The standard algebraic decoding algorithm of cyclic codes [n, k, d] up to the BCH bound δ = 2t + 1 is very efficient and practical for relatively small n while it becomes unpractical for large n as its computational complexity is O(nt). Aim of this paper is to show how to make this algebraic decoding computationally more efficient: in the case of binary codes, for example, the complexity of the syndrome computation drops from O(nt) to O(t√n), while the average complexity of the error location drops from O(nt) to max{O(t√n), O(t log2(t)log log(t)log(n))}.

A lower bound to the number of AND gates used in parallel multipliers for GF(2/sup m/), under the condition that time complexity be minimum, is determined. In particular, the exact minimum number of AND gates for primitive normal bases and optimal normal bases of Type II multipliers is evaluated. This result indirectly suggests that space complexity is essentially a quadratic function of m when time complexity is kept minimum.

We propose new classes of linear codes over integer rings of quadratic extensions of Q, the field of rational numbers. The codes are considered with respect to a Mannheim metric, which is a Manhattan metric module a two-dimensional (2-D) grid, in particular, codes over Gaussian integers and Eisenstein-Jacobi integers are extensively studied. Decoding algorithms are proposed for these codes when up to two coordinates of a transmitted code vector are affected by errors of arbitrary Mannheim weight. Moreover, we show that the proposed codes are maximum-distance separable (MDS), with respect to the Hamming distance. The practical interest in such Mannheim-metric codes is their use in coded modulation schemes based on quadrature amplitude modulation (QAM)-type constellations, for which neither the Hamming nor the Lee metric is appropriate.

Maximization of the energy transfer ratio for time-limited signals over linear channels is considered. The case of linear channels with rational transfer function is addressed and a general procedure for the analytic solution of the maximization problem is outlined. The maximum energy transfer ratio over a fixed time interval is evaluated in some cases of interest. Two performance metrics-the energy transfer ratio and the energy intersymbol interference (ISI) ratio-are evaluated for optimal signals and compared to those of commonly used rectangular and sinusoidal pulses in order to determine the achievable gain.

We present a decoding procedure for Reed-Solomon (RS) and BCH codes defined over an integer residue ring pgZ/sub q/, where q is a power of a prime p. The proposed decoding procedure, as for RS and BCH codes over fields, consists of four major steps: (1) calculation of the syndromes; (2) calculation of the "elementary symmetric functions," by a modified Berlekamp-Massey (1968, 1969) algorithm for commutative rings; (3) calculation of the error location numbers; and (4) calculation of the error magnitudes. The proposed decoding procedure also applies to the synthesis of a shortest linear-feedback shift register (LFSR), capable of generating a prescribed finite sequence of elements lying in a commutative ring with identity.

It is known that group block codes over nonabelian groups of order less than 32 are asymptotically bad. In this correspondence we extend that result, showing that group block codes over nonabelian groups are asymptotically bad, no matter what the order of the group under consideration is.

The filter minimizes the mean-square error between the sampled input to the decision device and the corresponding transmitted symbols. It is shown that the structure of this optimal receiving filter corresponds to a bank of matched filters, each followed by a linear transversal filter. Applications of this theory cover nonlinear channels with memory, linearly or nonlinearly modulated signals, and disturbances other than Gaussian noise (e.g. interference from other data sources).>

A class of multidimensional signals, based on generalized group alphabets, is introduced and its basic properties are derived. The combination of generalized group alphabets and coding is also examined: two coding schemes are considered-G. Ungerboeck's scheme (1982) for combination with convolutional codes, and V.V. Ginzburg's (1984) scheme for combination with block codes. It is concluded that the performance of these schemes makes them attractive for transmission over band-limited digital channels.>

It is shown that the algebraic method for decoding three-error-correcting BCH codes is also applicable to complete decoding of the(23,12,7)Golay code.

Complete decoding of systematic linear block codes by standard arrays with coset leaders not necessarily of minimal weight gives, in several interesting cases on the binary symmetric channel with error probabilityp, a smaller symbol error probability than decoding using the coset leaders of minimum weight. The proposed choice of the coset leaders allows for the classes of uniformly packed and nearly perfect binary codes to achieve a lower information bit error probability, at least forpsufficiently near zero.

The performance of a TFM scheme applied to a satellite downlink with an on-board regenerative repeater is evaluated by using a time domain simulation software package TOPSIM III. The TFM bit error rate in the presence of nonlinearities and adjacent channel interference is evaluated by means of semianalytic methods. The capabilities of convolutional codes for reducing the overall channel bit error rate are also investigated. Design guidelines are finally outlined for advanced satellite communication systems in which the joint use of nonconventional modulation schemes (like TFM) and of convolutional codes results in improved performance with respect to both bandwidth exploitation and system flexibility.

TOPSIM III is a flexible, easy to use, stand-alone Fortran package for the digital simulation of communication systems in the time domain. TOPSIM III provides a simple high-level language for the description of the system. Programs can be written following the same structure used in drawing the system block diagram. The paper discusses the time domain simulation approach used in the design of the package and describes its capabilities. The contents of the subprogram libraries are presented, describing some of the building blocks that can be used in the simulation as well as the performance evaluation facilities available in the package. Some simple examples illustrate the use of TOPSIM III as a tool for analysis and design of digital communication systems.

A mixed analytic and simulation method is described for the study of a DCPSK up-link used in a regenerative satellite transmission system. A great variety of impairments is considered and their effects on error probability are evaluated, thus emphasizing the performance evaluation in a "real situation." A case study is worked out in detail to stress the design-oriented approach.

A lower bound is given on the size of a linear code of given length and distance which improves the Varshamov bound in certain cases, though not asymptotically. Specific triples(n,k,d)are given with values larger than those guaranteed by the Varshamov bound.

The theory of cyclic-group codes for the Gaussian channel is reviewed, and the following new results are shown: i) the initial vector problem for these codes can be formulated as a linear programming problem, and ii) the codeword generation and the

Permutation modulation codes are a class of group codes for the Gaussian channel whose codewords are obtained by permuting the components of a given initial vectorXin Euclideann-dimensional space. In this paper, the problem of choosing the components ofXin such a way that the minimum distance between any two codewords is maximized is solved. In particular, a closed-form expression is obtained for this minimum distance and is used to investigate the asymptotic behavior of some selected codes.

This paper presents a new method to study arithmetic combinatorial circuits. Using polynomial associated to the input-output sequences and to the systems it is possible to solve the problem of minimization of the number of the component blocks. Particularly, the important case of the multiple outputs elementary units can be treated. Applications of the introduced procedures to multiplier and to fast networks for performing convolution are presented.

A number of theorems are given, which give insight into the algebraic structure of group codes for the Gaussian channel, as defined by Slepian. The problem of the existence of group codes for a given pair of defining parameters (i.e., number of codewordsM, and dimension of the signal spacen) is partially solved.