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Two new approaches to the optimal synthesis of difference patterns are proposed that can deal in an effective fashion with arbitrary sidelobe bounds. The first approach, which amounts to solving a convex programming problem, can be applied to completely arbitrary (fixed geometry) arrays and it is capable of taking into account additional constraints. The second approach, which amounts to solving a simpler linear programming problem, can be applied to uniformly spaced linear or planar arrays, and allows results about the uniqueness of the solution to be inferred. Numerical examples show the flexibility and effectiveness of the proposed procedures.