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Abstract: The unique Steiner system S(5, 8, 24) was discovered by Ernst Witt in 1938 and is known as the Witt design. The blocks of the Witt design are the Golay codewords of weight 8. Witt's original construction of S(5, 8, 24) used the largest of the Mathieu groups, M(24), that turns out to be the automorphism group of the Witt design. The group M(24) was the first sporadic simple group discovered in 1860 amongst exactly 26 that were eventually found, many of them arising from the exceptional geometry of the Witt design. This led to the complete classification of all finite simple groups accomplished during the 1980's. The combinatorial properties of the Witt design also played a major role in the proof of the Four Colour Theorem in 2000 and the Grand Unification of the Standard Model with Quantum Gravity in 2008. This website presents a straightforward and explicit construction of the Witt design, listing the blocks in dictionary order.

Acknowledgements: Thanks to The Design Research Group at Queen Mary for featuring this website in their Design Resources. Thanks to The Math Forum Internet Mathematics Library for writing a review of this website in their Combinatorics category. Thanks to Steven Klee and Leah Yates for citing this website in their recent work on the Higman-Sims Design.


 

A Steiner system S(t, k, v) is a set P of points together with a set B of blocks such that

The unique Steiner system S(5, 8, 24) was discovered by Ernst Witt in 1938 and is known as the Witt design. The set P consists of 24 points
 
α
β
γ
δ
ε
ζ
η
θ
ι
κ
λ
μ
ν
ξ
ο
π
ρ
σ
τ
υ
φ
χ
ψ
ω

A subset of points is written as its characteristic function, for example, the subset
 

_
_
_
_
ε
_
_
_
_
_
_
_
ν
_
_
_
ρ
σ
τ
_
_
_
_
_

is written as
 

0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
1
1
1
0
0
0
0
0

with a one indicating the presence and a zero indicating the absence of the corresponding point in the subset. With this notation, the set B of 759 blocks is given below in dictionary order.
 

Dictionary of Blocks
111111110000000000000000
111100001010100100000000
111100000101011000000000
111100000000000010001101
111100000000000001110010
111010001000001000100100
111010000110000010010000
111010000001000101000001
111010000000110000001010
111001001000010011000000
111001000100000100101000
111001000011000000000110
111001000000101000010001
111000101100000000000011
111000100010001001001000
111000100001100010100000
111000100000010100010100
111000011001000000011000
111000010100100001000100
111000010010010000100001
111000010000001110000010
110110001000010000010001
110110000100000100000110
110110000011000000101000
110110000000101011000000
110101001000001000001010
110101000110000001000001
110101000001000110010000
110101000000110000100100
110100101001000001000100
110100100100100000011000
110100100010010010000010
110100100000001100100001
110100011100000010100000
110100010010001000010100
110100010001100000000011
110100010000010101001000
110011001101100000000000
110011000010011100000000
110011000000000010100011
110011000000000001011100
110010101000000110001000
110010100100010001100000
110010100010100000000101
110010100001001000010010
110010011010000001000010
110010010100001000001001
110010010001010010000100
110010010000100100110000
110001101010000000110000
110001100100001010000100
110001100001010000001001
110001100000100101000010
110001011000000100000101
110001010100010000010010
110001010010100010001000
110001010001001001100000
110000111000111000000000
110000110111000100000000
110000110000000011010001
110000110000000000101110
110000001110010000001100
110000001100001101010000
110000001011001010000001
110000001001010100100010
110000001000100010010110
110000001000100001101001
110000000110101000100010
110000000101000011001010
110000000101000000110101
110000000100110110000001
110000000011110001010000
110000000010000111100100
110000000010000100011011
110000000001101100001100
110000000000011010111000
110000000000011001000111
101110001001000010000010
101110000100100000100001
101110000010010001000100
101110000000001100011000
101101001100000000010100
101101000010001010100000
101101000001100001001000
101101000000010100000011
101100101000010000101000
101100100100000111000000
101100100011000000010001
101100100000101000000110
101100011000001001000001
101100010110000000001010
101100010001000100100100
101100010000110010010000
101011001010000000001001
101011000100001001000010
101011000001010000110000
101011000000100110000100
101010101000100001010000
101010100101000000001100
101010100010000100100010
101010100000011010000001
101010011100010100000000
101010010011101000000000
101010010000000011101000
101010010000000000010111
101001101001001100000000
101001100110110000000000
101001100000000010011010
101001100000000001100101
101001011000100000100010
101001010101000010000001
101001010010000101010000
101001010000011000001100
101000111010000010000100
101000110100001000110000
101000110001010001000010
101000110000100100001001
101000001111000001100000
101000001100101010001000
101000001010011000010010
101000001001110000000101
101000001000000110110001
101000001000000101001110
101000000110001100000101
101000000101100100010010
101000000100010010100110
101000000100010001011001
101000000011010110001000
101000000010100011000011
101000000010100000111100
101000000001001011010100
101000000001001000101011
101000000000111101100000
100111001000000101100000
100111000100010010001000
100111000010100000010010
100111000001001000000101
100110101110001000000000
100110100001110100000000
100110100000000010110100
100110100000000001001011
100110011000100000001100
100110010101000001010000
100110010010000110000001
100110010000011000100010
100101101000100010000001
100101100101000000100010
100101100010000100001100
100101100000011001010000
100101011011010000000000
100101010100101100000000
100101010000000011000110
100101010000000000111001
100100111000000100010010
100100110100010000000101
100100110010100001100000
100100110001001010001000
100100001101000100001001
100100001100110001000010
100100001010000011011000
100100001010000000100111
100100001001101000110000
100100001000011110000100
100100000111100010000100
100100000110010100110000
100100000100001010010011
100100000100001001101100
100100000011001101000010
100100000010111000001001
100100000001010011100001
100100000001010000011110
100100000000100110101010
100100000000100101010101
100011101000010000000110
100011100100000100010001
100011100011000011000000
100011100000101000101000
100011011000001010010000
100011010110000000100100
100011010001000100001010
100011010000110001000001
100010111001000000100001
100010110100100010000010
100010110010010000011000
100010110000001101000100
100010001100000011000101
100010001100000000111010
100010001011000100010100
100010001010110010100000
100010001001011001001000
100010001000101100000011
100010000111010000000011
100010000110100101001000
100010000101001110100000
100010000100111000010100
100010000010001010001110
100010000010001001110001
100010000001100010011001
100010000001100001100110
100010000000010111010010
100010000000010100101101
100001111100000001001000
100001110010001000000011
100001110001100000010100
100001110000010110100000
100001001110000110000010
100001001100011000100001
100001001010101001000100
100001001001000010101100
100001001001000001010011
100001001000110100011000
100001000111001000011000
100001000101010101000100
100001000100100011110000
100001000100100000001111
100001000011100100100001
100001000010010010010101
100001000010010001101010
100001000001111010000010
100001000000001111001001
100001000000001100110110
100000101101010010010000
100000101100100100100100
100000101011100000001010
100000101010010101000001
100000101000001011100010
100000101000001000011101
100000100110000010101001
100000100110000001010110
100000100101101001000001
100000100100011100001010
100000100011011000100100
100000100010101110010000
100000100001000110000111
100000100001000101111000
100000100000110011001100
100000100000110000110011
100000011110100000010001
100000011101001000000110
100000011010001100101000
100000011001100111000000
100000011000010010001011
100000011000010001110100
100000010110011011000000
100000010101110000101000
100000010100000110011100
100000010100000101100011
100000010011000010110010
100000010011000001001101
100000010010110100000110
100000010001011100010001
100000010000101010100101
100000010000101001011010
011110001100000001001000
011110000010001000000011
011110000001100000010100
011110000000010110100000
011101001001000000100001
011101000100100010000010
011101000010010000011000
011101000000001101000100
011100101000001010010000
011100100110000000100100
011100100001000100001010
011100100000110001000001
011100011000010000000110
011100010100000100010001
011100010011000011000000
011100010000101000101000
011011001000000100010010
011011000100010000000101
011011000010100001100000
011011000001001010001000
011010101011010000000000
011010100100101100000000
011010100000000011000110
011010100000000000111001
011010011000100010000001
011010010101000000100010
011010010010000100001100
011010010000011001010000
011001101000100000001100
011001100101000001010000
011001100010000110000001
011001100000011000100010
011001011110001000000000
011001010001110100000000
011001010000000010110100
011001010000000001001011
011000111000000101100000
011000110100010010001000
011000110010100000010010
011000110001001000000101
011000001101000110000100
011000001100110000110000
011000001010000010101010
011000001010000001010101
011000001001101001000010
011000001000011100001001
011000000111100000001001
011000000110010101000010
011000000100001011100001
011000000100001000011110
011000000011001100110000
011000000010111010000100
011000000001010010010011
011000000001010001101100
011000000000100111011000
011000000000100100100111
010111001010000010000100
010111000100001000110000
010111000001010001000010
010111000000100100001001
010110101000100000100010
010110100101000010000001
010110100010000101010000
010110100000011000001100
010110011001001100000000
010110010110110000000000
010110010000000010011010
010110010000000001100101
010101101100010100000000
010101100011101000000000
010101100000000011101000
010101100000000000010111
010101011000100001010000
010101010101000000001100
010101010010000100100010
010101010000011010000001
010100111010000000001001
010100110100001001000010
010100110001010000110000
010100110000100110000100
010100001111000000010010
010100001100101000000101
010100001010011001100000
010100001001110010001000
010100001000000111000011
010100001000000100111100
010100000110001110001000
010100000101100101100000
010100000100010011010100
010100000100010000101011
010100000011010100000101
010100000010100010110001
010100000010100001001110
010100000001001010100110
010100000001001001011001
010100000000111100010010
010011101000001001000001
010011100110000000001010
010011100001000100100100
010011100000110010010000
010011011000010000101000
010011010100000111000000
010011010011000000010001
010011010000101000000110
010010111100000000010100
010010110010001010100000
010010110001100001001000
010010110000010100000011
010010001110000100100001
010010001100011010000010
010010001010101000011000
010010001001000011110000
010010001001000000001111
010010001000110101000100
010010000111001001000100
010010000101010100011000
010010000100100010101100
010010000100100001010011
010010000011100110000010
010010000010010011001001
010010000010010000110110
010010000001111000100001
010010000000001110010101
010010000000001101101010
010001111001000010000010
010001110100100000100001
010001110010010001000100
010001110000001100011000
010001001100000010011001
010001001100000001100110
010001001011000101001000
010001001010110000000011
010001001001011000010100
010001001000101110100000
010001000111010010100000
010001000110100100010100
010001000101001100000011
010001000100111001001000
010001000010001011010010
010001000010001000101101
010001000001100011000101
010001000001100000111010
010001000000010110001110
010001000000010101110001
010000101110100011000000
010000101101001000101000
010000101010001100000110
010000101001100100010001
010000101000010010100101
010000101000010001011010
010000100110011000010001
010000100101110000000110
010000100100000110110010
010000100100000101001101
010000100011000010011100
010000100011000001100011
010000100010110100101000
010000100001011111000000
010000100000101010001011
010000100000101001110100
010000011101010001000001
010000011100100100001010
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010000010001000110101001
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010000010000110000011101
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001111000000000011010001
001111000000000000101110
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001010000100000110001011
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001010000011000010100101
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001010000010110100010001
001010000001011100000110
001010000000101010110010
001010000000101001001101
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000000000011101011101000
000000000011101000010111
000000000010011110100011
000000000010011101011100
000000000001110110110100
000000000001110101001011
000000000000000011111111


ISBN 1466265302

Copyright © 2002 by Ashay Dharwadker. All rights reserved.