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In this note, we will be concerned with codes for a channel whose input alphabet is of size q; to fix ideas let the input alphabet be X = {0, 1, . . . , q − 1}. Codes for the case q = 2 are called ‘binary codes,’ on which we mostly restricted our attention to in the class. Codes for the alphabet X above are called q-ary codes. A block code of block length n with M codewords is a set of M sequences of length n from the alphabet X. For example the code C = {000, 011, 110, 101} is a code of block length 3 with 4 codewords on the alphabet {0, 1}. Given two sequences x = (x1, . . . , xn) ∈ X and y = (y1, . . . , yn) ∈ X, the Hamming distance dH(x,y) is the number of positions in which they differ: dH(x,y) = |{i : xi 6= yi}|. For example, dH(010, 111) = 2. Note that (i) dH(x,y) = 0 if and only if x = y, (ii) dH(x,y) = dH(y,x), and (iii) dH(x, z) ≤ dH(x,y) + dH(y, z), so dH is a metric. The property (iii) is called the triangle inequality. Given a code C, the minimum distance dmin(C) of the code is defined by dmin(C) = min x∈C,y∈C,x 6=y dH(x,y),

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