Construction

Construction

Methods for constructing representatives of classes of linear (n,k)-codes obviously do not reach as far as the enumerative methods, but the use of computers allows to get a complete overview of linear (n,k)-codes over GF(q) for quite a number of parameter triples (n,k,q). We have seen before that the isometry classes can be described as orbits of GL_k(q) on sets of mappings into the projective space. A very interesting and helpful constructive method for discrete structures which can be defined as orbits of finite groups on finite sets is based on the following fact. If G is a transitive permutation group on X, then the orbits of a subgroup U≤ G on X can be bijectively mapped onto double cosets as follows: For any x∈ X the mapping

U\\X→ U\G/G_x, U(gx)↦ UgG_x^._.

is a bijection. In the case of the (n,k)-codes we can use the fact that the general linear group GL_n(q) is transitive on the set S(n,k,q) of subspaces of dimension k in GF(q)ⁿ, so that the isometry classes of linear (n,k)-codes turn out to be in one-one-correspondence with the set of double cosets

GF(q)^*≀S_n\GL_n(q)/GL_n(q)_C₀,^._.

where C₀ is any linear (n,k)-code. A computer program due to Weinrich ([14]) allows to evaluate complete sets of representatives, and it was recently improved by using, besides of double cosets the combinatorial method of orderly generation. The work in this field of constructive theory is still in rapid progress, so that we cannot tell yet how far we can reach.

harald.fripertinger "at" uni-graz.at, May 10, 2016

Construction