Subfield codes of linear codes over finite fields have recently attracted great attention due to their wide applications in secret sharing schemes, authentication codes and association schemes. There are two major ingredients in this paper. The first ingredient is to solve four conjectures recently proposed by Heng and Ding (IEEE Trans. Inf. Theory, 68(6): 3643-3656, 2022). Besides, we also determine the weight distributions of subfield codes derived from$x^{3}$and oval polynomials over${\mathbb {F}}_{2^{m}}$. The second ingredient is to obtain two classes ofp-ary linear codes from monomials over${\mathbb {F}}_{p^{m}}$. We study the parameters of the constructed codes and determine their weight distributions. Notably, we show that the codes$\mathcal {C}_{(x^{p^{i}},p^{m})}^{(p)}$are optimal and almost optimal in many cases with respect to the online Database of Grassl. Finally, we observe that the derived linear codes also have the minimality property for most cases.

Minimal linear codes have important applications in secure communications, including in the framework of secret sharing schemes and secure multi-party computation. A lot of research have been carried out to derive codes with few weights (but more importantly, being minimal) using algebraic or geometric approaches. One of the main power and fructify algebraic methods is based on the design of those codes by employing functions over finite fields. Li et al. (2021) have recently identified some binary linear codes with few weights from two classes of two-to-one functions. In this paper, our ultimate objective is to expand the class of codes derived from the paper of Li et al. by proposing larger classes of binary linear codes with few weights via generic constructions involving other known families of two-to-one functions over the finite field$\mathbb {F}_{2^{n}}$of order$2^{n}$. We succeed in constructing such codes, and we also completely determine their weight distributions. The linear codes presented in this paper differ in parameters from those known in the literature. Besides, some of them are optimal concerning the well-known Griesmer bound. Notably, we prove that our codes are either optimal or almost optimal with respect to the online Database of Grassl. We next observe that the derived binary linear codes also have the minimality property for most cases. We then describe the access structures of the secret-sharing schemes based on their dual codes. Finally, we solve two problems left open in the paper by Li et al. (more specifically, a complete solution to Problem 2 and a partial solution to Problem 1).

Motivated by the recent studies on Z2Z4-additive cyclic codes, Z2Z2[u]-cyclic codes and Z2Z2(s)-additive codes have been introduced by Aydogdu et al.. In this paper, we study ZqR-linear cyclic codes where R = Zq+ uZq, q is a prime number and u2= 0. There are two major ingredients in this paper. The first is to investigate the algebraic structure of cyclic codes and their duals over ring ZqR, the spanning sets, the types and sizes of ZqR-cyclic codes and their duals as well. Based on this, the second ingredient is to present an infinite family of MDSS codes and obtain some illustrative examples of q-ary cyclic codes with optimal parameters derived from the ZqR-cyclic codes.