Uniqueness conditions for the solution of an inhomogeneous system of nonlinear random equations over the field GF(3)

Abstract

An inhomogeneous consistent system of second-order nonlinear random equations over a field consisting of three elements is considered. The necessary and sufficient condition is found under which the system has a unique solution belonging to a given set of n-dimensional vectors as n ® ¥ .

Estimates for the probability that a system of random equations is solvable in a given set of vectors over the field ${GF}(3)$

“…n , q = 1, 2. Let i c 1 c 2 , c 1 , c 2 ∈ GF(3), denote the number of pairs (c 1 , c 2 ) among n possible pairs x j (1) , x j (2) , 1 ≤ j ≤ n. Let i = i 01 + i 02 and l = i 10 + i 20 . By M n , we denote the maximal subset of the set V n (with respect to the inclusion) with the property that arbitrary vectorsx (1) ,x (2) ∈ V n belong to M n if and only if (2) i + l ≥ 1.…”

“…Remark 1.1. The existence of solutions belonging to a given set of vectors for a system of equations is considered in [2] for different right hand sides. In [1], special solutions of a homogeneous system of linear random equations over a finite field are studied and the study of special solutions for the random linear inclusion is considered.…”

“…where the summation 1 is considered with respect to all pairs of vectors x (1) ,x (2) such thatx (q) ∈ M n , q = 1, 2, andx (1) =x (2) . With the help of equality (17) we find that…”

“…Let γ (1) , γ (2) , and γ (3) be the number of elements of the sets E (1) , E (2) , and E (12) , respectively. Put Γ (1) = γ (1) + γ (2) , Γ (2) = γ (2) + γ (3) , (19) Γ (3) = γ (1) + γ (3) , Γ (4) = γ (1) + γ (2) + γ (3) . (20)…”

“…For arbitrary vectorsx (1) ,x (2) ∈ M n , denote by t the total number of pairs (c 1 , c 2 ), (c 3 , 0), (0, c 4 ) among n possible pairs x j (1) , x j (2) , 1 ≤ j ≤ n, with the property that c 1 c 2 c 3 c 4 = 0 for all c 1 , c 2 , c 3 , c 4 ∈ GF(3). Then t = i + l + h and the total number of pairs of vectors x (1) ,x (2) for which equality (21) holds is found from the following equation:…”