On the classification of perfect codes: Extended side class structures

The two 1-error correcting perfect binary codes, C and C^' are said to be equivalent if there exists a permutation @p of the set of the n coordinate positions and a word d@? such that C^'=@p(d@?+C). Hessler defined C and C^' to be linearly equivalent if there exists a non-singular linear map @f such that C^'=@f(C). Two perfect codes C and C^' of length n will be defined to be extended equivalent if there exists a non-singular linear map @f and a word d@? such that C^'=@f(d@?+C). Heden and Hessler, associated with each linear equivalence class an invariant L"C and this invariant was shown to be a subspace of the kernel of some perfect code. It is shown here that, in the case of extended equivalence, the corresponding invariant will be the extension of the code L"C. This fact will be used to give, in some particular cases, a complete enumeration of all extended equivalence classes of perfect codes.