## A conference matrix construction for matrices of order N≡15 mod 16

For details see ref. [OSDS].
Let k = (N-3) / 4. To perform the construction, a (k+1)×(k+1)
conference matrix is required. Since k+1 is a multiple of 4, this
conference matrix is antisymmetrizable, and we assume this has been
done. Furthermore, such a matrix can be normalized by multiplying
rows and the corresponding columns by -1 in such a way that the first
row contains 1s and the first column contains -1s (except for the
(1,1) element). Stripping off the first row and column yields the
k×k matrix c, which has the property that
c c^{T} = k I_{k-1} - J_{k-1}.

If k is a prime power, one matrix having all the properties that c is required
to have is the Jacobsthal matrix of the finite field GF(k).

Given such a matrix, c, one replaces 0s, 1s, and -1s by specified 4x4
matrices and then pads the result with 3 specified rows and columns.

The construction gives good lower bounds for small orders.
However, asymptotically the ratio of this lower bound to the
Ehlich bound tends to 0.

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Page created 11 April 2002.

Last modified 7 October 2012.

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