### Abstract

Let p be an odd characteristic and m be the degree of a primitive polynomial f (x) over the prime field F_{p}. Let ω be its zero, that is a primitive element in F^{∗} _{pm} , the sequence S = {si}, si = Tr (ω^{i}) for i = 0, 1, 2, ⋯ becomes a non-binary 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, autocorrelation, and linear complexity have not been discussed. Then, this paper shows that the generated binary sequence (geometric sequence by Legendre symbol) has the period n = 2(p^{m} ? 1)/(p ? 1) and a typical periodic autocorrelation. Moreover, it is experimentally observed that its linear complexity becomes the maximum, that is the period n. Among such experimental observations, especially in the case of m = 2, it is shown that the maximum linear complexity is theoretically proven. After that, this paper also demonstrates these properties with a small example.

Original language | English |
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Pages (from-to) | 2336-2342 |

Number of pages | 7 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E97A |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 1 2014 |

### Keywords

- Geometric sequence
- Legendre symbol
- Odd characteristic
- Primitive polynomial
- Trace

### ASJC Scopus subject areas

- Signal Processing
- Computer Graphics and Computer-Aided Design
- Electrical and Electronic Engineering
- Applied Mathematics