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11
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vol 62 / November, 2019
Article

DOI 10.17586/0021-3454-2018-61-3-240-248

UDC 519.725

FORMATION SEQUENCES OF GORDON — MILLS — WELCH WITH PERIOD N = 511

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


V. M. Kuznetsova
A. F. Mozhaisky Military Space Academy, Department of Technologies and Means for Automation of Processing and Analysis of Space Systems Information; Student


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Abstract. Based on developed algorithm for generating Gordon — Mills — Welch sequences, a full list of testing polynomials for GMW-sequences with the period N = 511 is obtained. Binary GMW-sequences are formed on the basis of base M-sequence over finite fields with double expansion GF[(2m)n] and can be represented as a matrix of dimension [JxL]=[(2m–1)x(2m+1)]. A qualitative specifics of sequences with the period N = 511 consists in the fact that they are formed over a finite field GF[(23)3] and are presented in the form of a matrix of dimension [JxL]=[7x73], but not in the form of quasi-quadratic matrix of dimension [(2m–1)x(2m+1)]. Equivalent linear complexity of these sequences corresponds to the degree of the testing polynomial which can be represented as a product of three irreducible polynomials of the ninth degree. GMW-sequences with period N = 511 are formed using M-sequences of the same period. There are 48 primitive polynomials of the ninth degree in the field GF(29), the full list also contains 48 test polynomials for GMW-sequences. 
Keywords: sequences of composite period, finite fields, indivisible and primitive polynomials, equivalent linear complexity

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