Tring GAPS on an eight variables problem, generated solver is inaccurate, and shows C0 ill-condtional with large contion number

[Chastj]

I used GAPS to generate a solver for an eight-variable problem. The problem has a quotient ring basis (degree = 80), and the template size is [1145, 1345]. While the generated solver can solve the problem, the results are not 100% accurate. I set the parameters to be solved as [0.5; 0.5; 100], but the closest solution returned is [0.49998; 0.49998; 100.0002]. As the amount of input data increases, the solution deviates further from the true values.

Here is a part of my code:
[C0, C1] = setup_elimination_template(H);
b_ind = [1102, 2248, 3394, 4540, 5686, 6832, 7978, 9124, 10270, 11416, 12562, 13708, 14854, 16000, 17146, 18292, 19438, 20584, 21730, 22876, 24022, 25168, 26314, 27460, 28606, 29752, 30898, 32044, 33190, 34336, 35482, 36628, 37774, 38920, 40066, 41212, 42358, 43504, 44650, 45796, 46942, 48088, 49234, 50380];
b = zeros(1145, 44);
b(b_ind) = -1;
alpha = C0' \ b;

At this line alpha = C0' \ b;, MATLAB indicates that the matrix C0 is ill-conditioned with a large condition number. This makes the solution numerically unstable.

Question:
What can I do to improve the accuracy of the solution, given the ill-conditioned nature of C0? Is there a way to mitigate the problem and make the solver more reliable as the data grows?

[prclibo]

Hi I havn't touched this code for long time and might forget the details. Probably two reasons related:

1. [1145, 1345] is very large and it is easy to have numerical issue with such large matrix.
2. Check if you generate your Zp samples correctly s.t. all constraints are obeyed.