Zhe-Xian Wan

A linear algebra approach to minimal convolutional encoders

The authors review the work of G.D. Forney, Jr., on the algebraic structure of convolutional encoders upon which some new results regarding minimal convolutional encoders rest. An example is given of a basic convolutional encoding matrix whose number of abstract states is minimal over all equivalent encoding matrices. However, this encoding matrix can be realized with a minimal number of memory elements neither in controller canonical form nor in observer canonical form. Thus, this encoding matrix is not minimal according to Forneys definition of a minimal encoder. To resolve this difficulty, the following three minimality criteria are introduced: minimal-basic encoding matrix, minimal encoding matrix, and minimal encoder. It is shown that all minimal-basic encoding matrices are minimal and that there exist minimal encoding matrices that are not minimal-basic. Several equivalent conditions are given for an encoding matrix to be minimal. It is proven that the constraint lengths of two equivalent minimal-basic encoding matrices are equal one by one up to a rearrangement. All results are proven using only elementary linear algebra. >

Some structural properties of convolutional codes over rings

Convolutional codes over rings have been motivated by phase-modulated signals. Some structural properties of the generator matrices of such codes are presented. Successively stronger notions of the invertibility of generator matrices are studied, and a new condition for a convolutional code over a ring to be systematic is given and shown to be equivalent to a condition given by Massey and Mittelholzer (1990). It is shown that a generator matrix that can be decomposed into a direct sum is basic, minimal, and noncatastrophic if and only if all generator matrices for the constituent codes are basic, minimal, and noncatastrophic, respectively. It is also shown that if a systematic generator matrix can be decomposed into a direct sum, then all generator matrices of the constituent codes are systematic, but that the converse does not hold. Some results on convolutional codes over Z(p/sup e/) are obtained.

On canonical encoding matrices and the generalized constraint lengths of convolutional codes

This paper is devoted to rational convolutional encoding matrices. Canonical encoding matrices are introduced and it is shown that every canonical encoding matrix is minimal but that there exist minimal encoding matrices that are not canonical. Some equivalent conditions for an encoding matrix to be canonical are given. The generalized constraint lengths are defined. They are invariants of equivalent canonical encoding matrices.

Group codes defined on posets

The present paper is devoted to the study of group codes defined on posets. Their natural state realizations are introduced and shown to be minimal. The trellis diagrams of the natural state realizations are also studied.<<ETX>>

A generalization of the predictable degree property to rational convolutional encoding matrices

The predictable degree property was introduced by Forney (1970) for polynomial convolutional encoding matrices. In this paper two generalizations to rational convolutional encoding matrices are discussed.<<ETX>>

On the Hensel lift of a polynomial

Denote by R the Galois ring of characteristic p/sup e/ and cardinality p/sup em/, where p is a prime and e and m are positive integers. Let g(x) be a monic polynomial over F/sub p/m. A polynomial f(x) over R is defined to be a Hensel lift of g(x) in R[x] if f~(x)=g(x), and there is a positive integer n not divisible by p such that f(x) divides x/sup n/-1 in R[x]. It is proved that g(x) has a unique Hensel lift in R[x] if and only if g(x) has no multiple roots and x/spl chi/g(x). An algorithm to compute the Hensel lift is also given.

Some results on convolutional codes over rings

Convolutional codes over rings were motivated from phase-modulated signals. Some structural properties of generator matrices of convolutional codes over rings have been studied. A condition for a convolutional code over a ring to be systematic is given and shown to be equivalent to the condition given by Massey and Mittelholzer (1989). Furthermore, the conditions of generator matrices over Z(p/sup e/) being catastrophic, basic, and minimal are considered, and the predictable degree property of polynomial generator matrices is considered.

Some remarks on convolutional codes

The author gives definitions of convolutional codes, discusses the dual code of a convolutional code and the minimality criterion of encoding matrices.