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 π of the set of the n coordinate positions and a word over(d, ̄) such that C^′ = π (over(d, ̄) + C). Hessler defined C and C^′ to be linearly equivalent if there exists a non-singular linear map φ such that C^′ = φ (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 φ and a word over(d, ̄) such that C^′ = φ (over(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.