TABLE OF CONTENTS While certain linear codes may not have a generator matrix in standard form, after a suitable permutation of the coordinates of the codewords and possibly multiplying certain coordinates with some nonzero scalars, one can always arrive at a new code which has a generator matrix in standard form.
Two ( n, M)-codes over F q are equivalent if one can be obtained from the other by a combination of operations of the following types:
permutation of the n digits of the codewords;
multiplication of the symbols appearing in a fixed position by a nonzero scalar.
(i) Let q = 2 and n = 4. Choosing to rearrange the bits in the order 2, 4, 1, 3, we see that the code
is equivalent to the code
(ii) Let q = 3 and n = 3. Consider the ternary code
Permuting the first and second positions, followed by multiplying the third position by 2, we obtain the equivalent code
Any linear code C is equivalent to a linear code C ? with a generator matrix in standard form.
Proof. If G is a generator matrix for C, place G in RREF. Rearrange the columns of the RREF so that the leading columns come first and form an identity matrix. The result is a matrix, G ?, in standard form which is a generator matrix for a code C ?
