StandardFormCode( C )
StandardFormCode returns C after putting it in standard form. If C
is a non-linear code, this means the elements are organized using
lexicographical order. This means they form a legal GAP Set.
If C is a linear code, the generator matrix and parity check matrix are
put in standard form. The generator matrix then has an identity matrix in
its left part, the parity check matrix has an identity matrix in its
right part. Although GUAVA always puts both matrices in a standard
form using BaseMat, this never alters the code. StandardFormCode even
applies column permutations if unavoidable, and thereby changes the
code. The column permutations are recorded in the construction history of
the new code (see Display). C and the new code are of course
equivalent.
If C is a cyclic code, its generator matrix cannot be put in the usual
upper triangular form, because then it would be inconsistent with the
generator polynomial. The reason is that generating the elements from the
generator matrix would result in a different order than generating the
elements from the generator polynomial. This is an unwanted effect, and
therefore StandardFormCode just returns a copy of C for cyclic codes.
gap> G := GeneratorMatCode( Z(2) * [ [0,1,1,0], [0,1,0,1], [0,0,1,1] ],
> "random form code", GF(2) );
a linear [4,2,1..2]1..2 random form code over GF(2)
gap> Codeword( GeneratorMat( G ) );
[ [ 0 1 0 1 ], [ 0 0 1 1 ] ]
gap> Codeword( GeneratorMat( StandardFormCode( G ) ) );
[ [ 1 0 0 1 ], [ 0 1 0 1 ] ]
GAP 3.4.4