scholarly journals Four Constructions of Asymptotically Optimal Codebooks via Additive Characters and Multiplicative Characters

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1144 ◽  
Author(s):  
Xia Wu ◽  
Wei Lu
Keyword(s):  
Cross Correlation ◽  
Irreducible Polynomials ◽  
Asymptotically Optimal ◽  
Welch Bound ◽  
Multiplicative Characters ◽  
Special Case

In this paper, we present four new constructions of complex codebooks with multiplicative characters, additive characters, and quadratic irreducible polynomials and determine the maximal cross-correlation amplitude of these codebooks. We prove that the codebooks we constructed are asymptotically optimal with respect to the Welch bound. Moreover, we generalize the result obtained by Zhang and Feng and contain theirs as a special case. The parameters of these codebooks are new.

scholarly journals Constructions of asymptotically optimal codebooks with respect to Welch bound and Levenshtein bound

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Gang Wang ◽  
Deng-Ming Xu ◽  
Fang-Wei Fu
Keyword(s):  
Code Division Multiple Access ◽  
Communication Systems ◽  
Multiple Access ◽  
Cross Correlation ◽  
Hadamard Matrices ◽  
Circulant Matrices ◽  
Asymptotically Optimal ◽  
Welch Bound ◽  
Construction Methods ◽  
Code Division Multiple

<p style='text-indent:20px;'>Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in code division multiple access communication systems. In this paper, several classes of codebooks are introduced, whose maximum cross-correlation amplitudes asymptotically achieve the corresponding Welch bound and Levenshtein bound. Specially, a class of optimal codebooks with respect to the Levenshtein bound is obtained. These classes of codebooks are constructed by selecting certain rows deterministically from circulant matrices, Fourier matrices and Hadamard matrices, respectively. The construction methods and parameters of some codebooks provided in this paper are new.</p>

New Constructions of Asymptotically Optimal Codebooks With Multiplicative Characters

2017 ◽  
Vol 63 (10) ◽  
pp. 6179-6187 ◽  
Author(s):  
Ziling Heng ◽  
Cunsheng Ding ◽  
Qin Yue
Keyword(s):  
Asymptotically Optimal ◽  
Multiplicative Characters

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

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Wanfeng Qi ◽  
Yueying Song ◽  
Rui Ma ◽  
Lingli Tang ◽  
Qian Wang
Keyword(s):  
Finite Fields ◽  
Difference Sets ◽  
Good Alternative ◽  
Asymptotically Optimal ◽  
Welch Bound ◽  
New Class ◽  
Almost Difference Sets

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.

scholarly journals Admissible Solutions of the Schwarzian Type Difference Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Baoqin Chen ◽  
Sheng Li
Keyword(s):  
Difference Equation ◽  
Rational Function ◽  
Admissible Solution ◽  
Irreducible Polynomials ◽  
Polynomial Coefficients ◽  
Admissible Solutions ◽  
Special Case

This paper is to investigate the Schwarzian type difference equationΔ3f/Δf-3/2Δ2f/Δf2k=Rz,f=P(z,f)/Q(z,f),whereR(z,f)is a rational function infwith polynomial coefficients,P(z,f), respectivelyQ(z,f)are two irreducible polynomials infof degreep, respectivelyq. Relationship betweenpandqis studied for some special case. Denoted=max⁡p,q. Letf(z)be an admissible solution of(*)such thatρ2(f)<1; then fors (≥2) distinct complex constantsα1,…,αs ,q+2k∑j=1sδ(αj,f)≤ 8k.In particular, ifN(r,f)=S(r,f), thend+2k∑j=1sδ (αj,f)≤4k.

Asymptotically Optimal Codebooks in Regard to the Welch Bound with Characters

2020 ◽  
Vol E103.A (11) ◽  
pp. 1292-1295
Author(s):  
Gang WANG ◽  
Min-Yao NIU ◽  
Lin-Zhi SHEN ◽  
You GAO
Keyword(s):  
Asymptotically Optimal ◽  
Welch Bound

scholarly journals A simpler linear-time algorithm for the common refinement of rooted phylogenetic trees on a common leaf set

2021 ◽  
Vol 16 (1) ◽  
Author(s):  
David Schaller ◽  
Marc Hellmuth ◽  
Peter F. Stadler
Keyword(s):  
Phylogenetic Trees ◽  
Linear Time ◽  
Time Algorithm ◽  
Input Tree ◽  
Linear Time Algorithm ◽  
Optimal Algorithms ◽  
Asymptotically Optimal ◽  
The Common ◽  
Mathematical Phylogenetics ◽  
Special Case

Abstract Background The supertree problem, i.e., the task of finding a common refinement of a set of rooted trees is an important topic in mathematical phylogenetics. The special case of a common leaf set L is known to be solvable in linear time. Existing approaches refine one input tree using information of the others and then test whether the results are isomorphic. Results An O(k|L|) algorithm, , for constructing the common refinement T of k input trees with a common leaf set L is proposed that explicitly computes the parent function of T in a bottom-up approach. Conclusion is simpler to implement than other asymptotically optimal algorithms for the problem and outperforms the alternatives in empirical comparisons. Availability An implementation of in Python is freely available at https://github.com/david-schaller/tralda.

scholarly journals Asymptotically Optimal Box Packing Theorems

10.37236/802 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Michael Reid
Keyword(s):  
Algebraic Geometry ◽  
Ad Hoc ◽  
Necessary Conditions ◽  
Combinatorial Proof ◽  
Original Proof ◽  
Asymptotically Optimal ◽  
Straightforward Calculation ◽  
Complete Answer ◽  
Special Case

Given a protoset of $d$-dimensional polyominoes, we ask which boxes can be packed by the protoset. In some cases, it may be too difficult to give a complete answer to this question, so we ask the easier question about determining all sufficiently large boxes that can be packed. (We say that a box is "sufficiently large" if all edge lengths are ${} \ge C$ for some large $C$.) We give numerous examples (mostly $2$-dimensional) where we can answer this easier question. The various techniques involved are: checkerboard-type colorings/numberings (tile homology), the boundary word method of Conway and Lagarias (tile homotopy), ad hoc geometric arguments, and a very nice theorem of Barnes. Barnes' Theorem asserts that all necessary conditions for a box to be packable can be given in a certain form, and these conditions are also sufficient for large boxes. Barnes' Theorem has not received the appreciation it deserves. We give a new, purely combinatorial proof of this important result. (Barnes' original proof uses techniques of algebraic geometry.) In the special case that all the prototiles are boxes themselves, we show how to determine all sufficiently large boxes that they pack. We prove a theorem based on Barnes' result that reduces this to a straightforward calculation.

A Modified Volterra Series Representation for a Class of Single-Valued, Continuous Nonlinear Systems

1980 ◽  
Vol 102 (3) ◽  
pp. 163-167 ◽  
Author(s):  
G. A. Parker ◽  
E. L. Moore
Keyword(s):  
Cross Correlation ◽  
Volterra Series ◽  
Series Representation ◽  
Weighting Coefficient ◽  
Functional Series ◽  
Level Sequence ◽  
Input Signals ◽  
Volterra Functional Series ◽  
Special Case ◽  
Modified Functional

A modification is presented to the Volterra functional series representation of the response from a cascaded linear-nonlinear-linear system in which the nonlinear element is single-valued, separable, and continuous. The particular advantage of this approach is that the dynamic effects represented in the convolution terms are independent of the bias or mean level of the input signal to the system. The effects of bias and element gain are included in a weighting coefficient βi (m) to each term in the series, with the first term representing the small signal gain of the system. The special case of pseudo-random input signals to the nonlinear system model is also examined using the modified functional series. It is concluded that the three-level sequence is particularly useful in producing a truncation in the modified cross-correlation series representation of the system.

A further construction of asymptotically optimal codebooks with multiplicative characters

2019 ◽  
Vol 30 (6) ◽  
pp. 453-469
Author(s):  
Wenjuan Yin ◽  
Can Xiang ◽  
Fang-Wei Fu
Keyword(s):  
Asymptotically Optimal ◽  
Multiplicative Characters

Generalization of cross-correlation in Boolean functions and some generalized constructions in gbent functions

2016 ◽  
Vol 09 (01) ◽  
pp. 1650019
Author(s):  
P. L. Sharma ◽  
Neetu Dhiman
Keyword(s):  
Boolean Functions ◽  
Cross Correlation ◽  
Generalized Boolean Functions ◽  
The Cross ◽  
Special Case ◽  
Existing Result

The cross-correlation of two generalized Boolean functions defined on [Formula: see text] with values in [Formula: see text] is discussed in literature for [Formula: see text]. We generalize this result for [Formula: see text] [Formula: see text] and show that the existing result is a special case of our work. Further, we give some constructions of gbent functions.

Sign in / Sign up

Export Citation Format

Share Document