John J. Komo

Nonbinary Kasami sequences over GF(p)

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. >

Maximal length sequences for frequency hopping

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. >

Relationships between m-sequences over GF(q) and GF(q/sup m/)

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). >

Primitive polynomials and m-sequences over GF(q/sup m/)

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. >

Implementation consideration of code division multiple access sequences

The linear span of a sequence is defined as the minimum number of stages of a linear feedback shift register required to generate the given sequence. Thus, the linear span is a measure of the complexity of the sequence structure. A large complexity or linear span is desired for security of the sequence. The binary No sequence requires nonlinear connections in addition to the linear feedback shift registers. Thus, No sequences have a large complexity and are suited for high security systems. The implementation of Gold, Kasami, and No sequences which are directly applicable for CDMA are presented and compared for complexity. An alternative to the nonlinear connection of the linear feedback shift register for the No sequences is also presented. This alternative is obtained as a direct linear feedback shift register of length equal to the linear span of the No sequence set.<<ETX>>

Decoding binary BCH codes

BCH codes are powerful error-correcting codes. Algorithms used for decoding must be able to find the error locations, and for nonbinary codes, the error magnitudes. One of the most efficient algorithms for decoding BCH codes is Berlekamps algorithm. To find the error locations the algorithm must solve a set of t equations in t unknowns. This paper explores, for binary BCH codes, a new method that uses half of the unknowns to determine the other unknowns, thus solving t/2 equations in t/2 unknowns. However, because of the reduced number of equations, the algorithm only iterates half the number of times. The performance of the new algorithm is shown to be superior to both Berlekamps algorithm and a simplified algorithm in terms of execution times, which includes the field multiplications and additions and required memory.

Soft decision decoding of BCH codes using error magnitudes

An efficient method for soft decision decoding of binary BCH codes using error magnitude calculations its presented. The 2t syndromes for a t-error correcting BCH code are used to solve 2t modified error magnitude type equations to obtain maximum likelihood decoding.

Fast error magnitude evaluations for Reed-Solomon codes

A fast algorithm for the evaluation of error magnitudes for Reed-Solomon codes is obtained here in terms of the error locations and syndromes. This fast algorithm is compared to the Forney algorithm (1965) in terms of required additions and multiplications and implementation speed.

Improved performance of reduced M-ary orthogonal signaling using Reed-Solomon codes

This paper presents a comparison of communication systems using different signal constellation sizes and Reed-Solomon code sizes 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 more than one channel symbol. 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 derived to compare the bit error probability performance of comparable systems based on individual codewords.

Sequential decoding of Reed-Solomon codes in an incremental redundancy system

A method of soft-decision decoding of Reed-Solomon codes is presented. This method is an iterative procedure where erasures only decoding is performed followed by a sequential procedure to determine the maximum likelihood estimate of the transmitted word. Results are given that compare this new method of soft decision decoding to more traditional soft decision decoding algorithms. The use of the sequential decoding method in an incremental redundancy system is discussed.