### Abstract

Let p be an odd characteristic and m be the degree of primitive polynomial f(x). Let ω be its zero, that is a primitive element in F _{pm}*, then the sequence S =s_{i}, s_{i} = Tr (ω^{i}) for i = 0, 1, 2,... becomes a maximum length sequence, where Tr (·) is the trace function over F_{p}. On this fact, this paper proposes to binarize the sequence by using Legendre symbol. It will be a class of geometric sequences but its properties such as the period and autocorrelation has not been discussed. Then, it is shown that the obtained binary sequence (geometric sequence with Legendre symbol) has the period L given by 2(p^{m} - 1)/(p-1) and a certain periodic autocorrelation. After that, this paper also shows the numbers of ones and minus ones in the proposed binary sequence per a period together with some small examples.

Original language | English |
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Title of host publication | IWSDA 2013 - 6th International Workshop on Signal Design and Its Applications in Communications |

Publisher | IEEE Computer Society |

Pages | 28-31 |

Number of pages | 4 |

ISBN (Print) | 9781467364812 |

DOIs | |

Publication status | Published - 2013 |

Event | 6th International Workshop on Signal Design and Its Applications in Communications, IWSDA 2013 - Tokyo, Japan Duration: Oct 27 2013 → Nov 1 2013 |

### Other

Other | 6th International Workshop on Signal Design and Its Applications in Communications, IWSDA 2013 |
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Country | Japan |

City | Tokyo |

Period | 10/27/13 → 11/1/13 |

### Fingerprint

### Keywords

- Legendre symbol
- odd characteristic
- primitive polynomial
- trace

### ASJC Scopus subject areas

- Computer Networks and Communications
- Signal Processing

### Cite this

*IWSDA 2013 - 6th International Workshop on Signal Design and Its Applications in Communications*(pp. 28-31). [6849054] IEEE Computer Society. https://doi.org/10.1109/IWSDA.2013.6849054

**A binarization of geometric sequences with Legendre symbol and its autocorrelation.** / Nogami, Yasuyuki; Tada, Kazuki; Uehara, Satoshi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*IWSDA 2013 - 6th International Workshop on Signal Design and Its Applications in Communications.*, 6849054, IEEE Computer Society, pp. 28-31, 6th International Workshop on Signal Design and Its Applications in Communications, IWSDA 2013, Tokyo, Japan, 10/27/13. https://doi.org/10.1109/IWSDA.2013.6849054

}

TY - GEN

T1 - A binarization of geometric sequences with Legendre symbol and its autocorrelation

AU - Nogami, Yasuyuki

AU - Tada, Kazuki

AU - Uehara, Satoshi

PY - 2013

Y1 - 2013

N2 - Let p be an odd characteristic and m be the degree of primitive polynomial f(x). Let ω be its zero, that is a primitive element in F pm*, then the sequence S =si, si = Tr (ωi) for i = 0, 1, 2,... becomes a maximum length sequence, where Tr (·) is the trace function over Fp. On this fact, this paper proposes to binarize the sequence by using Legendre symbol. It will be a class of geometric sequences but its properties such as the period and autocorrelation has not been discussed. Then, it is shown that the obtained binary sequence (geometric sequence with Legendre symbol) has the period L given by 2(pm - 1)/(p-1) and a certain periodic autocorrelation. After that, this paper also shows the numbers of ones and minus ones in the proposed binary sequence per a period together with some small examples.

AB - Let p be an odd characteristic and m be the degree of primitive polynomial f(x). Let ω be its zero, that is a primitive element in F pm*, then the sequence S =si, si = Tr (ωi) for i = 0, 1, 2,... becomes a maximum length sequence, where Tr (·) is the trace function over Fp. On this fact, this paper proposes to binarize the sequence by using Legendre symbol. It will be a class of geometric sequences but its properties such as the period and autocorrelation has not been discussed. Then, it is shown that the obtained binary sequence (geometric sequence with Legendre symbol) has the period L given by 2(pm - 1)/(p-1) and a certain periodic autocorrelation. After that, this paper also shows the numbers of ones and minus ones in the proposed binary sequence per a period together with some small examples.

KW - Legendre symbol

KW - odd characteristic

KW - primitive polynomial

KW - trace

UR - http://www.scopus.com/inward/record.url?scp=84904540690&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904540690&partnerID=8YFLogxK

U2 - 10.1109/IWSDA.2013.6849054

DO - 10.1109/IWSDA.2013.6849054

M3 - Conference contribution

AN - SCOPUS:84904540690

SN - 9781467364812

SP - 28

EP - 31

BT - IWSDA 2013 - 6th International Workshop on Signal Design and Its Applications in Communications

PB - IEEE Computer Society

ER -