Low Correlation Sequences From Linear Combinations of Characters

Abstract

Abstract. Pairs of binary sequences formed using linear combinations of multiplicative characters of finite fields are exhibited that, when compared to random sequence pairs, simultaneously achieve significantly lower mean square autocorrelation values (for each sequence in the pair) and significantly lower mean square crosscorrelation values. If we define crosscorrelation merit factor analogously to the usual merit factor for autocorrelation, and if we define demerit factor as the reciprocal of merit factor, then randomly selected binary sequence pairs are known to have an average crosscorrelation demerit factor of 1. Our constructions provide sequence pairs with crosscorrelation demerit factor significantly less than 1, and at the same time, the autocorrelation demerit factors of the individual sequences can also be made significantly less than 1 (which also indicates better than average performance). The sequence pairs studied here provide combinations of autocorrelation and crosscorrelation performance that are not achievable using sequences formed from single characters, such as maximal linear recursive sequences (msequences) and Legendre sequences. In this study, exact asymptotic formulae are proved for the autocorrelation and crosscorrelation merit factors of sequence pairs formed using linear combinations of multiplicative characters. Data is presented that shows that the asymptotic behavior is closely approximated by sequences of modest length.

Sequences with Low Correlation

“…Consider the group homomorphism ε : Z/mZ → µ m with ε(x) = e 2πix/m . If the sequences a and b of (1) and (2) are related by b k = ϕ(a k ) for every k, then we say that b is the multiplicative version of a, and equivalently, that a is the additive version of b.…”

“…Typically, our constructions produce families with unbounded lengths, and we rate a family by the asymptotic demerit factors, that is, the limit of the autocorrelation or crosscorrelation demerit factor as the length of the sequences tends to infinity. For sequences derived from finite field characters, it has been observed in many cases [27,2] that the limiting behavior of families is approached quite rapidly, so that even sequences of quite modest length (of the order of a hundred or more) already have demerit factors close to the limiting values.…”

“…. .. Boothby and Katz [2,Theorem 10] give conditions under which one can achieve limiting autocorrelation demerit factor 0.157 . .…”

“…for sequences derived from linear combinations of multiplicative characters, which includes the binary cyclotomic sequences as a proper subclass. Boothby and Katz's conditions are met by the cyclotomic sequences derived from quartic characters described at (16) and (17), and these sequences were used by both Boothby and Katz [2,Theorem 19] and Günther and Schmidt [16, p. 347] as examples showing that one can achieve asymptotic demerit factor 0.157 . .…”

“…. ., one must restrict one's sequence family to contain only sequences derived from fields F p whose orders fulfill exacting number-theoretic conditions (see [2,Theorem 19] and [16, Corollaries 2.5 and 2.6]). As such, if one adopts one of these cyclotomic sequence designs, the sequences produced will, at most lengths, fall short of Legendre sequences in terms of autocorrelation performance.…”