Canonical kernel representations for behaviors over finite abelian groups

In this paper the problem of constructing canonical kernel representations for systems whose alphabet set is an Abelian group is addressed. In standard linear system theory, there exist two important canonical kernel representations: minimal and row reduced kernel representations. In this paper we propose an extension of such canonical kernel representations to the group case. This topic has applications in the theory of convolutional codes over groups where kernel representations correspond to syndrome formers which are essential in decoding.