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Mathematical Problems in Engineering
Volume 2011, Article ID 783516, 12 pages
http://dx.doi.org/10.1155/2011/783516
Research Article

On the Uncontrollability of Nonabelian Group Codes with Uncoded Group 𝑝

Technological Center of Alegrete, Federal University of Pampa (UNIPAMPA) 97546-550 Alegrete, RS, Brazil

Received 5 November 2010; Revised 24 August 2011; Accepted 2 September 2011

Academic Editor: Gerhard-Wilhelm Weber

Copyright © 2011 Jorge Pedraza Arpasi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Error-correcting encoding is a mathematical manipulation of the information against transmission errors over noisy communications channels. One class of error-correcting codes is the so-called group codes. Presently, there are many good binary group codes which are abelian. A group code is a family of bi-infinite sequences produced by a finite state machine (FSM) homomorphic encoder defined on the extension of two finite groups. As a set of sequences, a group code is a dynamical system and it is known that well-behaved dynamical systems must be necessarily controllable. Thus, a good group code must be controllable. In this paper, we work with group codes defined over nonabelian groups. This necessity on the encoder is because it has been shown that the capacity of an additive white Gaussian noise (AWGN) channel using abelian group codes is upper bounded by the capacity of the same channel using phase shift keying (PSK) modulation eventually with different energies per symbol. We will show that when the trellis section group is nonabelian and the input group of the encoder is a cyclic group with, 𝑝 elements, 𝑝 prime, then the group code produced by the encoder is noncontrollable.