We provide an exposition of three Lemmas which relate
general properties of distributions
with the exclusive-or of certain bit locations.

The first XOR-Lemma, commonly attributed to U.V. Vazirani,
relates the statistical distance of a distribution from uniform
to the maximum bias of the xor of certain bit positions.
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A binary code \text{Enc}:\{0,1\}^k \rightarrow \{0,1\}^n is (\frac{1}{2}-\varepsilon,L)-list decodable if for every w \in \{0,1\}^n, there exists a set \text{List}(w) of size at most L, containing all messages m \in \{0,1\}^k such that the relative Hamming distance between \text{Enc}(m) and w is at most \frac{1}{2}-\varepsilon.
A q-query local list-decoder is ... more >>>