THE WITT DESIGN

DISTINGUISHED PROFESSOR OF
MATHEMATICS & NATURAL SCIENCES

ENDOWED CHAIR

INSTITUTE OF MATHEMATICS
H-501 PALAM VIHAR
DISTRICT  GURGAON
HARYANA  1 2 2 0 1 7
INDIA

Program & I/O Files

 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 paper 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. The Endowed Chair of the Institute of Mathematics was bestowed upon Distinguished Professor Ashay Dharwadker in 2012 to honour his fundamental contributions to Mathematics and Natural Sciences.

A Steiner system S(t, k, v) is a set P of points together with a set B of blocks such that
• There are v points.
• Each block consists of k points.
• Every set of t points is contained in a unique block.
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

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