Linear codes over a mixed-alphabet ring and their Gray images with applications to projective and locally repairable codes
Let $m \geq 2$ be an integer, and let $\mathbb{F}_q$ be the finite field of prime power order $q.$ Let $\mathcal{R}=\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}\times \mathbb{F}_q$ be the mixed-alphabet ring, where $\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}$ is the quasi-Galois ring with maximal ideal $\langle u\rangle$ of nilpotency index $2$ and residue field $\mathbb{F}_q.$ 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 non-empty 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 identify several infinite families of few-weight codes over $\mathbb{F}_q,$ as well as an infinite family of minimal, near-Griesmer and distance-optimal codes over $\mathbb{F}_q.$ We also observe that their Gray images are self-orthogonal codes for $q=2$ or $3.$ We determine spanning matrices of these codes. Leveraging this result, we provide two constructions of infinite families of projective few-weight codes over $\mathbb{F}_q$ with new parameters. As an application of our newly constructed minimal codes over $\mathbb{F}_q,$ we examine the minimal access structures of Masseys 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 and show that these codes have locality either $2$ or $3.$ As a consequence, we obtain four infinite families of $q$-ary locally repairable codes (LRCs) with locality $2,$ and two infinite families of binary LRCs with locality $3.$