Let C be an [n, k] linear code and G a permutation group of degree n. Then G acts on C in the following way: for a codeword v of C and a permutation x of G, the image of v under x is obtained from v by permuting the coordinate positions of v according to x. The following functions allow one to investigate such actions. For more information on permutation group actions and orbits, see the chapter on permutation groups.

v ^ x : ModTupFldElt, GrpPermElt -> ModTupFldElt

Given a codeword v belonging to the [n, k] code C and an element x belonging to a permutation group of degree n, construct the vector w obtained from v by permuting the coordinate positions of v according to the permutation x.

v ^ G : ModTupFldElt, GrpPerm -> GSet{ ModTupFldElt }

Given a codeword v belonging to the [n, k] code C and a permutation group G of degree n, construct the vector orbit Y of v under the action of G. Y is then a G-set for the group G.

C ^ x : Code, GrpPermElt -> Code

Given an [n, k] code C and an element x belonging to a permutation group G of degree n, construct the code consisting of all the images of the codewords of C under the action of x.

C ^ G : Code, GrpPerm -> GSet{ Code }

Given an [n, k] code C and a permutation group G of degree n, construct the orbit Y of C under the action of G. Y is then a G-set for the group G.

S ^ x : { ModTupFldElt }, GrpPermElt -> { ModTupFldElt }

S ^ x : [ModTupFldElt], GrpPermElt -> [ModTupFldElt]

Given a set or sequence S of codewords belonging to the [n, k] code C and an element x belonging to a permutation group of degree n, construct the set or sequence of the vectors obtained by permuting the coordinate positions of v, for each v in S, according to the permutation x.

S ^ x : { Code }, GrpPermElt -> { Code }

S ^ x : [Code], GrpPermElt -> [Code]

Given a set or sequence S of codes of length n and an element x belonging to a permutation group of degree n, construct the set or sequence of the codes consisting of all the images of the codewords of C under the action of x.

The automorphism group A of the binary [n, k] linear code C. A is the largest permutation group of degree n such that C ^ x equals C for each x in A.

Given an [n, k] code C and a permutation group G of degree n, find the subcode of C which consists of those vectors of C which are fixed by the elements of G.

Given [n, k] binary codes C and D, this function returns true iff C is isomorphic to D. If C is isomorphic to D, a permutation x of degree n is also returned such that C ^ x is equal to D.

> C := ReedMullerCode(2, 6);
> aut := AutomorphismGroup(C);
> print FactoredOrder(aut);
[ <2, 21>, [3, 4>, <5, 1>, <7, 2>, <31, 1> ]
> print CompositionFactors(aut);
    G
    |  A(5, 2)                = L(6, 2)
    *
    |  Cyclic(2)
    *
    |  Cyclic(2)
    *
    |  Cyclic(2)
    *
    |  Cyclic(2)
    *
    |  Cyclic(2)
    *
    |  Cyclic(2)
    1