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RESEARCH ARTICLE

Pinhui KE, Shengyuan ZHANG

New classes of sequence families with low correlation by

using multiplicative and additive characters

Abstract For an odd prime p, a new sequence family of

period p

– 1, size ðM – 1Þp

is proposed using multi-

plicative and additive characters. The upper bound for the

maximum magnitude of nontrivial correlations of the

sequence family is derived using well-known character

sums. The upper bound is shown to be ðr þ1Þ

ﬃﬃﬃﬃﬃﬃ

þ 3,

which meets the Welch bound asymptotically.

Keywords ﬁnite ﬁeld, character sum, correlation, poly-

phase sequence, Welch bound

1 Introduction

In wireless communications, sequences with low correla-

tion are widely used to distinguish multiple users or

channels with low mutual-access interference (MAI) [1].

To maximize the achievable data rate, polyphase sequence

sets with a variety of lengths and alphabet sizes are more

desirable than binary sequence sets for most wireless

mobile communication systems employing adaptive mod-

ulation schemes. In addition, a large number of distinct

sequences may be needed for supporting user requirements

that rapidly increase.

A sequence S ¼fsðtÞg is called a polyphase sequence

with alphabet q if sðtÞ is a qth root of unity for all t. Let

F ¼fs

,:::,s

g beasetofM cyclically distinct

polyphase sequence with period N, where s

¼fs

ðtÞg for

1£i£M. The periodic cross correlation function between

sequence s

and s

at the shift phase τ is given by

ðτÞ¼

N – 1

t¼0

ðtÞs

ðt þ τÞ:

If s

¼ s

, we call it the periodic autocorrelation function

of sequence s

at the shift phase τ, and it is denoted as

ðτÞ. Let R

max

ðFÞ be the maximal correlation of F, i.e.,

max

ðFÞ¼maxjR

ðτÞj

for any 0£τ£N – 1if1£i≠j£M and 0<τ£N – 1if

i ¼ j. The sequence family F is said to have low

correlation if R

max

ðFÞ£v

ﬃﬃﬃﬃ

for a small constant v.

There exist many designs of sequence families that meet

this asymptotic bound with equality. Readers may refer to

Refs. [1,2] for an overview on this topic.

In Ref. [3], polyphase sequence families with low

correlation were constructed from the shift and addition of

the power residue sequence (Sidelnikov sequence, respec-

tively) and its constant multiple sequences, whose

maximum correlation were shown to be upper bounded

by 2

ﬃﬃﬃ

þ 5or3

ﬃﬃﬃ

þ 4(2

ﬃﬃﬃ

þ 6or3

ﬃﬃﬃ

þ 5for

sequence families from Sidelnikov sequence, respec-

tively), where L is the period of the constructed sequences.

In Ref. [4], more generalized constructions were consid-

ered by the addition of multiple cyclic shifts of power

residue and Sidelnikov sequences, which can be repre-

sented by multiplicative character. Recently, sequence

family of prime period p, family size ðp – 2Þp

, and

maximum correlation at most ðr þ1Þ

ﬃﬃﬃ

þ 2 was obtained

by using multiplicative and additive characters in Ref. [5].

In this paper, we generalize the constructions in Ref. [5]

to the general ﬁnite ﬁeld. Our constructions are also based

on multiplicative character and additive character. How-

ever, instead of using the traditional multiplicat ive

character directly, we use it in a way similar to the one

use in the construction of Sidelnikov sequence. Before

presenting our constructions, we review some related

deﬁnitions and properties of ﬁnite ﬁeld.

Let p be a prime and q ¼ p

, where m is a positive

integer. Let F

be a ﬁnite ﬁeld containing q elements.

Given a 2 F

,anadditive character of F

is a homo-

morphism mapping from additive group ðF

,þÞ to ðC

, ⋅Þ

deﬁned by

Received February 15, 2012; accepted July 16, 2012

Pinhui KE (✉), Shengyuan ZHANG

Key Laboratory of Network Security and Cryptology, Fujian Normal

University, Fuzhou 350007, China

E-mail: keph@fjnu.edu.cn

Front. Electr. Electron. Eng.

DOI 10.1007/s11460-012-0205-z

FEE-12205-KPH.3d 3/8/012 13:49:34

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