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 journalArticlepeer-review

23 Citations (Scopus)


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
Issue number2
Publication statusPublished - 2016 Feb 1

Bibliographical note

Funding Information:
This work was supported by the Ministry of Science, ICT and Future Planning under Grant 10047212. This paper was presented at the 2015 IEEE International Symposium on Information Theory. (Corresponding author: Hong-Yeop Song.).

Publisher Copyright:
© 2015 IEEE.

All Science Journal Classification (ASJC) codes

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


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