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May  2021, 15(2): 279-289. doi: 10.3934/amc.2020066

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

 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

Received  March 2017 Revised  June 2017 Published  January 2020

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

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.

Citation: Gaojun Luo, Xiwang Cao. Two classes of near-optimal codebooks with respect to the Welch bound. Advances in Mathematics of Communications, 2021, 15 (2) : 279-289. doi: 10.3934/amc.2020066
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##### References:
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 $p=3$
 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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