| 44 | 0 | 87 |
| 下载次数 | 被引频次 | 阅读次数 |
线性互补对偶(Linear Complementary Dual, LCD)码因其在数据存储和密码学方面的显著作用而受到广泛研究.极大距离可分(Maximum Distance Separable, MDS)码具有最优的纠错能力,因而构造LCD MDS码是编码理论研究的一个热点.利用Goppa码、扭曲广义Reed-Solomon码和斜群码等可以给出LCD MDS码的显式构造.本文回顾了2020年以来LCD MDS码研究的最新进展,并总结了该领域一些悬而未决的问题.
Abstract：Linear complementary dual(LCD) codes have been extensively studied due to their significant applications in data storage and cryptography.Maximum distance separable(MDS) codes possess optimal error-correction capability,therefore,constructing LCD MDS codes is a prominent research topic in coding theory.The explicit constructions of LCD MDS codes can be given using Goppa codes,twisted generalized Reed-Solomon codes,and skew group codes.This paper reviews recent advances on LCD MDS codes since 2020,and proposes some unresolved problems in this field.
[1]MASSEY J L. Reversible codes[J].Inform Control,1964,7:369.
[2]MASSEY J L.Linear codes with complementary duals[J].Discrete Math,1992,106-107:337.
[3]CARLET C, GUILLEY S.Complementary dual codes for counter-measures to side-channel attacks[A].Coding Theory and Applications[C].Cham:Springer,2015:97.
[4]CARLET C, GUILLEY S.Complementary dual codes for counter-measures to side-channel attacks[J].Adv Math Commun,2016,10:131.
[5]LI S T,SHI M J,LIU H Z.Several constructions of optimal LCD codes over small finite fields[J].Cryptogr Commun,2024,16:779.
[6]WANG G D, LIU S W, LIU H W. New constructions of optimal binary LCD codes[J].Finite Fields Appl,2024,95:102381.
[7]LI X R,HENG Z L.Binary and ternary leadingsystematic LCD codes from special functions[J].Finite Fields Appl,2024,97:102441.
[8]WANG X R,HENG Z L,LI F W,et al.LCD codes and almost optimally extendable codes from self-orthogonal codes[J].Des Codes Cryptogr,2024,92:2901.
[9]FU Y Q,LIU H W.Two classes of LCD BCH codes over finite fields[J].Finite Fields Appl,2024,99:102478.
[10]LEI H Y,TANG C M,ZHANG J.Ternary LCD codes and their connection to bent functions[J].Adv Math Commun,2025,19(6):1713.
[11]XIE D C,ZHU S X.Two families of self-orthogonal codes with applications in LCD codes and optimally extendable codes[J].Finite Fields Appl,2025,108:102659.
[12]MESNAGER S,SINAK A.Further designs for self-orthogonal and LCD codes developed from functions over finite fields[J].Comp Appl Math,2025,44:341.
[13]HUFFMAN W C,PLESS V.Fundamentals of Error-correcting Codes[M].Cambridge:Cambridge University Press,2003.
[14]LI F G, CHEN Y Y, CHEN H. Non-ReedSolomon type cyclic MDS codes[J].IEEE Trans Inf Theory,2025,71(5):3489.
[15]WU Y S,DING C S,CHEN T F.When does the extended code of an MDS code remain MDS?[J].IEEE Trans Inf Theory,2025,71(1):263.
[16]ZHI Y J,ZHU S X.New MDS codes of non-GRS type and NMDS codes[J].Finite Fields Appl,2025,348(5):114436.
[17]LI F W,JIANG R Y,LIU Y T.New MDS and self-dual generalized Roth-Lample codes as well as their deep holes[J].Des Codes Cryptogr,2025,93:5079.
[18]MENG W R,FANG W J,FU F W,et al.Construction of MDS Euclidean self-dual codes via multiple subsets[J].Finite Fields Appl,2026,109:102718.
[19]金玲飞,孙中华,滕佳明.LCD MDS码的几类构造方法[J].中国科学:数学,2021,51(10):1463.
[20]GAO Y Y,YUE Q,HUANG X M,et al.Hulls of generalized Reed-Solomon codes via Goppa codes and their applications to quantum codes[J].IEEE Trans Inf Theory,2021,67(10):6619.
[21]SARI M,KOROGLU M E. New entanglementassisted quantum MDS codes with maximal entanglement[J].Internat J Theoret Phys,2021,60:243.
[22]ZHU C Z, LIAO Q Y. The[1,0]-twisted generalized Reed-Solomon code[J]. Cryptogr Commun,2024,16:857.
[23]LIANG Z H,LIAO Q Y.Four classes of LCD codes from(*)-(L,P)-twisted generalized ReedSolomon codes[J].ar Xvi:2509.14878.
[24]MEENA K C,PACHAURI P, AWASTHI A,et al.A class of triple-twisted GRS codes[J].Des Codes Cryptogr,2025,93:2369.
[25]YANG S D,WANG J L,WU Y S.Two classes of twisted generalized Reed-Solomon codes with two twists[J]. Finite Fields Appl,2025,104:102595.
[26]WU Y S,HYUN J Y,LEE Y J.New LCD MDS codes of non-Reed-Solomon type[J].IEEE Trans Inf Theory,2021,67(8):5069.
[27]LUO G J,CAO X W,EZERMAN M F,et al.Two new classes of Hermitian self-orthogonal nonGRS MDS codes and their applications[J].Adv Math Commun,2022,16(4):921.
[28]HUANG D T,YUE Q,NIU Y F.MDS or NMDS LCD codes from twisted Reed-Solomon codes[J].Cryptogr Commun,2023,15:221.
[29]XU H Q,DU W.Constructions of symplectic LCD MDS codes[J].Bull Malays Math Sci Soc,2021,44:3377.
[30]HUANG X,LI J, HUANG S.Constructions of symplectic LCD MDS codes from quasi-cyclic codes[J]. Adv Math Commun,2022,16(4):779.
[31]ALI S, MOHAMMAD G,JEELANI,et al.On quantum and LCD codes over the ring Fq+v Fq+v2Fq[J].Quantum Inf Process,2022,21:306.
[32]El BADRY M,HAILY A,MOUNIR A.On LCD skew group codes[J].Des Codes Cryptogr,2025,93(5):1487.
基本信息:
DOI：10.16783/j.cnki.nwnuz.2026.01.002
中图分类号:O157.4
引用信息:
[1]乔兴斌,杜小妮.LCD MDS码的最新研究进展[J].西北师范大学学报(自然科学版),2026,62(01):15-22.DOI:10.16783/j.cnki.nwnuz.2026.01.002.
基金信息:
国家自然科学基金资助项目（62562055,62172337）; 甘肃省自然科学基金重点资助项目（23JRRA685）; 甘肃省基础研究创新群体资助项目（23JRRA684）
2026-01-15
2026-01-15