### Abstract

Let p be a prime, q = p^{m} and F_{q} be the finite field with q elements. In this paper, we will consider q-ary sequences of period q^{n} - 1 for q > 2 and study their various balance properties: symbol-balance, difference-balance, and two-tuple-balance properties. The array structure of the sequences is introduced, and various implications between these balance properties and the array structure are proved. Specifically, we prove that if a q-ary sequence of period q^{n} - 1 is difference-balanced and has the "cyclic" array structure then it is two-tuple-balanced. We conjecture that a difference-balanced q-ary sequence of period q^{n} - 1 must have the cyclic array structure. The conjecture is confirmed with respect to all of the known q-ary sequences which are difference-balanced, in particular, which have the ideal two-level autocorrelation function when q = p.

Original language | English |
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Pages (from-to) | 2590-2598 |

Number of pages | 9 |

Journal | Discrete Applied Mathematics |

Volume | 154 |

Issue number | 18 |

DOIs | |

Publication status | Published - 2006 Dec 1 |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Discrete Applied Mathematics*,

*154*(18), 2590-2598. https://doi.org/10.1016/j.dam.2006.04.025