Selectin elements that satisfy a certain condition with modular arithmetic

2024 7 9
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Suppose I have the finite field F_3={0,1,2}. Suppose I have the following equations:
1=x_1+x_4+x_5
0=x_2+x_5+x_6
0=x_3+x_4+x_6
where each x_i is a member of F_3. How Can I obtain the set of all possible combinations for the solutions of the above equations?
Thanks for any help in advance.

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Voss
Voss 2024-7-9
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Ran in:
N = 6;
m = 3;
M = dec2base(0:m^N-1,m)-'0';
idx = mod(M(:,1)+M(:,4)+M(:,5),m) == 1 ...
& mod(M(:,2)+M(:,5)+M(:,6),m) == 0 ...
& mod(M(:,3)+M(:,4)+M(:,6),m) == 0;
sol = M(idx,:);
disp(sol)
0 0 0 2 2 1 0 0 1 0 1 2 0 0 2 1 0 0 0 1 0 1 0 2 0 1 1 2 2 0 0 1 2 0 1 1 0 2 0 0 1 0 0 2 1 1 0 1 0 2 2 2 2 2 1 0 0 0 0 0 1 0 1 1 2 1 1 0 2 2 1 2 1 1 0 2 1 1 1 1 1 0 0 2 1 1 2 1 2 0 1 2 0 1 2 2 1 2 1 2 1 0 1 2 2 0 0 1 2 0 0 1 1 2 2 0 1 2 0 0 2 0 2 0 2 1 2 1 0 0 2 0 2 1 1 1 1 1 2 1 2 2 0 2 2 2 0 2 0 1 2 2 1 0 2 2 2 2 2 1 1 0

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