A new family of binary pseudorandom sequences having optimal periodic correlation properties and large linear span

Abstract

A collection of families of binary {O,l} pseudorandom sequences (referred to as the families of No sequences) is introduced in this paper. Each sequence within a family has period N = 2"-1, where n = 2.m is an even integer. There are 2"' sequences within a family and the maximum over all (nontrivial) auto and cross-correlation values equals 2"+ 1. Thus these sequences are optimum with respect to the Welch bound on the maximum correlation value.Each family contains a Gordon-Mills-Welch (GMW)sequence and the collection of families includes as a special case, the small set of Kasami sequences. The linear span of these sequences is large. The balance properties of such families are evaluated and a count of the number of distinct families of given period N that can be contructed also provided.

d-form sequences: families of sequences with low correlation values and large linear spans

“…These are exactly the sequences that are simultaneously d-form sequences and generalized No sequences [13]. The results in this section were proved independently by No in his thesis.…”

“…The results are often closely related to results from coding theory, as in the case of −1 decimations of m-sequences [18]. In a few cases families of sequences with good correlation properties have been found, such as Kasami sequences [4,5], bent function sequences [10,11,15,16], and No sequences [14]. Table (1) summarizes the properties of some of these families.…”

“…Proof: The proof of this theorem is an extension of the proof of the corresponding theorem on No sequences [14]. Let L = (q e − 1)/(q − 1).…”

“…The families of sequences that arise include No sequences [14]. We then exhibit a particular family of quadratic forms with the required property, and show that the linear spans of the resulting sequences, trace norm (or TN) sequences, are larger (in some cases, asymptotically exponentially larger) than the linear spans of No sequences.…”

“…In his thesis No introduced a class of sequences called generalized No sequences [13]. These sequences are formed by iteratively applying functions of the form "trace of a power of In what follows, let e and m be positive integers, let n = em, and let T r n m be the trace function from GF (2 …”