This paper proposes a new Zernike modal gray map reconstruction algorithm used in the nematic liquid crystal adaptive optics system.Firstly,the new modal algorithm is described.Secondly,a single loop correction experi...This paper proposes a new Zernike modal gray map reconstruction algorithm used in the nematic liquid crystal adaptive optics system.Firstly,the new modal algorithm is described.Secondly,a single loop correction experiment was conducted,and it showed that the modal method has a higher precision in gray map reconstruction than the widely used slope method.Finally,the contrast close-loop correction experiment was conducted to correct static aberration in the laboratory.The experimental results showed that the average peak to valley (PV) and root mean square (RMS) of the wavefront corrected by mode method were reduced from 2.501λ (λ=633 nm) and 0.610λ to 0.0334λ and 0.00845λ,respectively.The corrected PV and RMS were much smaller than those of 0.173λ and 0.048λ by slope method.The Strehl ratio and modulation transfer function of the system corrected by mode method were much closer to diffraction limit than with slope method.These results indicate that the mode method can take good advantage of the large number of pixels of the liquid crystal corrector to realize high correction precision.展开更多
A uniform experimental design(UED)is an extremely used powerful and efficient methodology for designing experiments with high-dimensional inputs,limited resources and unknown underlying models.A UED enjoys the followi...A uniform experimental design(UED)is an extremely used powerful and efficient methodology for designing experiments with high-dimensional inputs,limited resources and unknown underlying models.A UED enjoys the following two significant advantages:(i)It is a robust design,since it does not require to specify a model before experimenters conduct their experiments;and(ii)it provides uniformly scatter design points in the experimental domain,thus it gives a good representation of this domain with fewer experimental trials(runs).Many real-life experiments involve hundreds or thousands of active factors and thus large UEDs are needed.Constructing large UEDs using the existing techniques is an NP-hard problem,an extremely time-consuming heuristic search process and a satisfactory result is not guaranteed.This paper presents a new effective and easy technique,adjusted Gray map technique(AGMT),for constructing(nearly)UEDs with large numbers of four-level factors and runs by converting designs with s two-level factors and n runs to(nearly)UEDs with 2^(t−1)s four-level factors and 2tn runs for any t≥0 using two simple transformation functions.Theoretical justifications for the uniformity of the resulting four-level designs are given,which provide some necessary and/or sufficient conditions for obtaining(nearly)uniform four-level designs.The results show that the AGMT is much easier and better than the existing widely used techniques and it can be effectively used to simply generate new recommended large(nearly)UEDs with four-level factors.展开更多
Gray mapping is a well-known way to improve the performance of regular constellation modulation,but it is challenging to be applied directly for irregular alternative.To address this issue,in this paper,a unified bit-...Gray mapping is a well-known way to improve the performance of regular constellation modulation,but it is challenging to be applied directly for irregular alternative.To address this issue,in this paper,a unified bit-to-symbol mapping method is designed for generalized constellation modulation(i.e.,regular and irregular shaping).The objective of the proposed approach is to minimize the average bit error probability by reducing the hamming distance(HD)of symbols with larger values of pairwise error probability.Simulation results show that the conventional constellation modulation(i.e.,phase shift keying and quadrature amplitude modulation(QAM)with the proposed mapping rule yield the same performance as that of classical gray mapping.Moreover,the recently developed golden angle modulation(GAM)with the proposed mapping method is capable of providing around1 d B gain over the conventional mapping counterpart and offers comparable performance to QAM with Gray mapping.展开更多
Let m ≥ 2 be any natural number and let be a finite non-chain ring, where and q is a prime power congruent to 1 modulo (m-1). In this paper we study duadic codes over the ring and their extensions. A Gray map from to...Let m ≥ 2 be any natural number and let be a finite non-chain ring, where and q is a prime power congruent to 1 modulo (m-1). In this paper we study duadic codes over the ring and their extensions. A Gray map from to is defined which preserves self duality of linear codes. As a consequence self-dual, formally self-dual and self-orthogonal codes over are constructed. Some examples are also given to illustrate this.展开更多
Let Z_(4) be the ring of integers modulo 4.We study the Λ-constacyclic and(θ,Λ)-cyclic codes over the non-chain ring R=Z_(4)[u,v]/<u^(2)=1,v^(2)=0,uv=vu=0>for a unit Λ=1+2u+2v in R.We define several Gray map...Let Z_(4) be the ring of integers modulo 4.We study the Λ-constacyclic and(θ,Λ)-cyclic codes over the non-chain ring R=Z_(4)[u,v]/<u^(2)=1,v^(2)=0,uv=vu=0>for a unit Λ=1+2u+2v in R.We define several Gray maps and find that the respective Gray images of a quasi-cyclic code over Z_(4) are cyclic,quasi-cyclic or permutation equivalent to this code.For an odd positive integer n,we determine the generator polynomials of cyclic and Λ-constacyclic codes of length n over R.Further,we prove that a(θ,Λ)-cyclic code of length n is a Λ-constacyclic code if n is odd,and a Λ-quasi-twisted code if n is even.A few examples are also incorporated,in which two parameters are new and one is best known to date.展开更多
By constructing a Gray map, constacyclic codes of arbitrary lengths over ring R =Z p m +vZ pmare studied, wherev 2=v. The structure of constacyclic codes over R and their dual codes are obtained. A necessary and suffi...By constructing a Gray map, constacyclic codes of arbitrary lengths over ring R =Z p m +vZ pmare studied, wherev 2=v. The structure of constacyclic codes over R and their dual codes are obtained. A necessary and sufficient condition for a linear code to be self-dual constacyclic is given. In particular,(1 +(v +1)ap)-constacyclic codes over R are classified in terms of generator polynomial, where a is a unit of Z m.展开更多
We study the structure of cyclic codes of an arbitrary length n over the ring F2+ uF2+ vF2, which is not a finite chain ring. We prove that the Gray image of a cyclic code length n over F2+ uF2+ vF2 is a 3-quasi-cycli...We study the structure of cyclic codes of an arbitrary length n over the ring F2+ uF2+ vF2, which is not a finite chain ring. We prove that the Gray image of a cyclic code length n over F2+ uF2+ vF2 is a 3-quasi-cyclic code length 3n over F2.展开更多
In this work, we investigate the cyclic codes over the ring F2+ uF2+ vF2. We first study the relationship between linear codes over F2+ uF2+ vF2 and that over F2.Then we give a characterization of the cyclic codes ove...In this work, we investigate the cyclic codes over the ring F2+ uF2+ vF2. We first study the relationship between linear codes over F2+ uF2+ vF2 and that over F2.Then we give a characterization of the cyclic codes over F2+ uF2+ vF2. Finally, we obtain the number of the cyclic code over F2+ uF2+ vF2 of length n.展开更多
Let <i>f</i>(u) and <i>g</i>(v) be two polynomials of degree <i>k</i> and <i>l</i> respectively, not both linear which split into distinct linear factors over F<sub&g...Let <i>f</i>(u) and <i>g</i>(v) be two polynomials of degree <i>k</i> and <i>l</i> respectively, not both linear which split into distinct linear factors over F<sub>q</sub>. Let <img src="Edit_83041428-d8b0-4505-8c3c-5e29f2886159.png" width="160" height="15" alt="" /> be a finite commutative non-chain ring. In this paper, we study polyadic codes and their extensions over the ring <i>R</i>. We give examples of some polyadic codes which are optimal with respect to Griesmer type bound for rings. A Gray map is defined from <img src="Edit_c75f119d-3176-4a71-a36a-354955044c09.png" width="50" height="15" alt="" /> which preserves duality. The Gray images of polyadic codes and their extensions over the ring <i>R</i> lead to construction of self-dual, isodual, self-orthogonal and complementary dual (LCD) codes over F<i><sub>q</sub></i>. Some examples are also given to illustrate this.展开更多
This paper firstly gives some necessary conditions on one-Gray weight linear codes. And then we use these results to construct several classes of one-Gray weight linear codes over Z_4 +uZ_4(u^2=u) with type 16^(k_1)8^...This paper firstly gives some necessary conditions on one-Gray weight linear codes. And then we use these results to construct several classes of one-Gray weight linear codes over Z_4 +uZ_4(u^2=u) with type 16^(k_1)8^(k_2)8^(k_3)4^(k_4)4^(k_5)4^(k_6)2^(k_7)2^(k_8) based on a distance-preserving Gray map from(Z4 + u Z4)n to Z2n4. Secondly, the authors use the similar approach to do works on two-Gray(projective) weight linear codes. Finally, some examples are given to illustrate the construction methods.展开更多
This paper is devoted to the construction of one-Lee weight codes and two-Lee weight codes over IF_p+vIF_p(v^2=v) with type p^(2 k_1)p^(k2)p^(k3) based on two different distance-preserving Gray maps from((IF_p+vIF_p)~...This paper is devoted to the construction of one-Lee weight codes and two-Lee weight codes over IF_p+vIF_p(v^2=v) with type p^(2 k_1)p^(k2)p^(k3) based on two different distance-preserving Gray maps from((IF_p+vIF_p)~n, Lee weight) to(IF_p^(2 n), Hamming weight), where p is a prime. Moreover, the authors prove that the obtained two-Lee weight codes are projective only when p=2.展开更多
基金Project supported by the National Natural Science Foundation of China (Grants Nos.60736042,60578035 and 50703039)Science and Technology Cooperation Project between Chinese Academy of Sciences and Jilin Province (Grant No.2008SYHZ0005)
文摘This paper proposes a new Zernike modal gray map reconstruction algorithm used in the nematic liquid crystal adaptive optics system.Firstly,the new modal algorithm is described.Secondly,a single loop correction experiment was conducted,and it showed that the modal method has a higher precision in gray map reconstruction than the widely used slope method.Finally,the contrast close-loop correction experiment was conducted to correct static aberration in the laboratory.The experimental results showed that the average peak to valley (PV) and root mean square (RMS) of the wavefront corrected by mode method were reduced from 2.501λ (λ=633 nm) and 0.610λ to 0.0334λ and 0.00845λ,respectively.The corrected PV and RMS were much smaller than those of 0.173λ and 0.048λ by slope method.The Strehl ratio and modulation transfer function of the system corrected by mode method were much closer to diffraction limit than with slope method.These results indicate that the mode method can take good advantage of the large number of pixels of the liquid crystal corrector to realize high correction precision.
基金supported by the UIC Research Grants with No.of(R201912 and R202010)the Curriculum Development and Teaching Enhancement with No.of(UICR0400046-21CTL)+1 种基金the Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science,BNU-HKBU United International College with No.of(2022B1212010006)Guangdong Higher Education Upgrading Plan(2021-2025)with No.of(UICR0400001-22).
文摘A uniform experimental design(UED)is an extremely used powerful and efficient methodology for designing experiments with high-dimensional inputs,limited resources and unknown underlying models.A UED enjoys the following two significant advantages:(i)It is a robust design,since it does not require to specify a model before experimenters conduct their experiments;and(ii)it provides uniformly scatter design points in the experimental domain,thus it gives a good representation of this domain with fewer experimental trials(runs).Many real-life experiments involve hundreds or thousands of active factors and thus large UEDs are needed.Constructing large UEDs using the existing techniques is an NP-hard problem,an extremely time-consuming heuristic search process and a satisfactory result is not guaranteed.This paper presents a new effective and easy technique,adjusted Gray map technique(AGMT),for constructing(nearly)UEDs with large numbers of four-level factors and runs by converting designs with s two-level factors and n runs to(nearly)UEDs with 2^(t−1)s four-level factors and 2tn runs for any t≥0 using two simple transformation functions.Theoretical justifications for the uniformity of the resulting four-level designs are given,which provide some necessary and/or sufficient conditions for obtaining(nearly)uniform four-level designs.The results show that the AGMT is much easier and better than the existing widely used techniques and it can be effectively used to simply generate new recommended large(nearly)UEDs with four-level factors.
基金supported in part by the National Key Research and Development Program of China under Grant 2021YFB2900502in part by the National Science Foundation of China under Grant 62001179in part by the Fundamental Research Funds for the Central Universities under Grant 2020kfy XJJS111。
文摘Gray mapping is a well-known way to improve the performance of regular constellation modulation,but it is challenging to be applied directly for irregular alternative.To address this issue,in this paper,a unified bit-to-symbol mapping method is designed for generalized constellation modulation(i.e.,regular and irregular shaping).The objective of the proposed approach is to minimize the average bit error probability by reducing the hamming distance(HD)of symbols with larger values of pairwise error probability.Simulation results show that the conventional constellation modulation(i.e.,phase shift keying and quadrature amplitude modulation(QAM)with the proposed mapping rule yield the same performance as that of classical gray mapping.Moreover,the recently developed golden angle modulation(GAM)with the proposed mapping method is capable of providing around1 d B gain over the conventional mapping counterpart and offers comparable performance to QAM with Gray mapping.
文摘Let m ≥ 2 be any natural number and let be a finite non-chain ring, where and q is a prime power congruent to 1 modulo (m-1). In this paper we study duadic codes over the ring and their extensions. A Gray map from to is defined which preserves self duality of linear codes. As a consequence self-dual, formally self-dual and self-orthogonal codes over are constructed. Some examples are also given to illustrate this.
文摘Let Z_(4) be the ring of integers modulo 4.We study the Λ-constacyclic and(θ,Λ)-cyclic codes over the non-chain ring R=Z_(4)[u,v]/<u^(2)=1,v^(2)=0,uv=vu=0>for a unit Λ=1+2u+2v in R.We define several Gray maps and find that the respective Gray images of a quasi-cyclic code over Z_(4) are cyclic,quasi-cyclic or permutation equivalent to this code.For an odd positive integer n,we determine the generator polynomials of cyclic and Λ-constacyclic codes of length n over R.Further,we prove that a(θ,Λ)-cyclic code of length n is a Λ-constacyclic code if n is odd,and a Λ-quasi-twisted code if n is even.A few examples are also incorporated,in which two parameters are new and one is best known to date.
基金Supported by the National Natural Science Foundation of China(No.61370089)
文摘By constructing a Gray map, constacyclic codes of arbitrary lengths over ring R =Z p m +vZ pmare studied, wherev 2=v. The structure of constacyclic codes over R and their dual codes are obtained. A necessary and sufficient condition for a linear code to be self-dual constacyclic is given. In particular,(1 +(v +1)ap)-constacyclic codes over R are classified in terms of generator polynomial, where a is a unit of Z m.
基金Foundation item： Supported by the Scientific Research Foundation of Education Department of Hubei Province（B2013069） Supported by the National Science Foundation of Hubei Polytechnic University of China （12xjzl4A, llyjz37B）
文摘We study the structure of cyclic codes of an arbitrary length n over the ring F2+ uF2+ vF2, which is not a finite chain ring. We prove that the Gray image of a cyclic code length n over F2+ uF2+ vF2 is a 3-quasi-cyclic code length 3n over F2.
基金Foundation item： Supported by the Scientific Research Foundation of Education Department of Hubei Province（B2013069） Supported by the National Science Foundation of Hubei Polytechnic University of China（12xjz14A,11yjz37B）
文摘In this work, we investigate the cyclic codes over the ring F2+ uF2+ vF2. We first study the relationship between linear codes over F2+ uF2+ vF2 and that over F2.Then we give a characterization of the cyclic codes over F2+ uF2+ vF2. Finally, we obtain the number of the cyclic code over F2+ uF2+ vF2 of length n.
文摘Let <i>f</i>(u) and <i>g</i>(v) be two polynomials of degree <i>k</i> and <i>l</i> respectively, not both linear which split into distinct linear factors over F<sub>q</sub>. Let <img src="Edit_83041428-d8b0-4505-8c3c-5e29f2886159.png" width="160" height="15" alt="" /> be a finite commutative non-chain ring. In this paper, we study polyadic codes and their extensions over the ring <i>R</i>. We give examples of some polyadic codes which are optimal with respect to Griesmer type bound for rings. A Gray map is defined from <img src="Edit_c75f119d-3176-4a71-a36a-354955044c09.png" width="50" height="15" alt="" /> which preserves duality. The Gray images of polyadic codes and their extensions over the ring <i>R</i> lead to construction of self-dual, isodual, self-orthogonal and complementary dual (LCD) codes over F<i><sub>q</sub></i>. Some examples are also given to illustrate this.
基金supported by the National Natural Science Foundation of China under Grant Nos.61672036 and 61202068the Open Research Fund of National Mobile Communications Research Laboratory,Southeast University under Grant No.2015D11+1 种基金Technology Foundation for Selected Overseas Chinese Scholar,Ministry of Personnel of China under Grant No.05015133Key projects of support program for outstanding young talents in Colleges and Universities under Grant No.gxyq ZD2016008
文摘This paper firstly gives some necessary conditions on one-Gray weight linear codes. And then we use these results to construct several classes of one-Gray weight linear codes over Z_4 +uZ_4(u^2=u) with type 16^(k_1)8^(k_2)8^(k_3)4^(k_4)4^(k_5)4^(k_6)2^(k_7)2^(k_8) based on a distance-preserving Gray map from(Z4 + u Z4)n to Z2n4. Secondly, the authors use the similar approach to do works on two-Gray(projective) weight linear codes. Finally, some examples are given to illustrate the construction methods.
基金supported by the National Natural Science Foundation of China under Grant No.61202068Technology Foundation for Selected Overseas Chinese Scholar,Ministry of Personnel of China under Grant No.05015133+1 种基金the Open Research Fund of National Mobile Communications Research Laboratory,Southeast University under Grant No.2015D11Key Projects of Support Program for Outstanding Young Talents in Colleges and Universities under Grant No.gxyqZD2016008
文摘This paper is devoted to the construction of one-Lee weight codes and two-Lee weight codes over IF_p+vIF_p(v^2=v) with type p^(2 k_1)p^(k2)p^(k3) based on two different distance-preserving Gray maps from((IF_p+vIF_p)~n, Lee weight) to(IF_p^(2 n), Hamming weight), where p is a prime. Moreover, the authors prove that the obtained two-Lee weight codes are projective only when p=2.
