# On Pellegrino's 20-Caps in $S_{4,3}$

• Published in 1983
Although Pellegrino demonstrated that every 20-cap in $S_{4,3}$ is one of two geometric types, but it is by no means clear how many inequivalent 20-caps are there in each type. This chapter demonstrates that there are in all exactly nine inequivalent 20-caps in $S_{4,3}$. It also shows that just two of these occur as the intersection of a 56-cap in $S_{5,3}$ with a hyperplane. Because any 10-cap in $S_{3,3}$ is an elliptic quadric and is unique up to equivalence, it follows that any choice of E and V is equivalent to any other. However, for a given choice of E and V, there are 310 different r-caps. The seemingly difficult task of finding how many of these are inequivalent is made relatively simple by using the triple transitivity of the group Aut E on the points of E, together with the uniqueness of the ternary Golay code. The chapter identifies those 20-caps that occur as the intersection of a 56-cap in $S_{5,3}$ with a hyperplane and shows that caps of both these types do occur as sections of a 56-cap in $S_{5,3}$.

@article{OnPellegrinos20CapsinS43,
title = {On Pellegrino's 20-Caps in {\$}S{\_}{\{}4,3{\}}{\$}},
abstract = {Although Pellegrino demonstrated that every 20-cap in {\$}S{\_}{\{}4,3{\}}{\$} is one of two geometric types, but it is by no means clear how many inequivalent 20-caps are there in each type. This chapter demonstrates that there are in all exactly nine inequivalent 20-caps in {\$}S{\_}{\{}4,3{\}}{\$}. It also shows that just two of these occur as the intersection of a 56-cap in {\$}S{\_}{\{}5,3{\}}{\$} with a hyperplane. Because any 10-cap in {\$}S{\_}{\{}3,3{\}}{\$} is an elliptic quadric and is unique up to equivalence, it follows that any choice of E and V is equivalent to any other. However, for a given choice of E and V, there are 310 different r-caps. The seemingly difficult task of finding how many of these are inequivalent is made relatively simple by using the triple transitivity of the group Aut E on the points of E, together with the uniqueness of the ternary Golay code. The chapter identifies those 20-caps that occur as the intersection of a 56-cap in {\$}S{\_}{\{}5,3{\}}{\$} with a hyperplane and shows that caps of both these types do occur as sections of a 56-cap in {\$}S{\_}{\{}5,3{\}}{\$}.},
}