Abstract: A
very important class of error
correcting codes (ECC) are
the group
codes.
These group codes are produced by the wide-sense
homomorphic encoders.
These encoders are automata whose input set U(uncoded),
and state set Sare
groups such that the next state mapping and encoding mapping are homomorphisms
over the extension of the input group Uby
state group S.
When the extension is abelian then the group code also is abelian, in other case
the group code is non-abelian. Abelian group codes are produced when and and
the extension is a direct product, where is
the binary group, also known as the module-2 addition group. These codes are
more known as convolutional
codes and
there is a lot of literature about them. In this work we study non-abelian group
codes, that is, non-abelian extensions of by
any finite group S.
Then we will show that these extensions when encoded by the wide-sense
homomorphic encoder yield bad group codes. More precisely, we will show that
when Sis
abelian the code has free distance limitations, and when Sis
non-abelian the code is non-controllable.
Keywords and phrases: error correcting codes, extension of groups, homomorphic encoder, group codes, controllability, convolutional codes over groups.
