Selectin elements that satisfy a certain condition with modular arithmetic

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.

Answer 1

[Voss]

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