Formation of quinary Gordon–Mills–Welch sequences for discrete information transmission systems

An algorithm for the formation of the quinary Gordon–Mills–Welch sequences (GMWS) with a period of $N=624$ over a finite field with a double extension is proposed. The algorithm is based on a matrix representation of a basic M-sequence (MS) with a primitive verification polynomial and a similar period. The transition to non-binary sequences is determined by the increased requirements for the information content of the information transfer processes, the speed of transmission through communication channels and the structural secrecy of the transmitted messages. It is demonstrated that the verification polynomial of the GMWS can be represented as a product of fourth-degree polynomials-factors that are indivisible over a simple field GF(5). The relations between roots of the polynomial of the basic MS and roots of the polynomials-factors are obtained. The entire list of GMWS with a period $N=624$ can be formed on the basis of the obtained ratios. It is demonstrated that for each of the 48 primitive fourth-degree polynomials that are test polynomials for basis MS, three GMWS with equivalent linear complexity (ELC) of 12, 24, 40 can be formed. The total number of quinary GMWS with period of $N=624$ is equal to 144. A device for the formation of a GMWS as a set of shift registers with linear feedbacks is presented. The mod5 multipliers and summators in registers are arranged in accordance with the coefficients of indivisible polynomials-factors. The symbols from the registers come to the adder mod5, on the output of which the GMWS is formed. Depending on the required ELC, the GMWS forming device consists of three, six or ten registers. The initial state of cells of the shift registers is determined by the decimation of the symbols of the basic MS at the indexes of decimation, equal to the minimum of the exponents of the roots of polynomials polynomials-factors. A feature of determining the initial States of the devices for the formation of quinary GMWS with respect to binary sequences is the presence of cyclic shifts of the summed sequences by a multiple of $N/(p-1)$. The obtained results allow to synthesize the devices for the formation of a complete list of 144 quinary GMWS with a period of $N=624$ and different ELC. The results can also be used to construct other classes of pseudo-random sequences that allow analytical representation in finite fields.