This paper deals with the problem of constructing superregular matrices that lead to MDP convolutional codes. These matrices are a type of lower block triangular Toeplitz matrices with the property that all the square submatrices that can possibly be nonsingular due to the lower block triangular structure are nonsingular. We present a new class of matrices that are superregular over a sufficiently large finite field F. Such construction works for any given choice of characteristic of the field F and code parameters (n, k, δ) such that (n − k)|δ. Finally, we discuss the size of F needed so that the proposed matrices are superregular.

No abstract

The main results of this paper are twofold: the first one is a matrix theoretical result. We say that a matriz is superregular if all of its minors that are not trivially zero are nonzero. Given a a × b, a ≥ b, superregular matrix over a field, we show that if all of its rows are nonzero then any linear combination of its columns, with nonzero coefficients, has at least a − b + 1 nonzero entries. Secondly, we make use of this result to construct convolutional codes that attain the maximum possible distance for some fixed parameters of the code, namely, the rate and the Forney indices. These results answer some open questions on distances and constructions of convolutional codes posted in the literature [6,9].

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 = Zpr by introducing a socalled "p-encoder". We show how to manipulate a polynomial encoding of a noncatastrophic convolutional code over Zpr 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 Zpr 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 Zpr admits a noncatastrophic p-encoder.

In this paper, polynomial matrix fraction descriptions (MFD’s) are used as a tool for investigating the structure of a (linear) convolutional code and the family of its encoders and syndrome formers. As static feedback and precompensation allow to obtain all minimal encoders (in particular, polynomial encoders and decoupled encoders) of a given code, a sim- ple parametrization of their MFD’s is provided. All minimal syndrome formers, by a duality argument, are obtained by resorting to output injection and postcompensation. Decoupled encoders are finally discussed as well as the possibility of representing a convolutional code as a direct sum of smaller ones