作者：
Dinh, Hai Q.;Wang, Xiaoqiang*;Liu, Hongwei;Yamaka, Woraphon
期刊：
Finite Fields and Their Applications,2021年70:101794 ISSN：1071-5797
通讯作者：
Wang, Xiaoqiang
作者机构：
[Dinh, Hai Q.] Ton Duc Thang Univ, Inst Computat Sci, Div Computat Math & Engn, Ho Chi Minh City, Vietnam.;[Wang, Xiaoqiang] Hubei Univ, Fac Math & Stat, Hubei Key Lab Appl Math, Wuhan 430062, Peoples R China.;[Dinh, Hai Q.] Ton Duc Thang Univ, Fac Math & Stat, Ho Chi Minh City, Vietnam.;[Liu, Hongwei] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Yamaka, Woraphon] Chiang Mai Univ, Fac Econ, Ctr Excellence Econometr, Chiang Mai 52000, Thailand.
通讯机构：
[Wang, Xiaoqiang] H;Hubei Univ, Fac Math & Stat, Hubei Key Lab Appl Math, Wuhan 430062, Peoples R China.
期刊：
Designs, Codes and Cryptography,2020年88(2):241-255 ISSN：0925-1022
作者机构：
[Hongwei Liu; Xu Pan] School of Mathematics and Statistics, Central China Normal University, Wuhan, China
摘要：
The \(\ell \)-Galois hull \(h_{\ell }(C)\) of an [n, k] linear code C over the finite field \({\mathbb {F}}_q\) is the intersection of C and \(C^{{\bot }_{\ell }}\), where \(C^{\bot _{\ell }}\) denotes the \(\ell \)-Galois dual of C which was introduced by Fan and Zhang in 2017. The \(\ell \)-Galois LCD code is a linear code C satisfying \(h_{\ell }(C)= C\bigcap C^{\bot _{\ell }}= \{0\}\). In this paper, we show that the dimension of the \(\ell \)-Galois hull of a linear code is invariant under permutation equivalences and we provide a method to calculate the dimension of the \(\ell \)-Galois hull through a generator matrix of the code. Moreover, we obtain that the dimension of the \(\ell \)-Galois hull of a ternary code is also invariant under monomial equivalences. We show that every [n, k] linear code over \({\mathbb {F}}_{q}\) is monomial equivalent to an \(\ell \)-Galois LCD code for any \(q>4\). We conclude that if there exists an [n, k] linear code over \({\mathbb {F}}_{q}\) for any \(q>4\), then there exists an \(\ell \)-Galois LCD code with the same parameters for any \(0\le \ell \le e-1\), where \(q=p^e\) for some prime number p. As an application, we characterize the \(\ell \)-Galois hulls of matrix product codes over finite fields.
期刊：
DESIGNS CODES AND CRYPTOGRAPHY,2020年88(6):1127-1138 ISSN：0925-1022
通讯作者：
Luo, Jinquan
作者机构：
[Fang, Xiaolei; Liu, Hongwei; Luo, Jinquan; Lebed, Khawla] Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.;[Fang, Xiaolei; Liu, Hongwei; Luo, Jinquan; Lebed, Khawla] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan, Peoples R China.
通讯机构：
[Luo, Jinquan] C;Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan, Peoples R China.
期刊：
DESIGNS CODES AND CRYPTOGRAPHY,2020年88(2):241-255 ISSN：0925-1022
通讯作者：
Pan, Xu
作者机构：
[Liu, Hongwei; Pan, Xu] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Pan, Xu] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
\(\ell \)-Galois hull of a linear code;Monomial equivalence;\(\ell \)-Galois LCD code;Matrix product code
摘要：
The $$\ell $$-Galois hull $$h_{\ell }(C)$$ of an [n, k] linear code C over the finite field $${\mathbb {F}}_q$$ is the intersection of C and $$C^{{\bot }_{\ell }}$$, where $$C^{\bot _{\ell }}$$ denotes the $$\ell $$-Galois dual of C which was introduced by Fan and Zhang in 2017. The $$\ell $$-Galois LCD code is a linear code C satisfying $$h_{\ell }(C)= C\bigcap C^{\bot _{\ell }}= \{0\}$$. In this paper, we show that the dimension of the $$\ell $$-Galois hull of a linear code is invariant under permutation equivalences and we provide a method to calculate the dimension of the $$\ell $$-Galois hull through a generator matrix of the code. Moreover, we obtain that the dimension of the $$\ell $$-Galois hull of a ternary code is also invariant under monomial equivalences. We show that every [n, k] linear code over $${\mathbb {F}}_{q}$$ is monomial equivalent to an $$\ell $$-Galois LCD code for any $$q>4$$. We conclude that if there exists an [n, k] linear code over $${\mathbb {F}}_{q}$$ for any $$q>4$$, then there exists an $$\ell $$-Galois LCD code with the same parameters for any $$0\le \ell \le e-1$$, where $$q=p^e$$ for some prime number p. As an application, we characterize the $$\ell $$-Galois hulls of matrix product codes over finite fields.
期刊：
IEEE Transactions on Information Theory,2020年66(12):7333-7340 ISSN：0018-9448
通讯作者：
Zhang, Guanghui
作者机构：
[Zhang, Guanghui] Luoyang Normal Univ, Sch Math Sci, Luoyang 471934, Peoples R China.;[Liu, Hongwei] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Zhang, Guanghui] L;Luoyang Normal Univ, Sch Math Sci, Luoyang 471934, Peoples R China.
关键词：
Linear codes;Reed-Solomon codes;Maintenance engineering;Upper bound;Education;Electronic mail;Task analysis;Codes with locality;locally repairable codes;hierarchical locality;multiple erasures;distributed storage
摘要：
Locally repairable codes (LRCs) and codes with hierarchical locality (H-LRCs) have a great significance due to their applications in distributed storage systems. Constructing LRCs or H-LRCs achieving the Singleton-type bound is a challenging task and has received much attention. In this paper, we make use of optimal LRCs to construct a family of optimal H-LRCs. For this purpose we first construct a new class of optimal LRCs via generalized Reed-Solomon codes (GRS codes). Next, based on these optimal LRCs we present some new optimal H-LRCs, of which the parameters are more flexible than those given in [1] and [18].
作者机构：
[Liu, Hongwei; Ouyang, Jianxin] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Ouyang, Jianxin] Guizhou Educ Univ, Sch Math & Big Data, Guiyang 550018, Peoples R China.;[Wang, Xiaoqiang] Hubei Univ, Fac Math & Stat, Hubei Key Lab Appl Math, Wuhan 430062, Peoples R China.
通讯机构：
[Liu, Hongwei] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Gauss sum;Linear code;MDS code;Optimal code;Weight distribution
摘要：
Recently, linear codes with a few weights have been constructed and extensively studied due to their applications in secret sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, we construct several classes of linear codes with a few weights over IFp, where p is an odd prime. The weight distributions of these constructed codes are also settled by applications of the theory of quadratic forms and Gauss sums over finite fields. Some of the linear codes obtained are optimal or almost optimal. The parameters of these linear codes are new in most cases. Moreover, two classes of MDS codes are obtained. (c) 2020 Elsevier B.V. All rights reserved.
摘要：
In this paper, we give the dimension and the minimum distance of two subclasses of narrow-sense primitive BCH codes over F q with designed distance delta = aq(m-1) (resp. delta = a q(m)-1/q-1) for all 1 <= a <= q - 1, where q is a prime power and m > 1 is a positive integer. As a consequence, we obtain an affirmative answer to two conjectures proposed by C. Ding in 2015. Furthermore, using the previous part, we extend some results of Yue and Hu [16], and we give the dimension and, in some cases, the Bose distance for a large designed distance in the range [a q(m)-1/q-1, a q(m)-1/q-1 + T] for 0 <= a <= q - 2, where T = q(m+1/2)-1 if m is odd, and T = 2q(m/2 )-1 if m is even. (C) 2020 Elsevier Inc. All rights reserved.
作者机构：
[Dinh, Hai Q.] Ton Duc Thang Univ, Inst Computat Sci, Div Computat Math & Engn, Ho Chi Minh City, Vietnam.;[Dinh, Hai Q.] Ton Duc Thang Univ, Fac Math & Stat, Ho Chi Minh City, Vietnam.;[Wang, Xiaoqiang] Hubei Univ, Fac Math & Stat, Hubei Key Lab Appl Math, Wuhan 430062, Peoples R China.;[Liu, Hongwei] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Sriboonchitta, Songsak] Chiang Mai Univ, Fac Econ, Chiang Mai 52000, Thailand.
通讯机构：
[Wang, Xiaoqiang] H;Hubei Univ, Fac Math & Stat, Hubei Key Lab Appl Math, Wuhan 430062, Peoples R China.
摘要：
Let p be a prime, s, m be positive integers, lambda be a nonzero element of the finite field F-pm. The b-distance generalizes the Hamming distance (b = 1), and the symbol-pair distance (b = 2). While the Hamming and symbol-pair distances of all lambda-constacyclic codes of length p(s) are completely determined, the general b-distance of such codes was left opened. In this paper, we provide a new technique to establish the b-distance of all lambda-constacyclic codes of length p(s), where 1 <= b <= left perpendicular p/2 right perpendicular. As an application, all MDS b-symbol constacyclic codes of length p(s) over F-pm are obtained. (C) 2019 Elsevier B.V. All rights reserved.
作者机构：
[Dinh, Hai Q.] Ton Duc Thang Univ, Inst Computat Sci, Div Computat Math & Engn, Ho Chi Minh City, Vietnam.;[Dinh, Hai Q.] Ton Duc Thang Univ, Fac Math & Stat, Ho Chi Minh City, Vietnam.;[Liu, Hongwei; Wang, Xiaoqiang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Sriboonchitta, Songsak] Chiang Mai Univ, Fac Econ, Chiang Mai 52000, Thailand.
通讯机构：
[Dinh, Hai Q.] T;Ton Duc Thang Univ, Inst Computat Sci, Div Computat Math & Engn, Ho Chi Minh City, Vietnam.;Ton Duc Thang Univ, Fac Math & Stat, Ho Chi Minh City, Vietnam.
摘要：
Let p be an odd prime, s, m be positive integers, gamma, lambda be nonzero elements of the finite field F-pm such that (gamma)p(s) = A. In this paper, we show that, for any positive integer eta, the Hamming distances of all repeated-root lambda-constacyclic codes of length eta p(s) can be determined by those of certain simple-root gamma-constacyclic codes of length eta. Using this result, Hamming distances of all constacyclic codes of length 4p(s) are obtained. As an application, we identify all MDS lambda-constacyclic codes of length 4p(s). (C) 2019 Elsevier B.V. All rights reserved.
作者机构：
[Dinh, Hai Q.] Ton Duc Thang Univ, Inst Computat Sci, Div Computat Math & Engn, Ho Chi Minh City, Vietnam.;[Dinh, Hai Q.] Ton Duc Thang Univ, Fac Math & Stat, Ho Chi Minh City, Vietnam.;[Wang, Xiaoqiang] Hubei Univ, Fac Math & Stat, Hubei Key Lab Appl Math, Wuhan 430062, Hubei, Peoples R China.;[Liu, Hongwei] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Sriboonchitta, Songsak] Chiang Mai Univ, Fac Econ, Ctr Excellence Econometr, Chiang Mai 52000, Thailand.
通讯机构：
[Wang, Xiaoqiang] H;Hubei Univ, Fac Math & Stat, Hubei Key Lab Appl Math, Wuhan 430062, Hubei, Peoples R China.
摘要：
Let p be an odd prime, and lambda be a nonzero element of the finite field Fp(m). The lambda-constacyclic codes of length 2p(s) over Fp(m) are classified as the ideals of quotient ring Fp(m)[chi]/(chi(2)p(s)-lambda) in terms of their generator polynomials. Based on these generator polynomials, the symbol-pair distances of all such lambda-constacyclic codes of length 2p(s) are obtained in this paper. As an application, all MDS symbol-pair constacyclic codes of length 2p(s) over Fp(m) are established, which produce many new MDS symbol-pair codes with good parameters. (C) 2019 Elsevier B.V. All rights reserved.
期刊：
Finite Fields and Their Applications,2019年59:199-207 ISSN：1071-5797
通讯作者：
Luo, Jinquan
作者机构：
[Luo, Jinquan] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Luo, Jinquan] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
期刊：
Cryptography and Communications,2019年11(2):137-146 ISSN：1936-2447
通讯作者：
Maouche, Youcef
作者机构：
[Liu, Hongwei; Maouche, Youcef] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Maouche, Youcef] Univ Sci & Technol Houari Boumediene, Dept Math, Algiers, Algeria.
通讯机构：
[Maouche, Youcef] C;[Maouche, Youcef] U;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Univ Sci & Technol Houari Boumediene, Dept Math, Algiers, Algeria.
关键词：
Linear codes;Weight distribution;Gauss sum
摘要：
Let
$\phantom {\dot {i}\!}\mathbb {F}_{q}$
be a finite field of order q, where q = ps is a power of a prime number p. Let m and m1 be two positive integers such that m1 divides m. For any positive divisor e of q − 1, we construct an infinite family of codes with dimension m + m1 and few weights over
$\phantom {\dot {i}\!}\mathbb {F}_{q}$
. Using Gauss sum, their weight distributions are provided. When gcd(e, m) = 1, we obtain a subclass of optimal codes which attain the Griesmer bound. Moreover, when gcd(e, m) = 2 or 3 we construct new infinite families of codes with at most four weights.
作者机构：
[Liu, Hongwei; Wang, Xiaoqiang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Zheng, Dabin] Hubei Univ, Fac Math & Stat, Hubei Key Lab Appl Math, Wuhan 430062, Hubei, Peoples R China.
通讯机构：
[Liu, Hongwei] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
关键词：
Cyclic code;Weight distribution;Exponential sum;Quadratic form
摘要：
Let m and k be two positive integers such that v(2)(m) = v(2)(k), where v(2)(.) denotes the 2-valuation function, and let alpha be a primitive element of F-p2m. Let C be a cyclic code over F-p with a parity-check polynomial h(1)(X)h(2)(X)h(3)(X) is an element of F-p2m [x], where h(1)(x), h(2)(x) and h(3)(x) are the minimal polynomials of alpha(-pk+1/2), -alpha(-pk+1/2) and alpha(-pm+1/2) over F-p respectively. In this paper, we determine the weight distribution and the complete weight distribution of the code C. (C) 2017 Elsevier B.V. All rights reserved.
期刊：
IEEE Transactions on Information Theory,2018年64(8):5776-5782 ISSN：0018-9448
通讯作者：
Chen, Bocong
作者机构：
[Chen, Bocong] South China Univ Technol, Sch Math, Guangzhou 510641, Guangdong, Peoples R China.;[Liu, Hongwei] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Chen, Bocong] S;South China Univ Technol, Sch Math, Guangzhou 510641, Guangdong, Peoples R China.
期刊：
DESIGNS CODES AND CRYPTOGRAPHY,2018年86(6):1267-1279 ISSN：0925-1022
通讯作者：
Liu, Hongwei
作者机构：
[Chen, Bocong] South China Univ Technol, Sch Math, Guangzhou 510641, Guangdong, Peoples R China.;[Liu, Hongwei] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Liu, Hongwei] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
