Solomon-Stiffler codes and Belov codes are two wellknown families of Griesmer codes that have recently attracted significant attention in recent coding-theoretic literature, due to their optimality and special algebraic structures. In this paper, we investigate their subfield codes and hull dimensions. Firstly, we establish the parameters of Solomon-Stiffler and Belov codes by means of exponential sums. Following the approach of Hyun et al. (IEEE Trans. Inf. Theory, 71(6): 4267-4283, 2025), we determine the exact parameters of the subfield codes of Solomon-Stiffler codes and Belov codes; for the non-projective case, the parameters are fixed, while for the projective case, they depend on the number of distinct types of generator matrices of the mutually disjoint subspaces. We also derive an explicit formula for the weight enumerators of Solomon-Stiffler codes. Secondly, we characterize the hull dimensions of Solomon-Stiffler codes and Belov codes, thereby extending the results of Shi et al. (J. Combin. Theory, Ser. A, 214: 106027, 2025) on the self-orthogonality of binary Solomon-Stiffler codes and Belov codes. As a consequence, we also obtain several families of self-orthogonal codes.

Recently, constructions of optimal linear codes from simplicial complexes have attracted much attention and some related nice works were presented. Let q be a prime power. In this paper, by using the simplicial complexes of${\mathbb {F}}_{q}^{m}$with one single maximal element, we construct four families of linear codes over the ring${\mathbb {F}}_{q}+u{\mathbb {F}}_{q}$($u^{2}=0$), which generalizes the results of Wu et al. (2020). The parameters and Lee weight distributions of these four families of codes are completely determined. Most notably, via the Gray map, we obtain several classes of optimal linear codes over${\mathbb {F}}_{q}$, including (near) Griesmer codes and distance-optimal codes. Moreover, it is shown that most of the Gray images are minimal or self-orthogonal codes which are useful in applications.

Twisted generalized Reed-Solomon (TGRS) codes are an extension of generalized Reed-Solomon (GRS) codes, and have recently attracted significant attention due to their potential for constructing non-GRS MDS codes. This paper presents an in-depth and comprehensive investigation of TGRS codes in their most general form, allowing arbitrary twists at arbitrary positions. First, we introduce a more precise definition of TGRS codes, namely (L,P)-TGRS codes, and provide a concise necessary and sufficient condition for them to be MDS, thereby generalizing previous results. Second, we explicitly characterize the parity check matrices of (L,P)-TGRS codes, and provide a sufficient condition for them to be self-dual. Finally, we investigate the non-GRS properties of (L,P)-TGRS codes via two approaches: the dimensions of Schur squares and combinatorial techniques. As a result, we obtain an infinite family of non-GRS MDS codes.

The primary objective of this paper is the construction of optimal codes with self-orthogonality that can be used to construct quantum codes. Recently, Ding and Heng explored subfield codes, which can be viewed as trace codes. In this paper, we focus on investigating self-orthogonal optimal trace codes. First, we provide a novel description of trace codes by choosing suitable defining sets. Second, we determine the parameters of the codes and their trace codes whose defining sets are disjoint union of some affine subspaces in both non-projective cases and projective-cases. This result extends the main findings in (Hu, Li, Zeng, Wang, Tang, IEEE Trans. Inform. Theory, 68(7): 4408-4421, 2022). Third, we compute the parameters of trace codes for MacDonald codes, including the first order Reed-Muller codes and simplex codes as special cases. Finally, we examine their self-orthogonality and distance-optimality to find several classes of self-orthogonal Griesmer codes. Additionally, we resolve a problem proposed by Ding and Heng as a byproduct.

In this paper, we construct a large family of projective linear codes over${\mathbb F}_{q}$from the general simplicial complexes of${\mathbb F}_{q}^{m}$via the defining-set construction, which generalizes the results of [IEEE Trans. Inf. Theory 66(11):6762-6773, 2020]. The parameters and weight distributions of this class of codes are completely determined. By using the Griesmer bound, we give a necessary and sufficient condition such that the codes are Griesmer codes and a sufficient condition such that the codes are distance-optimal. For a special case, we also present a necessary and sufficient condition for the codes to be near Griesmer codes. Moreover, by discussing the cases of simplicial complexes with one, two and three maximal elements respectively, the parameters and weight distributions of the codes are given more explicitly, which shows that the codes are at most 2-weight, 5-weight and 19-weight respectively. By studying the optimality of the codes for the three cases in detail, many infinite families of optimal linear codes with few weights over${\mathbb F}_{q}$are obtained, including Griesmer codes, near Griesmer codes and distance-optimal codes.

In this paper, we construct four families of linear codes over finite fields from the complements of either the union of subfields or the union of cosets of a subfield, which can produce infinite families of optimal linear codes, including infinite families of (near) Griesmer codes. We also characterize the optimality of these four families of linear codes with an explicit computable criterion using the Griesmer bound and obtain many distance-optimal linear codes. In addition, by a more in-depth discussion on some special cases of these four families of linear codes, we obtain several classes of (distance-)optimal linear codes with few weights and completely determine their weight distributions. It is shown that most of our linear codes are self-orthogonal or minimal which are useful in applications.