Interesting Esoterica

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

Article by R. Hill
  • Published in 1983
  • Added on
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 $S_{5,3}$.

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BibTeX entry

@article{OnPellegrinos20CapsinS43,
	title = {On Pellegrino's 20-Caps in {\$}S{\_}{\{}4,3{\}}{\$}},
	author = {R. Hill},
	url = {http://www.sciencedirect.com/science/article/pii/S030402080873322X http://www.sciencedirect.com/science/article/pii/S030402080873322X/pdf?md5=d1b75feaabe33b62c6beb656d86a2a7d{\&}pid=1-s2.0-S030402080873322X-main.pdf},
	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{\}}{\$}.},
	comment = {},
	urldate = {2016-06-01},
	year = 1983
}