  Let C be a linear code with length n and minimum distance d. The stopping
redundancy of C is defined as the minimum number of rows in a parity-check
matrix for C such that the smallest stopping sets in the corresponding Tanner
graph have size d. We derive new upper bounds on the stopping redundancy of
linear codes in general, and of maximum distance separable (MDS) codes
specifically, and show how they improve upon previously known results. For MDS
codes, the new bounds are found by upper bounding the stopping redundancy by a
combinatorial quantity closely related to Turan numbers. (The Turan number,
T(v,k,t), is the smallest number of t-subsets of a v-set, such that every
k-subset of the v-set contains at least one of the t-subsets.) We further show
that the stopping redundancy of MDS codes is T(n,d-1,d-2)(1+O(n^{-1})) for
fixed d, and is at most T(n,d-1,d-2)(3+O(n^{-1})) for fixed code dimension
k=n-d+1. For d=3,4, we prove that the stopping redundancy of MDS codes is equal
to T(n,d-1,d-2), for which exact formulas are known. For d=5, we show that the
stopping redundancy of MDS codes is either T(n,4,3) or T(n,4,3)+1.
