On Four Conjectures of Heng-Ding and <i>p</i>-ary Linear Codes From Monomials

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x^{3}$ </tex-math></inline-formula> and oval polynomials over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{2^{m}}$ </tex-math></inline-formula>. The second ingredient is to obtain two classes of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i>-ary linear codes from monomials over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{p^{m}}$ </tex-math></inline-formula>. We study the parameters of the constructed codes and determine their weight distributions. Notably, we show that the codes <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}_{(x^{p^{i}},p^{m})}^{(p)}$ </tex-math></inline-formula> 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.