On the classification of perfect codes: Extended side class structures

The two 1-error correcting perfect binary codes, CC and C′C′ are said to be equivalent if there exists a permutation ππ of the set of the nn coordinate positions and a word d̄ such that C′=π(d̄+C). Hessler defined CC and C′C′ to be linearly equivalent if there exists a non-singular linear map φφ such that C′=φ(C)C′=φ(C). Two perfect codes CC and C′C′ of length nn will be defined to be extended equivalent   if there exists a non-singular linear map φφ and a word d̄ such that C′=φ(d̄+C).Heden and Hessler, associated with each linear equivalence class an invariant LCLC 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 LCLC.This fact will be used to give, in some particular cases, a complete enumeration of all extended equivalence classes of perfect codes.