Andries Brouwer, preprint, 1982:

"The Witt designs, Golay codes and Mathieu groups"
(unpublished as of 2013)

Pages 8-9:

Substructures of S(5, 8, 24)

An octad is a block of S(5, 8, 24).

Theorem 5.1

Let B₀ be a fixed octad. The 30 octads disjoint from B₀
form a self-complementary 3-(16,8,3) design, namely
the design of the points and affine hyperplanes in AG(4, 2),
the 4-dimensional affine space over F₂.

Proof....

... (iv) We have AG(4, 2).

(Proof: invoke your favorite characterization of AG(4, 2)
or PG(3, 2), say Dembowski-Wagner or Veblen & Young.
An explicit construction of the vector space is also easy....)