65.52 ConferenceCode

ConferenceCode( H )

ConferenceCode returns a code of length <n>-1 constructed from a symmetric conference matrix H. H must be a symmetric matrix of order n, which satisfies H*H^T = ((n-1)*I. n = 2 (mod 4). The rows of 1/2(H+I+J), 1/2(-H+I+J), plus the zero and all-ones vectors form the elements of a binary non-linear (n-1, 2*n, 1/2 * (n-2)) code.

    gap> H6 := [[0,1,1,1,1,1],[1,0,1,-1,-1,1],[1,1,0,1,-1,-1],
    > [1,-1,1,0,1,-1],[1,-1,-1,1,0,1],[1,1,-1,-1,1,0]];;
    gap> C1 := ConferenceCode( H6 );
    a (5,12,2)1..4 conference code over GF(2)
    gap> IsLinearCode( C1 );
    false 

ConferenceCode( n )

GUAVA constructs a symmetric conference matrix of order <n>+1 (<n> = 1 (mod 4)) and uses the rows of that matrix, plus the zero and all-ones vectors, to construct a binary non-linear (n, 2*(n+1), 1/2 * (n-1)) code.

    gap> C2 := ConferenceCode( 5 );
    a (5,12,2)1..4 conference code over GF(2)
    gap> Elements( C2 );
    [ [ 0 0 0 0 0 ], [ 1 1 0 1 0 ], [ 1 1 1 0 0 ], [ 0 1 1 0 1 ],
      [ 1 0 0 1 1 ], [ 0 0 1 1 1 ], [ 1 0 1 0 1 ], [ 0 1 0 1 1 ],
      [ 1 0 1 1 0 ], [ 0 1 1 1 0 ], [ 1 1 0 0 1 ], [ 1 1 1 1 1 ] ] 

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