## System of linear equalities and non-equalities

Hello,

I would like to know if the theory of linear equalities and non-equalities exists in this sense :

You have $n-$bits strings $P_1, ..., P_{2k+1}, \lambda_1, ..., \lambda_k$ and we consider the addition in $(\mathbb{Z}/2\mathbb{Z})^n$.

What is the minimum number of solution (or at least a good approximation) of the following system :

$$P_{2i}\oplus P_{2i+1}=\lambda_i, \forall i\leq k$$

$$\forall i,j, \qquad i\neq j\implies P_i\neq P_j$$

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 I don't understand your question. Are the lambdas given and you want to find the number of different solutions for the P? – Zsbán Ambrus Feb 15 2012 at 10:08