Full rank perfect codes and αα-kernels

A perfect 1-error correcting binary code CC, perfect code for short, of length n=2m−1n=2m−1 has full rank if the linear span 〈C〉〈C〉 of the words of CC has dimension nn as a vector space over the finite field F2F2. 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 nn. In this study we construct full rank perfect codes, the so-called normal  αα-codes, by first finding the superdual of the perfect code.The superdual of a perfect code consists of two matrices GG and TT in which simplex codes play an important role as subspaces of the row spaces of the matrices GG and TT. The main idea in our construction is the use of αα-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.The kernel of these normal αα-codes will also be considered. It will be proved that any subspace, of the kernel of a normal αα-code, that satisfies a certain property will be the kernel of another perfect code, of the same length nn. In this way, we will be able to relate some of the full rank perfect codes of length nn to other full rank perfect codes of the same length nn.