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MDS or NMDS LCD codes from twisted Reed-Solomon codes

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Abstract

Maximum distance separable (MDS) codes are optimal with parameters [n,k,n − k + 1]. Near MDS (NMDS) codes were introduced in 1995 by weakening the definition of MDS codes. NMDS codes also have applications in secret sharing scheme. Linear complementary dual (LCD) codes have been widely used in communications systems, consumer electronics, cryptography and so on. The construction of LCD MDS codes is thus interesting in coding theory. Twisted Reed Solomon (TRS) codes are generalized by Reed Solomon (RS) codes and are not equivalent to RS codes in general case. In this paper, we give parity check matrices of twisted generalized Reed-Solomon (TGRS) codes, show the sufficient and necessary condition that TGRS codes are MDS or NMDS codes, and construct several classes of LCD MDS or NMDS codes from two classes of TGRS codes.

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Authors and Affiliations

  1. College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, People’s Republic of China

    Daitao Huang & Yongfeng Niu

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, People’s Republic of China

    Qin Yue

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  1. Daitao Huang
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  2. Qin Yue
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Correspondence to Qin Yue.

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This work was supported in part by National Natural Science Foundation of China (No. 62172219, No. 12171420) and National Science Foundation of Shandong Province under Grant ZR2021MA046, and National Science Foundation of Jiangsu Province of China (No. BK20200268).

This article belongs to the Topical Collection: Sequences and Their Applications III

Guest Editors: Chunlei Li, Tor Helleseth and Zhengchun Zhou

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Huang, D., Yue, Q. & Niu, Y. MDS or NMDS LCD codes from twisted Reed-Solomon codes. Cryptogr. Commun. 15, 221–237 (2023). https://doi.org/10.1007/s12095-022-00564-9

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  • DOI: https://doi.org/10.1007/s12095-022-00564-9

Keywords

Mathematics Subject Classification (2010)