### Abstract

Let p = ef+1 be an odd prime for some e and f, and let F_{p} be the finite field with pelements. In this paper, we explicitly describe the trace representations of the binary characteristic sequences (of period p) of all the cyclic difference sets D which are some union of cosets of eth powers H _{e} in F_{p}^{*} (Δ\= F_{p} \{0}) for e≤ 12. For this, we define eth power residue sequences of period p, which include all the binary characteristic sequences mentioned above as special cases, and reduce the problem of determining their trace representations to that of determining the values of the generating polynomials of cosets of H _{e} in F_{p}^{*} at some primitive pth root of unity, and some properties of these values are investigated. Based on these properties, the trace representation and linear complexity not only of the characteristic sequences of all the known eth residue difference sets, but of all the sixth power residue sequences are determined. Furthermore, we have determined the linear complexity of a nonconstant eth power residue sequence for any e to be either p-1 or p whenever (e,(p-1)/n)= 1, where n is the order of 2 mod p.

Original language | English |
---|---|

Article number | 5714271 |

Pages (from-to) | 1530-1547 |

Number of pages | 18 |

Journal | IEEE Transactions on Information Theory |

Volume | 57 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2011 Mar 1 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*IEEE Transactions on Information Theory*,

*57*(3), 1530-1547. [5714271]. https://doi.org/10.1109/TIT.2010.2103757

}

*IEEE Transactions on Information Theory*, vol. 57, no. 3, 5714271, pp. 1530-1547. https://doi.org/10.1109/TIT.2010.2103757

**Trace representation and linear complexity of binary eth power residue sequences of period p.** / Dai, Zongduo; Gong, Guang; Song, Hong-Yeop; Ye, Dingfeng.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Trace representation and linear complexity of binary eth power residue sequences of period p

AU - Dai, Zongduo

AU - Gong, Guang

AU - Song, Hong-Yeop

AU - Ye, Dingfeng

PY - 2011/3/1

Y1 - 2011/3/1

N2 - Let p = ef+1 be an odd prime for some e and f, and let Fp be the finite field with pelements. In this paper, we explicitly describe the trace representations of the binary characteristic sequences (of period p) of all the cyclic difference sets D which are some union of cosets of eth powers H e in Fp* (Δ\= Fp \{0}) for e≤ 12. For this, we define eth power residue sequences of period p, which include all the binary characteristic sequences mentioned above as special cases, and reduce the problem of determining their trace representations to that of determining the values of the generating polynomials of cosets of H e in Fp* at some primitive pth root of unity, and some properties of these values are investigated. Based on these properties, the trace representation and linear complexity not only of the characteristic sequences of all the known eth residue difference sets, but of all the sixth power residue sequences are determined. Furthermore, we have determined the linear complexity of a nonconstant eth power residue sequence for any e to be either p-1 or p whenever (e,(p-1)/n)= 1, where n is the order of 2 mod p.

AB - Let p = ef+1 be an odd prime for some e and f, and let Fp be the finite field with pelements. In this paper, we explicitly describe the trace representations of the binary characteristic sequences (of period p) of all the cyclic difference sets D which are some union of cosets of eth powers H e in Fp* (Δ\= Fp \{0}) for e≤ 12. For this, we define eth power residue sequences of period p, which include all the binary characteristic sequences mentioned above as special cases, and reduce the problem of determining their trace representations to that of determining the values of the generating polynomials of cosets of H e in Fp* at some primitive pth root of unity, and some properties of these values are investigated. Based on these properties, the trace representation and linear complexity not only of the characteristic sequences of all the known eth residue difference sets, but of all the sixth power residue sequences are determined. Furthermore, we have determined the linear complexity of a nonconstant eth power residue sequence for any e to be either p-1 or p whenever (e,(p-1)/n)= 1, where n is the order of 2 mod p.

UR - http://www.scopus.com/inward/record.url?scp=79951924341&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79951924341&partnerID=8YFLogxK

U2 - 10.1109/TIT.2010.2103757

DO - 10.1109/TIT.2010.2103757

M3 - Article

VL - 57

SP - 1530

EP - 1547

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 3

M1 - 5714271

ER -