Optimal families of perfect polyphase sequences from the array structure of fermat-quotient sequences

Ki Hyeon Park, Hong Yeop Song, Dae San Kim, Solomon W. Golomb

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

We show that a p-ary polyphase sequence of period p2 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 p2 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.

Original languageEnglish
Article number2511780
Pages (from-to)1076-1086
Number of pages11
JournalIEEE Transactions on Information Theory
Volume62
Issue number2
DOIs
Publication statusPublished - 2016 Feb 1

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Autocorrelation
Phase shift
Byproducts

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences

Cite this

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abstract = "We show that a p-ary polyphase sequence of period p2 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 p2 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.",
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Optimal families of perfect polyphase sequences from the array structure of fermat-quotient sequences. / Park, Ki Hyeon; Song, Hong Yeop; Kim, Dae San; Golomb, Solomon W.

In: IEEE Transactions on Information Theory, Vol. 62, No. 2, 2511780, 01.02.2016, p. 1076-1086.

Research output: Contribution to journalArticle

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