New Constructions of Asymptotically Optimal Codebooks via Cyclotomic Classes of Order 8

Wanfeng Qi,Yueying Song,Rui Ma,Qian Wang,Lingli Tang

doi:10.1155/2019/8708161

Abstract

Asymptotically optimal codebooks are a family of codebooks that can approach an optimal codebook meeting the Welch bound when the lengths of codewords are large enough. They can be constructed easily and are a good alternative for optimal codebooks in many applications. In this paper, we construct a new class of asymptotically optimal codebooks by using the product of some special finite fields and almost difference sets, which are composed of cyclotomic classes of order eight.

Highlights

An (N, K) codebook C is a set {c0, c1, . . . , cN−1}, where each codeword cl, 0 ≤ l ≤ N − 1, is a unit norm complex vector in CK
If l is a power of a prime and l = t2 + 2 ≡ 3, for q = l2, Ding [17] proved that q ≡ 9, x = −l, y = 0, a = −l+ 4, b = ±2t, and 2 is a quartic residue in Fq∗, and D = C(08,q) ∪C(18,q) ∪C(28,q) ∪C(58,q) is a (q, (q−1)/2, (q−5)/4, (q− 1)/2) almost difference set of Fq
We construct a new class of asymptotically optimal codebooks based on the product of two classes of almost difference sets in finite fields

Summary

Introduction

An (N, K) codebook C is a set {c0, c1, . . . , cN−1}, where each codeword cl, 0 ≤ l ≤ N − 1, is a unit norm complex vector in CK. Asymptotically optimal codebooks have attracted widespread attention, since they can be constructed and Imax(C) can approach the Welch bound for large enough N. They are good alternatives in many applications. Many methods can be used to construct asymptotically optimal codebooks, such as almost difference sets [8,9,10], binary sequences [11], and character sums in finite fields [12, 13]. We generalize this method to the case of cyclotomic classes of order 8 in finite fields and propose new classes of codebooks asymptotically meeting the Welch bound.

Preliminaries

Main Result

Conclusions