), p.1076-1086
Main Authors: Ki-Hyeon Park, Hong-Yeop Song, Dae San Kim, Golomb, Solomon W.
Format: Article
Language:English
Subjects:
Algebra
Arrays
Autocorrelation
Byproducts
Collection
Construction
Correlation
Correlation analysis
Fermat-quotient sequences
Frank-Zadoff sequences
Generators
Generators of perfect sequences
Indexes
Multiaccess communication
Number theory
Optimal family of perfect sequences
Optimization
Perfect polyphase sequences
Periodic structures
Spread spectrum communication
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Summary:We show that a p-ary polyphase sequence of period p 2 from the Fermat quotients is perfect. That is, its periodic autocorrelation is zero for all non-trivial phase shifts. We call this Fermat-quotient sequence. We propose a collection of optimal families of perfect polyphase sequences using the Fermatquotient sequences in the sense of the Sarwate bound. That is, the cross correlation of two members in a family is upper bounded by p. To investigate some relation between Fermat-quotient sequences and Frank-Zadoff sequences and to construct optimal families including these sequences, we introduce generators of p-ary polyphase sequences of period p 2 using their p × p array structures. We call an optimal generator to be the generator of some p-ary polyphase sequences which are perfect and which gives an optimal family by the proposed construction. Finally, we propose an algebraic construction for optimal generators as another main result. A lot of optimal families of size p - 1 can be constructed from these optimal generators, some of which are known to be from the Fermat-quotient sequences or from the Frank-Zadoff sequences, but some families are new for p ≥ 11. The relation between the Fermat-quotient sequences and the Frank-Zadoff sequences is determined as a by-product.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2015.2511780