Editing Leech lattice (section)

===Witt's construction===

In 1972 Witt gave the following construction, which he said he found in 1940, on January 28. Suppose that ''H'' is an ''n'' by ''n'' [[Hadamard matrix]], where ''n''=4''ab''. Then the matrix <math>\begin{pmatrix} Ia&H/2\\H/2&Ib\end{pmatrix}</math> defines a bilinear form in 2''n'' dimensions, whose kernel has ''n'' dimensions. The quotient by this kernel is a nonsingular bilinear form taking values in (1/2)'''Z'''. It has 3 sublattices of index 2 that are integral bilinear forms. Witt obtained the Leech lattice as one of these three sublattices by taking ''a''=2, ''b''=3, and taking ''H'' to be the 24 by 24 matrix (indexed by '''Z'''/23'''Z''' ∪ ∞) with entries Χ(''m''+''n'') where Χ(∞)=1, Χ(0)=−1, Χ(''n'')=is the quadratic residue symbol mod 23 for nonzero ''n''. This matrix ''H'' is a [[Paley matrix]] with some insignificant sign changes.