General Topology-I

This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

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Infinite Abelian Groups

The structures of cyclic and negacyclic codes of length n and their duals over a finite chain ring R are established when n is not divisible by the characteristic of the residue field R~. Some cases where n is divisible by the characteristic of the residue field R~ are also considered. Namely, the structure of negacyclic codes of length 2/sup t/ over /spl Zopf//sub 2//sup m/ and that of their duals are derived.

Cyclic and negacyclic codes over finite chain rings

An encoder uses an input file of data and a key to produce an output symbol. An output symbol with key I is generated by determining a weight, W(I), for the output symbol to be generated, selecting W(I) of the input symbols associated with the output symbol according to a function of I, and generating the output symbol's value B(I) from a predetermined value function F(I) of the selected W(I) input symbols. An encoder can be called repeatedly to generate multiple output symbols. The output symbols are generally independent of each other, and an unbounded number (subject to the resolution of I) can be generated, if needed. A decoder receives some or all of the output symbols generated. The number of output symbols needed to decode an input file is equal to, or slightly greater than, the number of input symbols comprising the file, assuming that input symbols and output symbols represent the same number of bits of data.

Information additive code generator and decoder for communication systems

Cyclic Codes over the Integers Modulopm

A method of building systematically a multi-dimensional (n, D, L) circular trellis coded modulation (CTCM) encoder with properties of optimal energy efficiency, strong tail biting and maximum minimum distance (dmin) of trellis paths is disclosed. In addition, a communication system for use in a power limited channel application is disclosed comprising a circular trellis coded modulation (CTCM) encoder with permuted state structure for encoding based on a circular trellis path associated with a sequence of digital information bits and a set of simplexes identified for the path from a multi-dimensional signal constellation; a transmitter coupled to said CTCM encoder for transmitting said sequence of channel symbols over said channel; a receiver for receiving a transmission from said transmitter including said sequence of channel symbols and any noise induced therein; and a CTCM decoder coupled to said receiver for decoding the received transmission without knowledge of the starting state of the circular trellis path of the CTCM encoder to recover the sequence of information bits. Apparatus and methods of encoding and decoding are also disclosed utilizing a combination of circular trellis-coded modulation with permuted state structure and simplex signal constellation techniques for use in the digital communication system.

Apparatus and method of ctcm encoding and decoding for a digital communication system

Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes

Characterizing rings in terms of the extent of the injectivity and projectivity of their modules

In this paper we focus on two generalizations of the notion of
cyclicity of codes: polycyclic codes and sequential codes. We
establish a duality between these two generalizations and also show
connections between them and other well-known generalizations of
cyclicity such as the notions of negacyclicity and constacyclicity.
In particular, it is shown that a code $C$ is sequential and
polycyclic if and only if $C$ and its dual C⊥ are both
sequential if and only if $C$ and its dual C⊥ are both
polycyclic. Furthermore, any one of these equivalent statements
characterizes the family of constacyclic codes.

Sergio R. López-Permouth

Papers

Cyclic Codes over the Integers Modulopm

Apparatus and method of ctcm encoding and decoding for a digital communication system

Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes

Characterizing rings in terms of the extent of the injectivity and projectivity of their modules

Dual generalizations of the concept of cyclicity of codes