let me parse more carefully the # of errors we require the decoding algorithm to correct: it is delta_0^2 2 Lambda --------- ( 1 - ---------- ) * N 8 delta_0 in 2b you should prove the statement in the last sentence, namely that |C| <= beta(0). A current version that is linked from the homepage now incorporate this clarifications (by writing things more clearly). S_n is the set of all permutations on n elements (so |S_n| = n!). for a point x in {0,1}^n sigma (x) is the point in {0,1}^n that shuffle the coordinates according to sigma. So if sigma takes 1 to 2, 2 to 3, 3 to 4, etc, that sigma(01110) = (00111) and sigma (10001) = 11000 1_C is the characteristic function of C. So 1_C(x) = 1 if x\in C and 0 otherwise. * is convolution. Lambda_B was defined as the maximum over all real functions f from {0,1}^n that are supported on B of the quantity / where A is the adjacency matrix of the hamming cube. You don't really need to know the definition for hequestion though. Check the link in the home page to the paper of Navon and Samorodnitsky for more background.