Several Families of Binary Minimal Linear Codes From Two-to-One Functions
Several Families of Binary Minimal Linear Codes From Two-to-One Functions
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DOI：
10.1109/tit.2023.3236955
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发表时间：
2023-05
影响因子：
2.5
通讯作者：
Sihem Mesnager;Liqin Qian;X. Cao;Mu Yuan
中科院分区：
文献类型：
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作者：
Sihem Mesnager;Liqin Qian;X. Cao;Mu Yuan
最小线性码在保密通信中有重要的应用，包括在秘密共享方案和安全多方计算的框架中。已经进行了大量的研究来使用代数或几何方法导出具有很少权重（但更重要的是，最小）的代码。主要的幂代数方法之一是基于有限域上的函数来设计这些码。Li等人（2021）最近从两类二比一函数中确定了一些具有很少权重的二进制线性码。在本文中，我们的最终目标是扩大类的代码来自李等人的文件。提出更大的类的二元线性码的重量少，通过一般的建设，涉及其他已知的家庭的二比一功能的有限域$\mathbb {F}_{2^{n}}$的顺序为$2^{n}$。我们成功地构建这样的代码，我们也完全确定其重量分布。本文提出的线性码的参数不同于那些已知的文献中。此外，其中一些是关于著名的Griesmer界的最佳。值得注意的是，我们证明了我们的代码是最优或几乎最优的在线数据库的Grassl。接下来我们观察到，导出的二元线性码在大多数情况下也具有极小性。然后，我们描述了秘密共享方案的访问结构的基础上，他们的双重代码。最后，我们解决了李等人在论文中留下的两个问题（更具体地说，问题2的完整解决方案和问题1的部分解决方案）。
Minimal linear codes have important applications in secure communications, including in the framework of secret sharing schemes and secure multi-party computation. A lot of research have been carried out to derive codes with few weights (but more importantly, being minimal) using algebraic or geometric approaches. One of the main power and fructify algebraic methods is based on the design of those codes by employing functions over finite fields. Li et al. (2021) have recently identified some binary linear codes with few weights from two classes of two-to-one functions. In this paper, our ultimate objective is to expand the class of codes derived from the paper of Li et al. by proposing larger classes of binary linear codes with few weights via generic constructions involving other known families of two-to-one functions over the finite field $\mathbb {F}_{2^{n}}$ of order $2^{n}$ . We succeed in constructing such codes, and we also completely determine their weight distributions. The linear codes presented in this paper differ in parameters from those known in the literature. Besides, some of them are optimal concerning the well-known Griesmer bound. Notably, we prove that our codes are either optimal or almost optimal with respect to the online Database of Grassl. We next observe that the derived binary linear codes also have the minimality property for most cases. We then describe the access structures of the secret-sharing schemes based on their dual codes. Finally, we solve two problems left open in the paper by Li et al. (more specifically, a complete solution to Problem 2 and a partial solution to Problem 1).