Algorithm for Determining the Initial Shifts in the Formation of Non-binary Gordon–Mills–Welch Sequences

Based on a modification of the algorithm for determining the shift vector when forming non-binary Gordon–Mills–Welch sequences (GMWS) in finite fields GF(p^S), where S = mn, an algorithm for determining the shift vector C(m, n, r) = (c₁,c₂, …, c_M) for M summable sequences is developed. This algorithm is based on solving a system of linear equations for the p-adic representation of the parameter r with respect to the shift values of the sequences. It is shown that for a known value of the decimation index vector A(m, n, r) = (d₁,d₂, …, d_M), the number of possible values of the shift vector is L₁ = ((p – 2)!). For given vectors A(m, n, r), the shift vectors C(m, n, r) are given for the formation of GF(5⁴), GF(7⁴), GF(11⁴) GMWS.

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