65.122 KrawtchoukMat

KrawtchoukMat( n , q )

KrawtchoukMat returns the <n>+1 by <n>+1 matrix K=(k_{ij}) defined by k_{ij}=K_i(j) for i,j=0,...,n. K_i(j) is the Krawtchouk number (see Krawtchouk). n must be a positive integer and q a prime power. The Krawtchouk matrix is used in the MacWilliams identities, defining the relation between the weight distribution of a code of length n over a field of size q, and its dual code. Each call to KrawtchoukMat returns a new matrix, so it is safe to modify the result.

    gap> PrintArray( KrawtchoukMat( 3, 2 ) );
    [ [   1,   1,   1,   1 ],
      [   3,   1,  -1,  -3 ],
      [   3,  -1,  -1,   3 ],
      [   1,  -1,   1,  -1 ] ]
    gap> C := HammingCode( 3 );; a := WeightDistribution( C );
    [ 1, 0, 0, 7, 7, 0, 0, 1 ]
    gap> n := WordLength( C );; q := Size( Field( C ) );;
    gap> k := Dimension( C );;
    gap> q^( -k ) * KrawtchoukMat( n, q ) * a;
    [ 1, 0, 0, 0, 7, 0, 0, 0 ]
    gap> WeightDistribution( DualCode( C ) );
    [ 1, 0, 0, 0, 7, 0, 0, 0 ] 

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