基于<inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_{{q}^{2}}^{*} $\end{document}</tex-math></inline-formula>的循环子群的极大距离可分码和近极大距离可分码的构造

杜小妮; 薛婧; 乔兴斌; 赵紫薇

doi:10.11999/JEIT251204

基于$ \mathbb{F}_{{q}^{2}}^{*} $的循环子群的极大距离可分码和近极大距离可分码的构造

doi: 10.11999/JEIT251204 cstr: 32379.14.JEIT251204

杜小妮^{1, 2, 3},
薛婧^1, ,,
乔兴斌^{1, 2},
赵紫薇¹

1.
西北师范大学数学与统计学院兰州 730070
2.
西北师范大学密码技术与数据分析重点实验室兰州 730070
3.
甘肃省数学与统计学基础学科研究中心兰州 730070

基金项目: 国家自然科学基金 (62172337, 62562055)，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基金(23JRRA684)

详细信息

作者简介:
杜小妮：女，博士，教授，博士生导师，研究方向为密码学与信息安全等

薛婧：女，硕士生，研究方向为密码学与信息安全等

乔兴斌：男，博士，讲师，研究方向为密码学与信息安全等

赵紫薇：女，博士生，研究方向为密码学与信息安全等

通讯作者:
薛婧　xjingmath@163.com

中图分类号: TN918.1; O157.4
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- 文章访问数: 72
- HTML全文浏览量: 29
- PDF下载量: 12
- 被引次数: 0
出版历程
- 收稿日期: 2025-11-14
- 修回日期: 2026-01-08
- 录用日期: 2026-01-12
- 网络出版日期: 2026-01-25

Construction of Maximum Distance Separable Codes and Near Maximum Distance Separable Codes Based on Cyclic Subgroup of $ \mathbb{F}_{{q}^{2}}^{*} $

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)

摘要

摘要: 极大距离可分(MDS)码和近极大距离可分(NMDS)码因其具有良好的代数结构和纠错能力，在通信系统、数据存储和秘钥共享方案等领域有广泛的应用。该文利用偶特征有限域$ {\mathbb{F}}_{{{q}^{2}}} $的乘法群$ \mathbb{F}_{{q}^{2}}^{*} $的循环子群$ {U}_{q+1} $，构造了几类码长为$ q+3 $的MDS码和NMDS码，并运用$ {U}_{q+1} $的性质，确定了所构造码的参数和重量计数器，利用Magma 程序举例验证了结论的正确性，另外，计算了NMDS码的最小局部度，得到了几类最优的局部修复码。特别地，所构造的码均是关于Griesmer界的最优码。
- 极大距离可分码 /
- 近极大距离可分码 /
- 重量计数器 /
- 局部修复码 /
- Griesmer 界
Abstract: Objective The demand for higher performance and efficiency in error-correcting codes has increased with the rapid development of modern communication technologies. These codes detect and correct transmission errors. Because of their algebraic structure, straightforward encoding and decoding, and ease of implementation, linear codes are widely used in communication systems. Their parameters follow classical bounds such as the Singleton bound: for a linear code with length $ n $ and dimension $ k $, the minimum distance $ d $ satisfies $ d\leq n-k+1 $. When $ d=n-k+1 $, the code is a Maximum Distance Separable (MDS) code. MDS codes are applied in distributed storage systems and random error channels. If $ d=n-k $, the code is Almost MDS (AMDS); when both a code and its dual are AMDS, the code is Near MDS (NMDS). NMDS codes have geometric properties that are useful in cryptography and combinatorics. Extensive research has focused on constructing structurally simple, high-performance MDS and NMDS codes. 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 using the cyclic subgroup $ {U}_{q+1} $. Several families of optimal Locally Repairable Codes (LRCs) are also obtained. LRCs support efficient failure recovery by accessing a small set of local nodes, which reduces repair overhead and improves system availability in distributed and cloud-storage settings. Methods In 2021, Wang et al. constructed NMDS codes of dimension 3 using elliptic curves over $ {\mathbb{F}}_{q} $. In 2023, Heng et al. obtained several classes of dimension-4 NMDS codes by appending appropriate column vectors to a base generator matrix. In 2024, Ding et al. presented four classes of dimension-4 NMDS codes, determined the locality of their dual codes, and constructed four classes of distance-optimal and dimension-optimal LRCs. Building on these works, this paper uses the unit circle $ {U}_{q+1} $ in $ {\mathbb{F}}_{{{q}^{2}}} $ and 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. The locality of the constructed NMDS codes is also determined, yielding several classes of optimal LRCs. Results and Discussions In 2023, Heng et al. constructed generator matrices with second-row entries in $ \mathbb{F}_{q}^{*} $ and with the remaining entries given by nonconsecutive powers of the second-row elements. In 2025, Yin et al. extended this approach by constructing generator matrices using elements of $ {U}_{q+1} $ and obtained infinite families of MDS and NMDS codes. Following this direction, the present study expands these matrices by appending two column vectors whose elements lie in $ {\mathbb{F}}_{{{q}^{2}}} $. The resulting matrices generate several classes of MDS and NMDS codes of length $ q+3 $. Several classes of NMDS codes with identical parameters but different weight distributions are also obtained. Computing the minimum locality of the constructed NMDS codes shows that some are optimal LRCs satisfying the Singleton-like, Cadambe–Mazumdar, Plotkin-like, and Griesmer-like bounds. All constructed MDS codes are Griesmer codes, and the NMDS codes are near Griesmer. These results show that the proposed constructions are more general and unified than earlier approaches. Conclusions This paper constructs several families of MDS and NMDS codes of length $ q+3 $ over $ {\mathbb{F}}_{{{q}^{2}}} $ using elements of the unit circle $ {U}_{q+1} $ and oval polynomials, and by appending two additional column vectors with entries in $ {\mathbb{F}}_{q} $. The minimum locality of the constructed NMDS codes is analyzed, and some of these codes are shown to be optimal LRCs. The framework generalizes earlier constructions, and the resulting codes are optimal or near-optimal with respect to the Griesmer bound.
- Maximum Distance Separable(MDS) code /
- NMDS code /
- weight enumerator /
- locally recoverable code /
- Griesmer bound

HTML全文

表 1 NMDS码$ {C}_{1} $的重量分布(不包括$ {A}_{0} $)

$ {A}_{q} $	$ {A}_{q+1} $	$ {A}_{q+2} $	$ {A}_{q+3} $
$ q({q}^{2}-1) $	$ \dfrac{({q}^{2}-1)({q}^{2}-q+6)}{2} $	$ ({q}^{2}-1)({q}^{3}+2{q}^{2}-q-3) $	$ \dfrac{{(q-1)}^{2}(q+1)(2{q}^{3}-3q-2)}{2} $
$ 2({q}^{2}-1) $	$ \dfrac{({q}^{2}-1)({q}^{2}+5q-6)}{2} $	$ ({q}^{2}-1)({q}^{3}+2{q}^{2}-4q+3) $	$ \dfrac{({q}^{2}-1)(2{q}^{4}-2{q}^{3}-3{q}^{2}+3q-2)}{2} $
$ \dfrac{(q+2)({q}^{2}-1)}{2} $	$ \dfrac{q({q}^{2}-1)(q+2)}{2} $	$ \dfrac{q({q}^{2}-1)(2{q}^{2}+4q-5)}{2} $	$ \dfrac{q({q}^{2}-1)(2{q}^{3}-2{q}^{2}-3q+2)}{2} $
$ \dfrac{q({q}^{2}-1)}{2} $	$ \dfrac{({q}^{2}-1)({q}^{2}+2q+6)}{2} $	$ \dfrac{({q}^{2}-1)(2{q}^{3}+4{q}^{2}-5q-6)}{2} $	$ \dfrac{({q}^{2}-1)(2{q}^{4}-2{q}^{3}-3{q}^{2}+2q+2)}{2} $
$ {q}^{2}-1 $	$ \dfrac{({q}^{2}-1)({q}^{2}+5q)}{2} $	$ q({q}^{2}-1)({q}^{2}+2q-4) $	$ \dfrac{q({q}^{2}-1)(q-1)(2{q}^{3}-3)}{2} $

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