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 cT = k Ik-1 - Jk-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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