GMW Sequences Research Articles

Generalized GMW Quadriphase Sequences Satisfying the Welch Bound with Equality

This paper gives new families of quadriphase sequences obtained from large linear complexity sequences over Z4 by making use of generalized permutation monomials over Galois rings. The construction of these sequences can be seen as a generalization of the binary GMW sequence construction and hence they are referred to as GGMW sequences over Z4. The GGMW families satisfy the Welch bound on inner products with equality and it is shown that the root mean square of all the crosscorrelations and out-of-phase autocorrelations (θrms), is approximately equal to the quantity \(\); L being the period of the sequences. However, θmax, the maximum magnitude of periodic crosscorrelation and out-of-phase autocorrelation, deviates from the optimal value of \(\). Computer results suggest that the number of crosscorrelation values which deviate from the optimal value of \(\) is small. The weight structure of these sequences is the same as those of m-sequences over Z4. The linear complexity (LC) of the sequences is computed using a generalized Blahuts theorem on the LC of sequences over Z4.

Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings

We construct new sequences over finite rings having optimal Hamming correlation properties. These sequences are useful in frequency hopping multiple-access (FHMA) spread-spectrum communication systems. Our constructions can be classified into linear and nonlinear categories, both giving optimal Hamming correlations according to Lempel-Greenberger (1974) bound. The nonlinear sequences have large linear complexity and can be seen as a generalized version of GMW sequences over fields.

On GMW Designs and a Conjecture of Assmus and Key

We show that a family of cyclic Hadamard designs defined from regular ovals is a sub-family of a class of difference set designs due to B. Gordon, W. H. Mills and L. R. Welch [Can. J. Math.14(1962), 614–625]. Using a result of R. A. Scholtz and L. R. Welch [IEEE Trans. Inform. Theory30, No. 3 (1984), 548–553] on the linear span of GMW sequences, we give a short proof of a conjecture of Assmus and Key on the 2-rank of this family of designs.

Generalization of GMW sequences and No sequences

GMW sequences and families of No sequences are generalized. Generalized GMW sequences have ideal autocorrelation and balance properties and generalized No sequences also have optimal correlation properties in terms of Welch's lower bound. The linear spans of the generalized GMW and No sequences appear to be large although there is not at present a closed-form expression for the linear span. A count of the numbers of cyclically distinct generalized GMW sequences and generalized No sequences that can be constructed is provided. Generalized GMW sequences have also been found in the literature by Klapper et al. (1993) under the name "cascaded GMW sequences".

Q-ary cascaded GMW sequences

Cascaded GMW sequences over GF(q) of arbitrary characteristic are investigated. Their periods are derived. As compared with the existing results on such sequences, substantially extended and improved theorems concerning their linear spans and autocorrelation functions are obtained. A criterion and an algorithm for constructing cascaded GMW sequences that have much larger linear spans than GMW sequences with the same periods are given. These provide a new class of complex-valued sequences which have larger linear spans and two-valued autocorrelation functions.

Theory and applications of q-ary interleaved sequences

A new class of q-ary sequences, called interleaved sequence, is introduced. Their periods, shift equivalence, linear spans, and autocorrelation functions are derived. The interleaved sequences include a large number of popular sequences, such as multiplexed sequences, clock-controlled sequences, Kasami (1966) sequences, GMW sequences, geometric sequences, and No (1989) sequences. A special class of the interleaved sequences is constructed by mapping GF(q/sup m/) sequences into GF(q) sequences in terms of different bases of GF(q/sup m/) over GF(q). As an application of the theory of interleaved sequences, some new families of binary pseudo-random sequences are constructed, which have large linear spans, optimal periodic cross/autocorrelation functions, balance, and the rapidly hopped properties. A complete comparison of the new family of sequences with the Gold sequence family, the Kasami (small and large set) sequence families, the Bent-function sequence family, and the No sequence family is discussed. This shows that the new sequence families have important advantages for use in spread-spectrum multiple-access communication systems. >

Cross-correlation of p-ary GMW sequences

The cross-correlation function of p-ary (p prime) GMW sequences is investigated. The well-known m-sequences are incorporated in these sequences. GMW sequences also have the two-level autocorrelation function, but their linear span may be much larger than those of m sequences. The calculation of the cross-correlation function of GMW sequences is reduced to the cross-correlation of the m-sequences because many results on this kind of sequence are known. In the literature, only the cross-correlation function of an m sequence with a GMW sequence has been known up to now. >

Autocorrelation Function of a Family of Complex Sequences

In 1984, R. A. Scholtz and L. R. Welch constructed a series of new binary se-quences which are called GMW sequences by using the trace function as follows: LetM, J be positive integers, J|M, α be a primitive element of the finite field GF(2~M), rbe a positive integer, 1≤r≤2~J, and(r, 2~J-1)=1. Then the sequence b=b(0)b(1)…b(n)…is called GMW sequence over GF(2). Here b(n)=tr_1~J(tr_J~Mα~n)~r, for n=0, 1, 2….

Cross-correlations of linearly and quadratically related geometric sequences and GMW sequences

In this paper we study the cross-correlation function values of geometric sequences obtained from q -ary m-sequences whose underlying m-sequences are linearly or quadratically related. These values are determined by counting the points of intersection of pairs of hyperplanes or of hyperplanes and quadric hypersurfaces of a finite geometry. The results are applied to obtain the cross-correlations of m-sequences and GMW sequences with different primitive polynomials.

Cascaded GMW sequences

Pseudorandom binary sequences with high linear complexity and low correlation function values are sought in many applications of modern communication systems. A new family of pseudorandom binary sequences, cascaded GMW sequences, is constructed. These sequences are shown to share many desirable correlation properties with the GMW sequences of B. Gordon, W.A. Mills, and L.R. Welch (1962)-for example, high-shifted autocorrelation values and, in many cases, three-valued cross-correlation values with m-sequences. It is shown, moreover, that in many cases the linear complexities of cascaded GMW sequences are far greater than those of GMW sequences.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

GMW sequences (Corresp.)

The difference set design of Gordon, Mills, and Welch (GMW) is adapted for use as a pseudorandom number generator. Statistical properties of the generated binary sequences, including periodic correlation, linear span, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -tuple statistics, are derived. One mechanization of a GMW sequence generator is suggested, and the number of sequences that can be generated with a fixed number of shift-register stages and read-only memory (ROM) size is evaluated.