Lecturer : Madhu Sudan Scribe : Anastasios Sidiropoulos 1 Overview

n Proof Assume that m > 2n. We define vectors c̃1, . . . , c̃m ≈ R, such that for each i, with � {−1, 1} ci, and for each i, with 1 � j � n, the jth coordinate of ̃ 1 � i � n, ̃ ci is −1, iff the jth bit of ci is 1. Note that if �(ci, cj ) � n/2, then this implies →c̃i, c̃j � 0. Intuitively, this means that if two codewords ⊂ ci, and cj have large Hamming distance, then the angle between the corresponding vectors ̃ cj , ci, and ̃ should be large. Pick a random unit vector x � R . We have that w.h.p., →x, c̃i = 0, for all i, with 1 � i � m. ⊂ √ Moreover, since there are m codewords, either x, or −x has strictly positive inner product with at least m/2 of the c̃is. We can assume w.l.o.g., that this holds for x. Since m > 2n, it follows that there exist n+ 1 vectors having strictly positive inner product with x. W.l.o.g., assume that these are the vectors c̃1, . . . , c̃n+1. Observe that a set of n+ 1 vectors in an n-dimensional space, cannot be linear independent. Thus, we can assume that there exist �1, . . . , �n+1 � R, with �i > 0, for each i, with 1 � i � j, and �i � 0, for each i, with j < i � n+ 1, such that