Linear codes over $\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}$ with mixed-alphabet defining sets and their Gray images: Constructions of projective few-weight, distance-optimal and minimal codes

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.