Subspaces of GF(q)^w and Convolutional Codes

The present paper is a self-contained treatment of subspaces of the space GF(q)ω of all semi-infinite strings over GF(q). Some necessary and sufficient conditions which characterize those subspaces of GF(q)ω are derived which are convolutional codes, and the classes of subspaces defined by one or more of them are investigated. Moreover structural parameters of convolutional codes such as block length, rate, delay, and constraint length are considered as parameters of subspaces rather than parameters of an encoding device. As a conclusion it is obtained that for error-control purposes none of the investigated superclasses of the class of convolutional codes is better suited than the class of convolutional codes itself.

[1]  R. Lindner,et al.  Algebraische Codierungstheorie : Theorie d. sequentiellen Codierungen , 1977 .

[2]  J. Dieudonne.  Foundations of Modern Analysis , 1969 .

[3]  G. David Forney,et al.  Convolutional Codes II. Maximum-Likelihood Decoding , 1974, Inf. Control..

[4]  James L. Massey,et al.  Inverses of Linear Sequential Circuits , 1968, IEEE Transactions on Computers.

[5]  Werner Kuich,et al.  On the Entropy of Context-Free Languages , 1970, Inf. Control..

[6]  Daniel J. Costello.  A construction technique for random-error-correcting convolutional codes , 1969, IEEE Trans. Inf. Theory.

[7]  Ludwig Staiger,et al.  Finite-State omega-Languages , 1983, J. Comput. Syst. Sci..

[8]  K. Sh. Zigangirov.  Procedures of Sequential Decoding , 1975 .

[9]  Klaus W. Wagner,et al.  On omega-Regular Sets , 1979, Inf. Control..

[10]  Andrew J. Viterbi,et al.  Convolutional Codes and Their Performance in Communication Systems , 1971 .

[11]  Philippe Piret.  Generalized Permutations in Convolutional Codes , 1978, Inf. Control..

[12]  G. David Forney,et al.  Convolutional codes I: Algebraic structure , 1970, IEEE Trans. Inf. Theory.

[13]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[14]  Boris A. Trakhtenbrot,et al.  Finite automata : behavior and synthesis , 1973 .

[15]  John E. Savage,et al.  Sequential decoding — the computation problem , 1966 .

[16]  Ludwig Staiger,et al.  Recursive omega-Languages , 1977, FCT.

[17]  Matti Linna.  On omega-Sets Associated with Context-Free Languages , 1976, Inf. Control..

[18]  R. Blahut.  Theory and practice of error control codes , 1983 .

[19]  Lin-nan Lee,et al.  Short unit-memory byte-oriented binary convolutional codes having maximal free distance (Corresp.) , 1976, IEEE Trans. Inf. Theory.