Existence of Good Binary Codes
نویسندگان
چکیده
Proof Let C be a (n, k, d)q code. Then it has q k codewords of length n. Project C to a k − 1 dimensional space by only considering the first k− 1 entries of each code words (see Figure 1). There are at most qk−1 possible strings of length k − 1 in Fk−1 q . However, we have in total qk code words so there must exist at least two codewords with the same projections. Thus the distance between these two code words is n− (k − 1). Hence d ≤ n− (k − 1) = n− k + 1.
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