Page 66 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 66
2.7 References
Example 24. H (4):
+ + + +
+ + − −
+ − + −
+−−+
Necessary condition for the existence of an H (n ) is n ≡ 0 (mod 4) or n = 1, 2. The
famous conjecture, stated by Hadamard in 1893, claims that the above condition is also
sufficient. The smallest order for which the conjecture remains open is 668.
Definition 22. A Hadamard design is a symmetric (4m − 1, 2m − 1, m − 1) − BIBD.
The existence of a Hadamard design of order 4m − 1 is equivalent to the existence of
a Hadamard matrix of side 4m .
Example 25. (7, 3, 1) − BIBD and its corresponding H (8).
1101000 + + + + + + + +
0110100
0011010 + + + − + − − −
0001101
1000110
0100011 + − + + − + − −
1010001
+ − − + + − + −
+ − − − + + − +
+ + − − − + + −
+ − + − − − + +
++−+−−−+
Exercise 28.
Construct a Hadamard matrix H (12).
2.7 References
[1] C.J. Colbourn, J.H. Dinitz (eds.), Handbook of Combinatorial Designs, Second Edition,
Chapman & Hall/CRC, 2006.
[2] C.J. Colbourn, A. Rosa, Triple Systems, Clarendon Press, 1999.
[3] C.C. Lindner, C.A. Rodger, Design Theory, Second Edition, Chapman & Hall/CRC,
2009.
[4] D.R. Stinson, Combinatorial Designs, Constructions and Analysis, Springer, 2004.
[5] W.D. Wallis, Introduction to Combinatorial Designs, Chapman & Hall/CRC, 2007.
Example 24. H (4):
+ + + +
+ + − −
+ − + −
+−−+
Necessary condition for the existence of an H (n ) is n ≡ 0 (mod 4) or n = 1, 2. The
famous conjecture, stated by Hadamard in 1893, claims that the above condition is also
sufficient. The smallest order for which the conjecture remains open is 668.
Definition 22. A Hadamard design is a symmetric (4m − 1, 2m − 1, m − 1) − BIBD.
The existence of a Hadamard design of order 4m − 1 is equivalent to the existence of
a Hadamard matrix of side 4m .
Example 25. (7, 3, 1) − BIBD and its corresponding H (8).
1101000 + + + + + + + +
0110100
0011010 + + + − + − − −
0001101
1000110
0100011 + − + + − + − −
1010001
+ − − + + − + −
+ − − − + + − +
+ + − − − + + −
+ − + − − − + +
++−+−−−+
Exercise 28.
Construct a Hadamard matrix H (12).
2.7 References
[1] C.J. Colbourn, J.H. Dinitz (eds.), Handbook of Combinatorial Designs, Second Edition,
Chapman & Hall/CRC, 2006.
[2] C.J. Colbourn, A. Rosa, Triple Systems, Clarendon Press, 1999.
[3] C.C. Lindner, C.A. Rodger, Design Theory, Second Edition, Chapman & Hall/CRC,
2009.
[4] D.R. Stinson, Combinatorial Designs, Constructions and Analysis, Springer, 2004.
[5] W.D. Wallis, Introduction to Combinatorial Designs, Chapman & Hall/CRC, 2007.
