### 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 |
---|---|

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 1 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

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*IEEE Transactions on Information Theory*, vol. 64, no. 4, pp. 2901-2909. https://doi.org/10.1109/TIT.2018.2801796

**A Construction of Odd Length Generators for Optimal Families of Perfect Sequences.** / Song, Min Kyu; Song, Hong-Yeop.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A Construction of Odd Length Generators for Optimal Families of Perfect Sequences

AU - Song, Min Kyu

AU - Song, Hong-Yeop

PY - 2018/4/1

Y1 - 2018/4/1

N2 - In this paper, we give a construction of optimal families of N-ary perfect sequences of period N2, 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-1φ(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 N2, all of which contain pmin -1 perfect sequences, where pmin is the least positive prime factor of N.

AB - In this paper, we give a construction of optimal families of N-ary perfect sequences of period N2, 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-1φ(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 N2, all of which contain pmin -1 perfect sequences, where pmin is the least positive prime factor of N.

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U2 - 10.1109/TIT.2018.2801796

DO - 10.1109/TIT.2018.2801796

M3 - Article

AN - SCOPUS:85041671360

VL - 64

SP - 2901

EP - 2909

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 4

ER -