Optimal Linear Codes With Few Weights From Simplicial Complexes

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}^{m}$ </tex-math></inline-formula> with one single maximal element, we construct four families of linear codes over the ring <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}+u{\mathbb {F}}_{q}$ </tex-math></inline-formula> (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$u^{2}=0$ </tex-math></inline-formula>), 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula>, 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.