Let $\mathcal{R}=\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}\times \mathbb{F}_q$ be the mixed alphabet ring. In this paper, we construct four infinite families of linear codes over the ring $\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}$ whose defining sets are certain nonempty subsets of $\mathcal{R}^m$ associated with three simplicial complexes of $\mathbb{F}_q^m,$ each possessing a single maximal element. We explicitly determine the parameters and Lee weight distributions of these codes. We also study their Gray images and obtain three infinite families of few weight, near Griesmer, distance optimal and minimal codes over $\mathbb{F}_q$ with new parameters. We also provide two constructions of infinite families of projective few weight codes over $\mathbb{F}_q$ with new parameters, and observe that these codes are self orthogonal for $q=2$ or $3.$ Additionally, we obtain two infinite families of binary distance optimal projective codes and an infinite family of dimension optimal projective codes over $\mathbb{F}_q$ with new parameters. Apart from this, we construct an infinite family of quaternary projective $3$-weight codes whose non zero Hamming weights sum to $\frac{9}{4}$ times the code length, which give rise to strongly walk regular graphs. As an application of our newly constructed minimal codes over $\mathbb{F}_q$, we examine the minimal access structures of Massey's secret sharing schemes based on their duals and determine the number of dictatorial participants in these schemes. Finally, we investigate the locality properties of our newly constructed projective codes.

翻译：令$\mathcal{R}=\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}\times \mathbb{F}_q$为混合字母表环。本文在环$\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}$上构造了四个无限族线性码，其定义集为$\mathcal{R}^m$中与$\mathbb{F}_q^m$的三个单纯复形相关联的非空子集，每个单纯复形仅含单个极大元。我们精确确定了这些码的参数与Lee重量分布，并研究了它们的Gray像，从而在$\mathbb{F}_q$上获得了三个具有新参数的少重量、近Griesmer、距离最优且极小的无限族码。此外，我们提供了两个具有新参数的$\mathbb{F}_q$上射影少重量码无限族的构造，并观察到当$q=2$或$3$时这些码是自正交的。我们还得到了两个具有新参数的二进制距离最优射影码无限族，以及一个$\mathbb{F}_q$上具有新参数的维数最优射影码无限族。除此之外，我们构造了一个四进制射影$3$重量码无限族，其非零汉明重量之和为码长的$\frac{9}{4}$倍，这类码可导出强正则行走图。作为新构造的$\mathbb{F}_q$上极小码的应用，我们基于其对偶码分析了Massey秘密共享方案的极小访问结构，并确定了这些方案中专制参与者的数量。最后，我们研究了新构造射影码的局部性质。