conference matrices

[N. J. A. Sloane]

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* N. J. A. Sloane <njas at research.att.com> [Mar 14. 2008 08:31]:
> 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 ?

Yes, code attached.
You will also need this file:
http://www.jjj.de/pari/conference-mat-ff.inc.gp

The "sum of two squares" condition puzzles me:
isn't q^n+1 where q is an odd prime always of that form?

> 
> Thanks for catching this mistake!
> 
> 
>  Best regards
>  			 Neil

regards,   jj

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\r conference-mat-ff.inc.gp

\\ --- q an odd prime and fp a field polynomial:
\\q = 13; fp = x-1;    \\ simple case: GF(q)
\\q = 5;  fp = x^2-2;  \\ GF(q^2)
\\q = 3;  fp = x^2+1;  \\ GF(q^2) \\ symm
\\q = 3;  fp = x^3-x+1;  \\ GF(q^3) \\ antisymm

q = 53; fp = x-1;    \\ simple case: GF(q)

\\default(echo, 1);

print("q = ", q);
print("fp = ", fp);
n = poldegree(fp);
print("  GF(", q, "^", n,")");
fp *= Mod(1,q);
if ( !polisirreducible(fp),  error("fp reducible over GF(q)")); \\ check


sv = quadcharvec(fp, q);  \\ quadratic character
print("Table of quadratic characters:"); print1("  "); printmat01(Mat(sv));
if ( 0!=sum(j=1,length(sv), sv[j]), quit() );  \\ check
C=matconference(q, fp, sv);
N=q^n+1;

\\print("",N ,"x",N ," conference matrix C:");  printmat01(C);

matsize(C)

if ( C*C~ != (N-1)*matid(N), error("not a conference matrix!") );
\\C*C~

quit /* ++++++++++++++++++++++ */



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