Two classes of near-optimal codebooks with respect to the Welch bound

Gaojun Luo; Xiwang Cao

doi:10.3934/amc.2020066

Article Contents

2021, Volume 15, Issue 2: 279-289. Doi: 10.3934/amc.2020066

This issue Previous Article On the existence of PD-sets: Algorithms arising from automorphism groups of codes Next Article An explicit representation and enumeration for negacyclic codes of length

$ 2^kn $

over

$ \mathbb{Z}_4+u\mathbb{Z}_4 $

Two classes of near-optimal codebooks with respect to the Welch bound

Gaojun Luo^1, and
Xiwang Cao^1,2, ,

1.
Department of Math, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
2.
State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China

^* Corresponding author: Xiwang Cao
^* Corresponding author: Xiwang Cao

Received: March 2017

Revised: June 2017

Early access: January 2020

Published: May 2021

The second author is supported by the National Natural Science Foundation of China (Grant No. 11771007 and 61572027)

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Abstract

An $ (N,K) $ codebook $ {\mathcal C} $ is a collection of $ N $ unit norm vectors in a $ K $-dimensional vectors space. In applications of codebooks such as CDMA, those vectors in a codebook should have a small maximum magnitude of inner products between any pair of distinct code vectors. In this paper, we propose two constructions of codebooks based on $ p $-ary linear codes and on a hybrid character sum of a special kind of functions, respectively. With these constructions, two classes of codebooks asymptotically meeting the Welch bound are presented.

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Mathematics Subject Classification: Primary: 11T71; Secondary: 11T23.

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Table 1. Known weakly regular bent functions over $ {\mathbb F}_{p^m} $ with odd characteristic $ p $

Bent functions	$ m $	$ p $
$ \sum_{i=0}^{\lfloor m/2\rfloor}\operatorname{Tr}_{1}^{m}(a_ix^{p^i+1}) $	arbitrary	arbitrary
$ \sum_{i=0}^{p^k-1}\operatorname{Tr}^m_1(a_ix^{i(p^k-1)})+\operatorname{Tr}^l_1(\epsilon x^{\frac{p^m-1}{e}}) $, $ e\|p^k+1 $	$ m=2k $	arbitrary
$ \operatorname{Tr}_{1}^{m}(ax^{\frac{3^m-1}{4}+3^k+1}) $	$ m=2k $	$ p=3 $
$ \operatorname{Tr}_{1}^{m}(x^{p^{3k}+p^{2k}-p^k+1}+x^2) $	$ m=4k $	arbitrary
$ \operatorname{Tr}_{1}^{m}(ax^{\frac{3^i+1}{2}}) $; $ i $ odd, $ {\rm gcd}(i,m)=1 $	arbitrary	$ p=3 $

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