## 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 |
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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 |

### Bibliographical note

Funding Information:Manuscript received February 20, 2008; revised August 01, 2010; accepted August 21, 2010. Date of current version February 18, 2011. The work of G. Gong was supported by a NSERC Discovery Grant, Canada. The work of H.-Y. Song was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0083888). Part of the material in this paper was presented at the 2003 Workshop on Coding and Cryptography, INRIA, France, 2003.

## All Science Journal Classification (ASJC) codes

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