Hadamard matrices

  • It has been a common misapprehension over the last seven or eight years that the number of pairwise non-isomorphic 2-(23,11,5) designs is 1102. This notion was first put forward by H. Kimura (New Hadamard matrix of order 24, Graphs Combin. 2 (1986) 247-257) who found a Hadamard matrix of order 24 that had been missed by Ito et al (Classification of 3-(24,12,5) designs and 24-dimensional Hadamard matrices, J. Comb. Theory Ser. A 27 (1979) 289-306). It is a figure that appears even today in the new CRC Handbook of Combinatorial Designs. In my recent paper on Hadamard matrices of orders 24 and 28 that appeared in Discrete Math. 140 (1995) 185-243 , I also repeated this erroneous figure. Had I bothered to check the numbers of Hadamard designs that I had found from the 60 Hadamard matrices of order 24, I would have announced that the correct number of pairwise non-isomorphic 2-(23,11,5) designs is 1106. This number is implicit in the Table on pp 192-196 appearing in my paper cited above, and indeed all 1106 designs are contained in each of the files 23-11-5.gz* above.

  • Some people may be interested in having the electronic copies of the Hadamard matrices of orders up to 28. Since they are unique for orders 2, 4, 8 and 12, I only list those of orders 16, 20, 24 and 28. These can be found in the files Hadamard.16, Hadamard.20, Hadamard.24, Hadamard.28. These are gzipped files of zero-one matrices of orders 15, 19, 23 and 27, respectively. They are obtained from the normalised Hadamard matrix of order 4m by deleting the first row and column, and replacing the -1's by 0's.

    It is a well-known fact that regular two-graphs on 36 vertices correspond to regular symmetric Hadamard matrices with constant diagonal. In a recent investigation into such regular two-graphs (E. Spence, Regular two-graphs on 36 vertices, Lin. Alg. Appl., 226-228 (1995) 459-497), 136 new ones were found, giving a total of 227. Not all these non-isomorphic regular two-graphs were non-isomorphic as Hadamard matrices. In fact the number of pairwise non-isomorphic Hadamard matrices thus found is 180. These Hadamard matrices were examined for Hadamard designs (symmetric 2-(35,17,8) designs) and I discovered that they yielded 101,863 in number (NOT 108,131 as I wrote in the above paper!!). I include the 180 Hadamard matrices in the file Hadamard.36 in hexadecimal form. This is obtained from the greatest descendant of the Hadamard matrix by concatenating its rows, adding just enough zeros to make the length of the resulting binary integer divisible by 4 and then expressing it as a hexadecimal integer. I also include the order of the automorphism group of the Hadamard matrix as well as the number of non-isomorphic descendants that arise form each. In common with the listing of regular two-graphs that appear in Tables of Two-Graphs, the notation [x,y] means x descendants with automorphism group of order y.

    When we had nothing better to do Dick Turyn and I had a look at the numbers of Hadamard matrices of various orders of the Goethals-Seidel type. The numbers of such pairwise non-isomorphic matrices grows rapidly with the order, as one might expect. When the order is 36 we found 24, none of which appears among the 180 referred to above. For completeness they are listed here in (0,1) form G-S matrices of order 36 . They can also be found in hexadecimal form, along with the numbers of descendants and the orders of their automorphism groups. G-S matrices of order 36 in hexadecimal form . The hexadecimal form is that of the greatest descendant, so represents an incidence matrix of a 2-(35,17,8) design