Subfield Codes of Several Few-Weight Linear Codes Parameterized by Functions and Their Consequences

Subfield codes of linear codes over finite fields have recently received much attention since they can produce optimal codes, which may have applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction framework of 3-dimensional linear codes <inline-formula> <tex-math notation="LaTeX">${\mathcal{ C}}_{f,g}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{m}}$ </tex-math></inline-formula> parameterized by any two functions <inline-formula> <tex-math notation="LaTeX">$f, g$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{m}}$ </tex-math></inline-formula>, and then study the properties of six types of <inline-formula> <tex-math notation="LaTeX">${\mathcal{ C}}_{f,g}$ </tex-math></inline-formula>, its punctured code <inline-formula> <tex-math notation="LaTeX">${\mathcal{ C}}^{\ast}_{f,g}$ </tex-math></inline-formula> and their corresponding subfield codes over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula>. The classification of <inline-formula> <tex-math notation="LaTeX">${\mathcal{ C}}_{f,g}$ </tex-math></inline-formula> is based on special choices of <inline-formula> <tex-math notation="LaTeX">$f,g$ </tex-math></inline-formula> as trace function, norm function, almost bent function, Boolean bent function or a combination of these functions. For the first two types of <inline-formula> <tex-math notation="LaTeX">${\mathcal{ C}}_{f,g}$ </tex-math></inline-formula>, we explicitly determine the weight distributions and dualities of <inline-formula> <tex-math notation="LaTeX">${\mathcal{ C}}_{f,g}, {\mathcal{ C}}_{f,g}^{\ast}$ </tex-math></inline-formula> and their subfield codes over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula>. The remaining four types of <inline-formula> <tex-math notation="LaTeX">${\mathcal{ C}}_{f,g}$ </tex-math></inline-formula> are restricted to <inline-formula> <tex-math notation="LaTeX">$q=2$ </tex-math></inline-formula>, and the weight distributions and dualities of the subfields code <inline-formula> <tex-math notation="LaTeX">${\mathcal{ C}}_{f,g}^{(q)}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${\cal C_{f,g}^{\ast}}^{(q)}$ </tex-math></inline-formula> are completely determined. Most of the resultant linear codes (over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{m}}$ </tex-math></inline-formula> or over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula>) have few weights. Some of them are optimal and some have the best-known parameters according to the tables maintained at <uri>https://www.codetables.de</uri>. In fact, 16 infinite families of optimal linear codes are produced in this paper. As a byproduct, a family of <inline-formula> <tex-math notation="LaTeX">$[2^{4m-2},2m+1,2^{4m-3}]$ </tex-math></inline-formula> quaternary Hermitian self-orthogonal codes are obtained with <inline-formula> <tex-math notation="LaTeX">$m \geq 2$ </tex-math></inline-formula>. As an application, we present several infinite families of 2-designs or 3-designs with some of the codes presented in this paper.