On minimality of convolutional ring encoders
Abstract
Convolutional codes are considered with code sequences modelled as semi-infinite Laurent series. It is wellknown 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 wellknown 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 socalled "p-encoder". We show how to manipulate a polynomial encoding 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^gamma, where gamma 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.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2008
- DOI:
- 10.48550/arXiv.0801.3703
- arXiv:
- arXiv:0801.3703
- Bibcode:
- 2008arXiv0801.3703K
- Keywords:
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- Computer Science - Information Theory
- E-Print:
- 13 pages in v1, submitted