[seqfan] Re: Hadamard matrices of small order and Yang conjecture

Orrick, William P. worrick at indiana.edu
Sat Jan 2 16:15:23 CET 2010

> 191, 5767, 7081, 8249
> Actually, several sequences here, and/or corrections tol old seqs?
> arXiv:0912.5091 [ps, pdf, other]
>    Title: Hadamard matrices of small order and Yang conjecture
>    Authors: Dragomir Z. Djokovic
>    Comments: 6 pages, 1 table
>    Subjects: Combinatorics (math.CO)
>    We show that 138 odd values of n less than 10000 for which one
> knows how to construct a Hadamard matrix of order 4n have been
> overlooked in the recent handbook of combinatorial designs. There are
> four additional odd n, namely 191, 5767, 7081 and 8249, in that range
> for which we can construct a Hadamard matrix of order 4n. Our
> exhaustive computer searches show that the near-normal sequences NN(n)
> exist for n=36,38,40. Thus the Yang conjecture on the existence of
> NN(n) for all even n has been verified for n <= 40 but it still
> remains open.
> Dec 27, 2009

I gather from the Djokovic preprint that five additional terms are now 
known for A158763 (number of equivalence classes of near-normal 

1, 2, 2, 3, 8, 14, 11, 24, 20, 18, 32, 12, 3, 20, 9, 8, 5, 1, 1, 1.

That the numbers of equivalence classes in NN(32) and NN(34) are 8 and 
5 is stated in the author's earlier paper "A new Yang number and 
consequences", to appear in Designs, Codes and Cryptography.  (No 
volume or page numbers yet, but available at the journal's web site.)  
The new preprint states that there is a unique equivalence class for 
each of NN(36), NN(38), and NN(40).

It seems unlikely that any new sequences are defined in the preprint, 
which mostly consists of filling in gaps high up in the table of orders 
for which Hadamard matrices are known to exist.  (The lowest gap is 
still at 167.)  A number of different techniques, using existing 
results, are used to fill these gaps.  In particular

> 191, 5767, 7081, 8249

doesn't seem to form part of any interesting sequence.  The first 
number, 191, has a different origin from the last three numbers, which 
all derive from the new result for NN(36).

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