## Abstract

In this paper, we give a construction of optimal families of N-ary perfect sequences of period N^{2}, where N is a positive odd integer. For this, we re-define perfect generators and optimal generators of any length N which were originally defined only for odd prime lengths by Park, Song, Kim, and Golomb in 2016, but investigate the necessary and sufficient condition for these generators for arbitrary length N. Based on this, we propose a construction of odd length optimal generators by using odd prime length optimal generators. For a fixed odd integer N and its odd prime factor p, the proposed construction guarantees at least (N/p)^{p-1}φ(N/p)φ(p)φ(p-1)/φ(N)^{2} inequivalent optimal generators of length N in the sense of constant multiples, cyclic shifts, and/or decimations. Here, φ (·) is Euler's totient function. From an optimal generator one can construct lots of different N-ary optimal families of period N^{2}, all of which contain p_{min}^{-1} perfect sequences, where p_{min} is the least positive prime factor of N.

Original language | English |
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Pages (from-to) | 2901-2909 |

Number of pages | 9 |

Journal | IEEE Transactions on Information Theory |

Volume | 64 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2018 Apr |

### Bibliographical note

Funding Information:Manuscript received May 31, 2017; revised January 24, 2018; accepted January 24, 2018. Date of publication February 5, 2018; date of current version March 15, 2018. This work was supported by the National Research Foundation of Korea Grant through the Korea Government (MSIP) under Grant 2017R1A2B4011191. This paper was presented in part at the 2017 International Workshop on Signal Design and Its Applications in Communications.

Publisher Copyright:

© 2018 IEEE.

## All Science Journal Classification (ASJC) codes

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