The paper presents a comparison of communication systems using different signal constellation sizes and Reed-Solomon or Bose-Chaudhuri-Hocquengem codes with different rates so that the overall required bandwidth is the same for each system. In these comparisons, the channel symbol size is smaller than the code symbol size, so that a code symbol contains parts of multiple channel symbols. Thus, the normal assumption of independent code symbols does not apply. Instead, consideration must be taken to obtain the best arrangement of channel symbols in each code symbol. Analytical expressions are developed to compare the bit error probability performance of comparable systems, based on individual codewords using errors-only decoding and errors and erasures decoding with transmission over a Rayleigh fading channel.

An expression for the shift necessary to obtain a canonical M-sequence is obtained in terms of the coefficients of the primitive polynomial that generates this m-sequence. This shift for obtaining a canonical M-sequence is also directly related to the shift for obtaining the location of the span "100...0" in the M-sequence. In addition, a recursion is developed for obtaining the dual basis elements of the basis expressing GF(q/sup m/) over GF(q). Canonical M-sequences and dual bases are important in the design of sequences for spread-spectrum multiple-access systems as well as in the design of general decoder structures using bit-serial encoders.

Procedures for obtaining primitive polynomials and m-sequences over GF(q/sup m/) in terms of primitive polynomials and m-sequences over GF(q) are presented. Using a degree mn primitive polynomial g(x) in GF(g, x), an m-sequence over GF(q/sup m/) can be expressed as a vector m-sequence whose component m-sequences are shifted versions of the m-sequence generated by g(x). The degree-n primitive polynomials in GF(q/sup m/,x) with root alpha q/sup i/, that are factors of g(x) with root alpha when g(x) is viewed in GF(q/sup m/,x), are then developed from the m-sequence over GF(q/sup m/). Expressions for the shifts and corresponding primitive polynomial for the m-sequence generated by the uth decimation of the m-sequence generated by the polynomials are also given. Expressions for the shifts and corresponding primitive polynomial factors of g(x) for different bases expressing GF(q/sup m/) over GF(q) are presented.>

The correlation values and the distribution of these correlation values for the small set of nonbinary Kasami sequences over GF(p) (p prime) are presented. The correlation results are an extension of the binary results and have p+2 correlation levels. This nonbinary Kasami set is asymptotically optimum with respect to its correlation properties. These sequences are obtained, as in the binary case, from a large primitive polynomial of degree n=2 m and a small primitive polynomial of degree m that yields a sequence length of p/sup n/-1 and maximum nontrivial correlation value of 1+p/sup m/. Nonbinary Kasami sequences are directly implemented using shift registers and are applicable for code division multiple access systems.>

Normally, frequency hopping sequences for spread-spectrum communication systems are obtained by selecting groups of elements of binary m sequences. An alternative to using groups of binary m-sequence elements is developed. It involves obtaining nonbinary m sequences with the number of desired hopping frequencies equal to the number of symbols in the finite field which the nonbinary m sequence is over. The grouping of elements of binary m sequences does not necessarily have the m sequences. In addition, the autocorrelation function of the nonbinary m sequence has a maximal period, whereas the frequency hopping sequences obtained from the grouping of elements of binary m sequences may not have a maximal period.>

It is shown that m-sequences over GF(q/sup m/) of length q/sup nm/-1 corresponding to primitive polynomials in GF(q/sup m/,x) of degree n can be generated from known m-sequences over GF(q) of length q/sup nm/-1 obtained from primitive polynomials in GF(q,x) of degree mn. A procedure for generating the m-sequences over GF(q/sup 2/) from m-sequences over GF(q) was given which enables the generation of m-sequences over GF(p/sup 2n/). In addition it was shown that all of the primitive polynomials in GF(q,/sup m/,x) can be obtained from a complete set of the primitive polynomials in GF(q,x).>

This paper presents expressions for the cutoff rate R0diversity transmission over the Rayleigh and Rician fading channels withM-ary orthogonal signaling. These expressions include the unquantized R0forD-fold diversity, which upper bounds the channel performance, and hard decision and four- and eight-level soft decision quantized R0expressions. Tradeoffs betweenDand the number of quantization levels for equivalent performance are presented for the unquantized and quantized channels. These tradeoffs illustrate the reduction in signal energy, system bandwidth, and system complexity by increasing the number of quantization levels, thereby allowing a reduction inD.