automorphism group (linear code)

Let 𝔽q be the finite field with q elements. The group β„³n,q of nΓ—n monomial matrices with entries in 𝔽q acts on the set β„­n,q of linear codesMathworldPlanetmath over 𝔽q of block length n via the monomial transform: let M=(Mi⁒j)i,j=1nβˆˆβ„³n,q and Cβˆˆβ„­n,q and set


This definition looks quite complicated, but since M is , it really just means that CM is the linear code obtained from C by permuting its coordinatesPlanetmathPlanetmath and then multiplying each coordinate with some nonzero element from 𝔽q.

Two linear codes lying in the same orbit with respect to this action are said to be equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath. The isotropy subgroupPlanetmathPlanetmath of C is its automorphism groupMathworldPlanetmath, denoted by Aut⁑(C). The elements of Aut⁑(C) are the automorphismsPlanetmathPlanetmathPlanetmathPlanetmath of C.

Sometimes one is only interested in the action of the permutation matricesMathworldPlanetmath on β„­n,q. The permutation matrices form a subgroupMathworldPlanetmathPlanetmath of β„³n,q and the resulting subgroup of the automorphism group Aut⁑(C) of a linear code Cβˆˆβ„­n,q is called the permutation groupMathworldPlanetmath. In the case of binary codes, this doesn’t make any differencePlanetmathPlanetmath, since the finite field 𝔽2 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