Gorenstein Simplices and Even Binary Self-Complementary Codes

It is known that if a Gorenstein simplex of dimension $d$ and degree $s$ is not a lattice pyramid, then $d\le 2s-1$. In this paper, we study the extremal case $d=2s-1$. More precisely, we characterize Gorenstein simplices of dimension $2s-1$ and degree $s$ which are not lattice pyramids in terms of even binary self-complementary codes. As an application, combining this characterization with existing classification results on reflexive simplices, we classify Gorenstein simplices of degree $3$ and $4$. Equivalently, we classify polarized $d$-dimensional Gorenstein fake weighted projective spaces $(X,L)$ satisfying $-K_X=(d-2)L$ or $-K_X=(d-3)L$, where $-K_X$ is the anticanonical divisor of $X$ and $L$ is a Cartier divisor on $X$.