# Application: The automorphism group of a code

Let be a code. It is not necessary at this point for to be linear. Let

This is called the automorphism group of

Proposition 5.6.1   is a group under ordinary matrix multiplication.

proof: The identity matrix is in . Associativity is an inherited property from . The set is closed under multiplication since the defining property is preserved. The existence of inverses is inherited from .

Example 5.6.2   Let be the Hamming code over having generator matrix

The elements of are

The automorphism group is a group of order generated by

These are the permutation matrices associated with the coordinate permutations , , , and .

Let denote the group of of monomial matrices with entries in . These are the matrices which have exactly one non-zero element in each row and column.

Let be two linear codes. We say that is an isometry between and if

• is an isomorphism of vector spaces,
• for all ,

In this case, we say are isometric (with respect to the Hamming metric).

Exercise 5.6.3   Show that two codes are isometric if and only if they are equivalent.

Exercise 5.6.4   Find the automorphism group of the binary repetition code of length ,

david joyner 2008-04-20