The rate vs. distance problem is a long-standing open problem in coding theory. Recent papers have suggested a new way to tackle this problem by appealing to a new hierarchy of linear programs. If one can find good dual solutions to these LPs, this would result in improved upper bounds for the rate vs. distance problem of linear codes. In this work, we develop the first dual feasible solutions to the LPs in this hierarchy. These match the best-known bound for a wide range of parameters. Our hope is that this is a first step towards better solutions, and improved upper bounds for the rate vs. distance problem of linear codes.