Solve diophantine systems?

Questions
,
1

I have a Diophantine system Ax=b and Cx<=d where A and C are m*n matrices that I want to solve subject to the constraint that x consists of integers that are between 0 and k (inclusive). Looking through the documentation centre I have found the following tutorial regarding Diophantine polynomial systems (Diophantine Polynomial Systems—Wolfram Documentation) which uses the reduce function to solve the system. This works perfectly other than the fact that each equation and inequality must be entered separately, and I need to perform this for large systems. Ideally I would like to be able to call the function with the matrices directly, along the lines of

Reduce[A*x == b && C*x <= d && 0 <= x <= k, x, Integers]

to return all possible solutions for x. Is this possible with reduce, or any other function? Keeping in mind that I need all solutions to be returned. As a side note, I know that there are solutions to my system. Thanks.

2

In[3]:= mat = {{3, 4, 18, 24}, {27, 16, 28, 24}}; r = {30,
30}; vars = {a, b, c, d};

In[4]:= Reduce[Thread[mat.vars == r], vars, Integers]

Out[4]= (C[1] | C[2]) \[Element] Integers && a == 2 + 8 C[1] && 
 b == 6 + 9 C[1] + 15 C[2] && c == -12 - 30 C[1] - 18 C[2] && 
 d == 9 + 20 C[1] + 11 C[2]
3

In[5]:= Reduce[mat.vars == r, vars, Integers]

Out[5]= (C[1] | C[2]) \[Element] Integers && a == 2 + 8 C[1] && 
 b == 6 + 9 C[1] + 15 C[2] && c == -12 - 30 C[1] - 18 C[2] && 
 d == 9 + 20 C[1] + 11 C[2]
4

Check the Documentation Center page for LinearProgramming. It indicates how to specify Integers as the domain of interest.