On the classification of perfect codes: Extended side class structures

Olof Heden, Martin Hessler, Thomas Westerbäck
2010 Discrete Mathematics  
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 wordd such that C = π (d + C ). Hessler defined C and C to be linearly equivalent if there exists a nonsingular 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 wordd such that Heden and Hessler, associated with each linear
more » ... 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.
doi:10.1016/j.disc.2009.07.023 fatcat:767jnbrrxvfilffemlgoimezmy