We prove that on the Euclidean sphere S-3 there exist a unique up to isometry 120-point spherical 11-design and a maximal (4, 120, cos(pi /5)) -code. Both these are nothing but copies of a famous regular polytope in four dimensions-the 600-cell.

We describe three new methods for obtaining superimposed codes in Euclidean spaces. With help of them we construct codes with parameters improving upon known constructions. We also prove that the spherical simplex code is not optimal as superimposed code at least for dimensions greater than 9. © 2003 Elsevier Science B.V. All rights reserved.

We give a sufficient condition for a spherical code to be a (n, d, m, T)-spherical superimposed code. We apply this condition to obtain new spherical superimposed codes with help of known spherical codes and codes in the real projective spaces PRn-1. © 2001.

While multiple-access communication dates back to systems invented in the 1870's to transmit simultaneous data through a single wire, the foundation of the discipline now known as 'multiuser information theory' was laid in 1961, when Claude E. Shannon published his paper on two-way channels. Since then, multiuser information theory has been an extremely active research area, and has seen a large number of fundamental contributions, covering, besides the two-way channel studied in, multiple access, interference, broadcast, and wiretap channels. However, several key canonical problems have defied many efforts. This book brings together leading experts working in the fields of information theory, coding theory, multiple user communications, discrete mathematics, etc., who survey recent and general results on multiple-access channels (rate regions, rate splitting, etc.), and give an overview of the problems of current CDMA solutions (fading channels, multi-user detection, multiple-antenna systems, iterative joint decoding, OFDMA, etc.). This publication consist of three parts. The first part includes chapters devoted to the information-theoretical aspects of multiple-access communication. In the second part, multiple-access techniques are discussed and the third part of this volume covers coding techniques

In this paper we investigate reduced complexitydecoders for LDPC codes over M-PSK modulation. Capacitycalculations show only minor losses if only phase informationis considered in the decoder. We device a decoder based onphase angle summation, which unfortunately does not work forcodes over Galois fields (but shows promising results for codesover integer rings). For codes over Galois Fields we constructtable-based decoders which gives performance close to BeliefPropagation decoders at substantially reduced complexity.

A sequence of q-ary cyclic codes is analyzed. For each finite field GF (q), q=4, there is a code with parameters [n, k, d, q] = [q(q-1)+1, q(q-1)-6, 6,q]. All these codes are n-, k-, and d-optimal, with only one exception. The dual codes are considered. Their true minimum distances are calculated in the range 4=q=32.

While multiple-access communication dates back to systems invented in the 1870's to transmit simultaneous data through a single wire, the foundation of the discipline now known as 'multiuser information theory' was laid in 1961, when Claude E. Shannon published his paper on two-way channels. Since then, multiuser information theory has been an extremely active research area, and has seen a large number of fundamental contributions, covering, besides the two-way channel studied in, multiple access, interference, broadcast, and wiretap channels. However, several key canonical problems have defied many efforts. This book brings together leading experts working in the fields of information theory, coding theory, multiple user communications, discrete mathematics, etc., who survey recent and general results on multiple-access channels (rate regions, rate splitting, etc.), and give an overview of the problems of current CDMA solutions (fading channels, multi-user detection, multiple-antenna systems, iterative joint decoding, OFDMA, etc.). This publication consist of three parts. The first part includes chapters devoted to the information-theoretical aspects of multiple-access communication. In the second part, multiple-access techniques are discussed and the third part of this volume covers coding techniques

List decoding of block codes is an alternative approach to the decoding problem with appealing qualities. The fairly recent development of efficient algorithms for list decoding of Reed-Solomon codes spur new fuel to the study of this decoding strategy. In this paper we give a weight-based characterization of the set of correctable error patterns under list-of-2 decoding of (tau, 2)-list-decodable linear codes with known weight distribution. We apply our characterization of the set of correctable error patterns to a few codes in a family of low-rate list-of-2 decodable Reed-Solomon codes. We study the increase in error-correction performance obtained in a symmetric AWGN channel by using list-of-2 decoding instead of traditional decoding for these codes. Some simulation results for list-of-2 decoding on QAM channels using the Guruswami-Sudan algorithm for decoding of Reed-Solomon codes are also presented.

We study the problem of list decoding with focus on the case when we have a list size limited to two. Under this restriction we derive general lower bounds on the maximum possible size of a list-of-2-decodable code. We study the set of correctable error patterns in an attempt to obtain a characterization. For a special family of Reed-Solomon codes - which we identify and name 'class-I codes' - we give a weight-based characterization of the correctable error patterns under list-of-2 decoding. As a tool in this analysis we use the theoretical framework of Sudan's algorithm. The characterization is used in an exact calculation of the probability of transmission error in the symmetric channel when list-of-2 decoding is used. The results from the analysis and complementary simulations for QAM-systems show that a list-of-2 decoding gain of nearly 1 dB can be achieved.

Further we study Sudan's algorithm for list decoding of Reed-Solomon codes for the special case of the class-I codes. For these codes algorithms are suggested for both the first and second step of Sudan's algorithm. Hardware solutions for both steps based on the derived algorithms are presented.

A cyclically permutable code is a set of codewords having the property that no codeword is a cyclic shift of another codeword. We study the problem of constructing cyclically permutable codes of large size and low correlation.

Cyclically permutable codes are used in code-division multiple-access systems realized by e.g. direct-sequence modulation or frequency-hopping. Advantages of code-division multiple-access to conventional access methods, such as timedivision and frequency-division, include greater flexibility, better robustness and that no synchronization among the transmitters is needed.

Among our main results are an efficient method of selecting cyclically distinct codewords from linear cyclic codes, a new family of sequences for direct-sequence modulation, several constructions of hopping-sequences for multiple-access coupled with a decoding algorithm for asynchronous communication.

We have also constructed new binary constant-weight codes of high minimum distance.

We define a matrix representation of elements in finite extension fields and determine under what circumstances we can find regularities in these matrices. Especially, we determine all cases for which the matrices are Hankel. This is of special interest for the design of algorithms for inversion in finite extension fields.

An important operation in certain cryptosystems is exponentiation in large galois fields, which can be achieved by multiplication and squaring. This report deals first with the theory of Galois fields, and then with a number of architectures for multiplication, squaring and exponentiation in Galois fields. Various bases for Galois fields yield architectures with different degrees of regularity. Since the polynomial basis yields the best known architectures as far as performance, complexity and structure are concerned, we discuss solely this basis. We also present a VLSI implementation of a universal exponentiator over GF(2^m), m<337, which we use to study expected performance, complexity, chip size and power consumption as a function of m for an arbitrary exponentiator. An important conclusion is that the performance is not limited by the critical paath, but by the distribution of signals needed all over the chip. We also arrived at the conclusion that it is perfectly possible today to expand the exponentiator to m<1001.