利用单纯复形构造环上的无穷多族最优码与极小码
Infinite families of optimal and minimal codes over rings using simplicial complexes
AI总结：
该研究利用单纯复形在非单位非交换环扩环上构造无穷多族码，表征其参数、确定李重量分布，进一步得到大量满足极小性或距离最优的有限域线性码。
摘要：
本文利用一般单纯复形，在非单位非交换环的扩环上构造了若干无穷多族码。借助定义集的特殊结构，表征了这些码的主要参数。特别地，当所用单纯复形由单个极大元生成时，我们完全确定了它们的李重量分布。此外，通过研究Gray像码及相应的类子域码，还得到了大量$\u200b\mathbb{F}_q$上的线性码，其中q为素数幂。给出了在q=2、3、4情形下确保上述线性码为（埃尔米特）自正交码的特定条件。值得注意的是，大多数所得的$\u200b\mathbb{F}_q$上的码满足Ashikhmin-Barg极小性条件。此外，我们还得到了两族关于Griesmer界的距离最优$\u200b\mathbb{F}_q$码无穷族。
英文摘要：
In this paper, several infinite families of codes over the extension of non-unital non-commutative rings are constructed utilizing general simplicial complexes. Thanks to the special structure of the defining sets, the principal parameters of these codes are characterized. Specially, when the employed simplicial complexes are generated by a single maximal element, we determine their Lee weight distributions completely. Furthermore, by considering the Gray image codes and the corresponding subfield-like codes, numerous of linear codes over $\mathbb{F}_q$ are also obtained, where $q$ is a prime power. Certain conditions are given to ensure the above linear codes are (Hermitian) self-orthogonal in the case of $q=2,3,4$. It is noteworthy that most of the derived codes over $\mathbb{F}_q$ satisfy the Ashikhmin-Barg's condition for minimality. Besides, we obtain two infinite families of distance-optimal codes over $\mathbb{F}_q$ with respect to the Griesmer bound.
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