On minimality of convolutional ring encoders

Convolutional codes are considered with code sequences modeled as semi-infinite Laurent series. It is well known that a convolutional code C over a finite group G has a minimal trellis representation that can be derived from code sequences. It is also well known that, for the case that G is a finite field, any polynomial encoder of C can be algebraically manipulated to yield a minimal polynomial encoder whose controller canonical realization is a minimal trellis. In this paper we seek to extend this result to the finite ring case G = ℤ_{p^r} by introducing a so-called "p-encoder". We show how to manipulate a polynomial encoding scheme of a noncatastrophic convolutional code over ℤ_{p^r} to produce a particular type of p-encoder ("minimal p-encoder") whose controller canonical realization is a minimal trellis with nonlinear features. The minimum number of trellis states is then expressed as p^γ, where γ is the sum of the row degrees of the minimal p-encoder. In particular, we show that any convolutional code over ℤ_{p^r} admits a delay-free p-encoder which implies the novel result that delay-freeness is not a property of the code but of the encoder, just as in the field case. We conjecture that a similar result holds with respect to catastrophicity, i.e., any catastrophic convolutional code over ℤ_{p^r} admits a noncatastrophic p-encoder. © 2009 IEEE

The Dynamics of Group Codes: Dual Abelian Group Codes and Systems

Fundamental results concerning the dynamics of abelian group codes (behaviors) and their duals are developed. Duals of sequence spaces over locally compact abelian groups may be defined via Pontryagin duality; dual group codes are orthogonal subgroups of dual sequence spaces. The dual of a complete code or system is finite, and the dual of a Laurent code or system is (anti-)Laurent. If C and C^\perp are dual codes, then the state spaces of C act as the character groups of the state spaces of C^\perp. The controllability properties of C are the observability properties of C^\perp. In particular, C is (strongly) controllable if and only if C^\perp is (strongly) observable, and the controller memory of C is the observer memory of C^\perp. The controller granules of C act as the character groups of the observer granules of C^\perp. Examples of minimal observer-form encoder and syndrome-former constructions are given. Finally, every observer granule of C is an "end-around" controller granule of C.Comment: 30 pages, 11 figures. To appear in IEEE Trans. Inform. Theory, 200