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 d ̄ such that C ′ = π ( 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 d ̄ such that C ′ = φ ( d ̄ + C ) . 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 . act will be used to give, in some particular cases, a complete enumeration of all extended equivalence classes of perfect codes.