For a string a01n its k-fold direct sum encoding is a function fa that takes as input sets S[n] ofsize k and outputs fa(S)=iSai .In this paper we are interested in the Direct Sum Testing Problem,where we are given a function f, and our goal isto test whether f is close to a direct sum encoding,i.e., whether there exists some a01n such thatf(S)=iSai for most inputs S.By identifying the subsets of [n] with vectorsin 01n in the natural way, this problem can be thought of aslinearity testing of functions whose domain is restricted to thek'th layer of the hypercube.

We first consider the case k=n2 , and analyze for it avariant of the natural 3-query linearity test introducedby Blum, Luby, and Rubinfeld (STOC '90). Our analysisproceeds via a new proof for linearity testing onthe hypercube, which extends also to our setting.

We then reduce the Direct Sum Testing Problem for general kn2 to thecase k=n2 , and use a recent result on Direct Product Testingof Dinur and Steurer in order to analyze the test