# Solving linear inequalities over rings

The concrete problem: for any given $N\ge 1$ I have a system of $2^N-1$ linear inequalities over $\mathbb{Z}_6^N$ which looks like this: for every nonempty $S\subseteq[N]$ there is some $b_S\in\mathbb{Z}_6$ and the inequality $\sum_{i\in S}x_i\ne b_S$. I want to find a solution to all the inequalities at once, of course.

Is there an efficient way to find solution to such a system? To count the number of solutions? To check whether a solution exists? Also, what about more general cases (any number of inequalities, any type of ring, no restriction on the coefficients of the variables in the inequalities)?

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What order are you using in the inequalities? – user23211 May 13 '12 at 9:54
I'm afraid I don't understand the question. – Gadi A May 13 '12 at 10:02
In an inequality you use the symbol $\leq$ or a similar one. What does the symbol mean in this case? – user23211 May 13 '12 at 10:07
Maybe "inequality" is the wrong word (but I don't know what's the right word). There's no order here, only expressions of the form $x\ne y$ (not equal). – Gadi A May 13 '12 at 11:41
Oh, I see. Sorry, I misunderstood your question. – user23211 May 13 '12 at 11:48