### 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 |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*,

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

}

*Discrete Applied Mathematics*, vol. 154, no. 18, pp. 2590-2598. https://doi.org/10.1016/j.dam.2006.04.025

**Two-tuple balance of non-binary sequences with ideal two-level autocorrelation.** / Gong, Guang; Song, Hong Yeop.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Two-tuple balance of non-binary sequences with ideal two-level autocorrelation

AU - Gong, Guang

AU - Song, Hong Yeop

PY - 2006/12/1

Y1 - 2006/12/1

N2 - Let p be a prime, q = pm and Fq be the finite field with q elements. In this paper, we will consider q-ary sequences of period qn - 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 qn - 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 qn - 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.

AB - Let p be a prime, q = pm and Fq be the finite field with q elements. In this paper, we will consider q-ary sequences of period qn - 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 qn - 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 qn - 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.

UR - http://www.scopus.com/inward/record.url?scp=33750429950&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750429950&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2006.04.025

DO - 10.1016/j.dam.2006.04.025

M3 - Article

AN - SCOPUS:33750429950

VL - 154

SP - 2590

EP - 2598

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 18

ER -