conference matrices

[Joerg Arndt]

entry: A000952
lowest line says:
54 seems to be the smallest order for which it is not known if a matrix exists. 

...but with the quadratic characters in finite fields we get conference matrices
for all "Numbers of form p^m + 1, p odd prime, m >= 1." (A061344)

Am I confused here?

Anyway:

  GF(53^1)
Table of quadratic characters:
   0 + - - + - + + - + + + - + - + + + - - - - - - + + - - + + - - - - - - + + + - + - + + + - + + - + - - +
54x54 conference matrix C:
 0 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
 + 0 + - - + - + + - + + + - + - + + + - - - - - - + + - - + + - - - - - - + + + - + - + + + - + + - + - - +
 + + 0 + - - + - + + - + + + - + - + + + - - - - - - + + - - + + - - - - - - + + + - + - + + + - + + - + - -
 + - + 0 + - - + - + + - + + + - + - + + + - - - - - - + + - - + + - - - - - - + + + - + - + + + - + + - + -
 + - - + 0 + - - + - + + - + + + - + - + + + - - - - - - + + - - + + - - - - - - + + + - + - + + + - + + - +
 + + - - + 0 + - - + - + + - + + + - + - + + + - - - - - - + + - - + + - - - - - - + + + - + - + + + - + + -
 [--snip--]
 + - + - + + - + + + - + - + + + - - - - - - + + - - + + - - - - - - + + + - + - + + + - + + - + - - + 0 + -
 + - - + - + + - + + + - + - + + + - - - - - - + + - - + + - - - - - - + + + - + - + + + - + + - + - - + 0 +
 + + - - + - + + - + + + - + - + + + - - - - - - + + - - + + - - - - - - + + + - + - + + + - + + - + - - + 0

The method and pari/gp code is given in the fxtbook.




Dear Jeorg,

Yes, I agree that this line is wrong:

%E A000952 54 seems to be the smallest order for which it is not known if a matrix exists.

(For 54 is congruent to 2 mod 4, and 53 = 2^2 + 7^2, so
the matrix should exist)

By the way, did you verify directly that your matrix satisfies A A' = 53 I ?

Thanks for catching this mistake!