Construction of MDS Codes and NMDS Codes Based on Cyclic Subgroup of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{F}_{{q}^{2}}^{*} $\end{document}</tex-math></inline-formula>

DU Xiaoni; XUE Jing; QIAO Xingbin; ZHAO Ziwei

doi:10.11999/JEIT251204

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DU Xiaoni, XUE Jing, QIAO Xingbin, ZHAO Ziwei. Construction of MDS Codes and NMDS Codes Based on Cyclic Subgroup of $ \mathbb{F}_{{q}^{2}}^{*} $[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT251204

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DU Xiaoni, XUE Jing, QIAO Xingbin, ZHAO Ziwei. Construction of MDS Codes and NMDS Codes Based on Cyclic Subgroup of $ \mathbb{F}_{{q}^{2}}^{*} $[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT251204

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Construction of MDS Codes and NMDS Codes Based on Cyclic Subgroup of $ \mathbb{F}_{{q}^{2}}^{*} $

doi: 10.11999/JEIT251204 cstr: 32379.14.JEIT251204

DU Xiaoni^{1, 2, 3},
XUE Jing^{1
,
,},
QIAO Xingbin^{1, 2},
ZHAO Ziwei¹

1.
College of Mathematics and Statistic, Northwest Normal University, Lanzhou 730070, China
2.
Key Laboratory of Cryptography and Data Analytics, Northwest Normal University, Lanzhou 730070, China
3.
Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics, Lanzhou 730070, China

Funds: The National Natural Science Foundation of China (62172337, 62562055), The Key Project of Gansu Natural Science Foundation (23JRRA685), The Funds for Innovative Fundamental Research Group Project of Gansu Province (23JRRA684)

Accepted Date: 2026-01-12
Rev Recd Date: 2026-01-12

Available Online: 2026-01-25

Abstract

Abstract

Objective With the rapid development of modern communication technologies, the demand for higher performance and efficiency in error correcting codes has intensified. Error-correcting codes are used to detect and correct errors introduced during transmission. Thanks to their superior algebraic structure, straightforward encoding and decoding algorithms, and ease of implementation, linear codes have become the most widely used class of error-correcting codes in communication systems. Their parameters are constrained by classical bounds, such as the Singleton bound: for a linear code of length $ n $ and dimension $ k $, the minimum distance $ d $ satisfies $ d\leq n-k+1 $. When $ d=n-k+1 $, the code is known as maximum distance separable (MDS) code. MDS codes are widely used in distributed storage systems and random error channels. If $ d=n-k $, the code is termed almost MDS (AMDS), when both a code and its dual are AMDS, it is near MDS (NMDS). Owing to their distinctive geometric structure, NMDS codes have important applications in cryptography and combinatorics. There has been sustained, in-depth research worldwide on constructing structurally simple, high performance MDS and NMDS codes. Against this backdrop, this paper constructs several families of MDS and NMDS codes of length $ q+3 $ over the finite field $ {\mathbb{F}}_{{{q}^{2}}} $ of even characteristic, leveraging the cyclic subgroup $ {U}_{q+1} $. Furthermore, several families of optimal locally repairable codes (LRCs) are presented. LRCs enable efficient failure recovery by accessing only a small set of local nodes, thereby reducing repair overhead and improving system availability, which makes them attractive for distributed and cloud-storage settings. Methods In 2021, Wang et al. constructed NMDS codes of 3 dimension using elliptic curves over finite fields $ {\mathbb{F}}_{q} $. In 2023, Heng et al. obtained several classes of 4 dimensional NMDS codes by appending appropriate column vectors to a base generator matrix. In 2024, Ding et al. presented four classes of 4 dimensional NMDS codes. They also determined the locality of the corresponding dual codes and constructed four classes of distance optimal and dimension optimal LRCs. Building upon the aforementioned works, this paper combines the unit circle $ {U}_{q+1} $ in $ {\mathbb{F}}_{{{q}^{2}}} $ with elliptic curves to construct generator matrices. By augmenting these matrices with two additional column vectors, several classes of MDS and NMDS codes of length $ q+3 $ are obtained. Furthermore, the locality of the constructed NMDS codes is precisely determined, resulting in several classes of optimal LRCs. Results and Discussions In 2023, Heng et al. constructed a class of generator matrices whose second row entries range over $ \mathbb{F}_{q}^{*} $, while the elements of the other rows are composed of nonconsecutive powers of the elements of the second row. Building on this work, Yin et al. constructed generator matrices with elements taken from $ {U}_{q+1} $ in 2025, thereby obtaining new matrices and constructing infinite families of MDS and NMDS codes. Following this line of research, the present paper expands such generator matrices by appending two column vectors, whose elements are selected from the finite field $ {\mathbb{F}}_{{{q}^{2}}} $. The resulting matrices serve as generator matrices for several new classes of MDS and NMDS codes of length$ q+3 $. Notably, several classes of NMDS codes with identical parameters but distinct weight distributions are obtained. Computing the minimum locality of the NMDS codes constructed in this paper shows that some of them are optimal LRCs that satisfy the Singleton-like, Cadambe-Mazumdar, Plotkin-like and Griesmer-like bounds. Furthermore, all of the constructed MDS codes are Griesmer codes, whereas the NMDS codes are near Griesmer. The results indicate that the constructions presented in this paper are more general and unified in nature. Conclusions In this paper, several families of MDS and NMDS codes of length $ q+3 $ over $ {\mathbb{F}}_{{{q}^{2}}} $ are obtained by leveraging elements from the unit circle $ {U}_{q+1} $ in conjunction with oval polynomials, and by appending two additional column vectors with entries in $ {\mathbb{F}}_{q} $ to the resulting matrices. Furthermore, the minimum locality of the constructed NMDS codes is analyzed, revealing that some of them are optimal LRCs. The results demonstrate that the proposed framework generalizes previous constructions and that the obtained codes are either optimal or near-optimal with respect to the Griesmer bound.
- MDS code,
- NMDS code,
- weight enumerator,
- locally recoverable code,
- Griesmer bound

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References(19)

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