Binary sequences with Gold-like correlation but larger linear span

Abstract

A new construction of optimal binary sequences, identical to the well known family of Gold sequences in terms of maximum nontrivial correlation magnitude and family size, but having larger linear span is presented. The distribution of correlation values is determined. For every odd integer r 2 3, the construction provides a family that contains 2' + 1 cyclically distinct sequences, each of period 2'-1. The maximum nontrivial correlation magnitude equals 2(r+1)/2+ 1. With one exception, each of the sequences in the family has linear span at least (rZr) / 2 (compared to 2r for Gold sequences). The sequences are easily implemented using a quaternary shift register followed by a simple feedforward nonlinearity. Index Tenns-Binary sequences, Gold sequences, sequences with low correlation, large linear span.

New constructions of low-correlation sequences with high-linear complexity

“…However, only a few sequence sets can attain the Welch's bound or Sidelnokiv's bound , for example the Kasami sequences [1], the Gold sequences [2], the Sidelnikov sequences [3], the bent sequences [4], and the Gold-like sequences [6].…”

Generalized modified Gold sequences

“…9 · 2 5n−4 + 3 · 2 4n−3 + 2 3n − 9 · 2 2n−3 − 3 · 2 n−2 − 2 times −1 ± 2 m+1 , 1 3 · 2 n−2 ± 2 m−1 5 · 2 4n−1 + 2 3n+2 − 5 · 2 n − 8 times −1 ± 2 m+2 , 1 3 · 2 n−4 ± 2 m−2 2 4n−1 − 2 3n − 2 n + 2 times.…”

“…correlation have been reported, see [1,3,5,6,[10][11][12][14][15][16], for example. Among them, the large set of Kasami sequences [5], the set of modified Gold sequences [10], the set of Yu-Gong sequences [15], and the large set of generalized Kasami sequences [16], have large size as well.…”

“…. , and tr n 1 α 1+2 m+l t (1) where tr n 1 (x) is the trace function from the finite field F 2 n with 2 n elements to its subfield F 2 , and α is a primitive element of F 2 n . It was estimated in [10] that the nontrivial correlation of the sequences in F (l) lies between −1−2 m+l and −1+2 m+l .…”

On Plateaued Functions and Their Constructions

“…Because of Parseval's relation (2), N f is upper bounded by 2 n−1 − 2 n/2−1 . This bound is tight for every n even.…”