automorphism group (linear code)
Let be the finite field with elements. The group
of monomial matrices with entries in
acts on the set of linear codes over of
block length via the monomial transform: let and and set
This definition looks quite complicated, but since is , it
really just means that is the linear code obtained from by
permuting its coordinates and then multiplying each coordinate with
some nonzero element from .
Two linear codes lying in the same orbit with respect to this action
are said to be equivalent. The isotropy subgroup
of
is its automorphism group
, denoted by . The elements
of are the automorphisms
of .
Sometimes one is only interested in the action of the permutation
matrices on . The permutation matrices form a subgroup
of and the resulting subgroup of the automorphism group
of a linear code is called the
permutation group
. In the case of binary codes, this doesnβt
make any difference
, since the finite field contains only
one nonzero element.
Title | automorphism group (linear code) |
Canonical name | AutomorphismGrouplinearCode |
Date of creation | 2013-03-22 15:18:40 |
Last modified on | 2013-03-22 15:18:40 |
Owner | GrafZahl (9234) |
Last modified by | GrafZahl (9234) |
Numerical id | 5 |
Author | GrafZahl (9234) |
Entry type | Definition |
Classification | msc 94B05 |
Synonym | automorphism group |
Related topic | LinearCode |
Defines | monomial transform |
Defines | equivalent |
Defines | equivalent code |
Defines | automorphism |
Defines | permutation group |