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 |