Formation of Septenary Gordon–Mills–Welch Sequences for Digital Information Transmission Systems

Abstract

In this paper, we present septenary Gordon–Mills–Welch sequences (GMWSs) with a period of N = 2400 that are formed in finite GF[(7^m)]ⁿ = GF(7^S) fields. Checking polynomials h_GMWS(x) are obtained in the form of a product of both primitive and irreducible polynomials h_сi(x) with a degree of S = 4. The formation of GMWSs by summing sequences with polynomials h_сi(x) is shown to require knowledge of the symbols of the M-sequence (MS) with polynomial h_MS(x) and decimation indices determined by the exponents of the roots of polynomials h_сi(x). It is determined that, compared to the binary case, septenary summable sequences can have an initial shift that is a multiple of N/(p – 1) = 400. It is shown that for each of the 160 primitive polynomials of degree S = 4 in the GF(7⁴) field, it is possible to form seven GMWSs with equivalent linear complexity l_s from 12 to 84. Compared to septenary MSs, the maximal gain in structural secrecy is 21 times.