New quaternary sequences of even length with optimal auto-correlation

Abstract

Sequences with low auto-correlation property have been applied in code-division multiple access communication systems, radar and cryptography. Using the inverse Gray mapping, a quaternary sequence of even length N can be obtained from two binary sequences of the same length, which are called component sequences. In this paper, using interleaving method, we present several classes of component sequences from twin-prime sequences pairs or GMW sequences pairs given by Tang and Ding in 2010; two, three or four binary sequences defined by cyclotomic classes of order 4. Hence we can obtain new classes of quaternary sequences, which are different from known ones, since known component sequences are constructed from a pair of binary sequences with optimal auto-correlation or Sidel'nikov sequences.The cross-correlation function R s,t (τ ) between s and t is defined by

New optimal binary sequences with period 4p via interleaving Ding–Helleseth–Lam sequences

“…The correlation values of Ding-Helleseth-Lam sequences s 1 , s 2 , s 3 , s 4 have been determined in [11], which are useful for the main result of this paper. Here we list it as follows.…”

“…Example 1: Let p = 29, and α = 2 be a primitive element of the residue ring Z p . Then D 0 = {1, 7, 16, 20, 23, 24, 25}, 3,11,14,17,19, 21}, 5,6,9,13,22, 28}, 10, 12, 15, 18, 26, 27} are four cyclotimic classes of order 4 with respect to Z p . In this case, x = 5, y = −1, and f = 7.…”

“…The rest of this paper is organized as follows. In Section II, we recall the interleaving method, Ding-Helleseth-Lam sequences [3] and their correlation properties [11]. In Section III, we present eight classes of new interleaved sequences by choosing suitable four Ding-Helleseth-Lam sequences as column sequences.…”

Some new balanced and almost balanced quaternary sequences with low autocorrelation

“…It is desirable to have small autocorrelation, large linear complexity, and balancedness for applications in certain communications systems, cryptography and digital systems (see [4], [12], [13], [25] and [29]). Binary sequences with low autocorrelation are important building blocks for constructing quaternary sequences with low autocorrelation ( [2], [13], [31], [32]). The use of cyclotomic classes and generalized cyclotomic classes to construct binary sequences is a well-known and extensively used method [6], [7].…”

“…In [20], Kim et al constructed families of balanced quaternary sequences of even period with optimal autocorrelation, in [19], the same authors constructed quaternary sequences with ideal autocorrelation from Legendre sequences, and in [17], the same authors constructed new quaternary sequences with ideal autocorrelation from binary sequences with ideal autocorrelation. Autocorrelation of certain quaternary cyclotomic sequences of period 2p was studied in [18] by Kim, Hong and Song, and in [31], Su et al constructed new quaternary sequences of even length with optimal autocorrelation by interleaving certain combinations of binary sequences. In [9], Edemskiy and Ivanov constructed balanced quaternary sequences of period 2p with low autocorrelation, and this construction was further generalized in [27] by Shen et al Some recent progress in linear complexity of quaternary sequences over finite fields includes Edemskiy and Ivanov's work on quaternary sequences of period pq in [10], and the work of Kim et al on quaternary sequences constructed from binary Legendre sequences in [21].…”