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

Guang Gong, Hong Yeop Song

Research output: Contribution to journalArticle

22 Citations (Scopus)

### Abstract

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.

Original language English 2590-2598 9 Discrete Applied Mathematics 154 18 https://doi.org/10.1016/j.dam.2006.04.025 Published - 2006 Dec 1

### Fingerprint

Autocorrelation
p.m.
Autocorrelation Function
Galois field

### All Science Journal Classification (ASJC) codes

• Discrete Mathematics and Combinatorics
• Applied Mathematics

### Cite this

title = "Two-tuple balance of non-binary sequences with ideal two-level autocorrelation",
abstract = "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.",
author = "Guang Gong and Song, {Hong Yeop}",
year = "2006",
month = "12",
day = "1",
doi = "10.1016/j.dam.2006.04.025",
language = "English",
volume = "154",
pages = "2590--2598",
journal = "Discrete Applied Mathematics",
issn = "0166-218X",
publisher = "Elsevier",
number = "18",

}

In: Discrete Applied Mathematics, Vol. 154, No. 18, 01.12.2006, p. 2590-2598.

Research output: Contribution to journalArticle

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 -