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vol 62 / September, 2019
Article

DOI 10.17586/0021-3454-2017-60-4-318-330

UDC 519.725

GORDON—MILLS—WELCH SEQUENCES OF PERIOD N = 1023

V. G. Starodubtsev
Multiservice Nets and Telecommunications, Ltd., St. Petersburg; Head of Department


A. M. Popov
A. F. Mozhaisky Military State Academy, Department of Technologies and Means of Complex Processing and Transmission of Information in ACS; Student


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Abstract. А full list of testing polynomials for Gordon—Mills—Welch sequences of period N = 1023 are derived on the basis of a developed algorithm of forming data sequences. The principle dissimilarity from sequences with a smaller period is the possibility to create several GMW-sequences with different equivalent linear complexity (ELC) determined as the degree of testing polynomial hGMW(x) for each basic Msequence (MS) with the primitive testing polynomial hMS(x). This is a consequence of existence of six primitive polynomials in the finite field of GF(25), in contrast to the fields of GF(23) and GF(24) with two primitive polynomials in each. For each of the six MS of period N=31 acting as a characteristic sequence for MS matrix representation of period N=1023, it is possible to use the other five different MS to form five different GMW-sequences. It is shown that on the base of every MS with the period N=1023 it is possible to build five GMW-sequences. One of the GMW-sequences has a testing polynomial of the eightieth degree, two sequences — polynomials of fortieth degree, and two sequences — polynomials of the twentieth degree.
Keywords: sequence of composite period, finite fields, indivisible and primitive polynomials, equivalent linear complexity

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