Full rank perfect codes and -kernels

A perfect 1-error correcting binary code C , perfect code for short, of length n = 2 m − 1 has full rank if the linear span 〈 C 〉 of the words of C has dimension n as a vector space over the finite field F 2 . There are just a few general constructions of full rank perfect codes, that are not given by recursive methods using perfect codes of length shorter than n . In this study we construct full rank perfect codes, the so-called normal α -codes, by first finding the superdual of the perfect code. perdual of a perfect code consists of two matrices G and T in which simplex codes play an important role as subspaces of the row spaces of the matrices G and T . The main idea in our construction is the use of α -words. These words have the property that they can be added to certain rows of generator matrices of simplex codes such that the result will be (other) sets of generator matrices for simplex codes. rnel of these normal α -codes will also be considered. It will be proved that any subspace, of the kernel of a normal α -code, that satisfies a certain property will be the kernel of another perfect code, of the same length n . In this way, we will be able to relate some of the full rank perfect codes of length n to other full rank perfect codes of the same length n .