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4) Example: Hamming codes. Let Fq be the field with q elements, let k −1 ; note that gcd(q, n) = 1, k ≥ 2 such that gcd(k, q − 1) = 1, and let n := qq−1 and that the condition on k always holds for q = 2. Let C ≤ Fnq be the cyclic code with defining set {ζn }, that is V(C) = (ζn )Γ ⊆ V(fn ). 55) We show that the order of q ∈ Z∗n equals k: We have n | q k − 1, hence the order of q divides k; assume that n | q s − 1 for some s ∈ {1, . . , k − 1}, then we have q k − 1 | (q − 1)(q s − 1) = q s+1 − q s − q + 1, thus q k ≤ q k − 2q + 2 ≤ q k − 2, k−1 a contradiction.

S}. q Let C := ker((A ∗ B)tr ) ≤ Fnq ; note that A ∗ B in general does not have full rank. For v = [x1 , . . , xn ] ∈ C let J := {j ∈ {1, . . , n}; xj = 0}; hence we have r×|J | s×|J | wt(v) = |J | ∈ {0, . . , n}. Letting AJ ∈ Fq and BJ ∈ Fq be the submatrices of A and B, respectively, consisting of the columns in J , we have rkFq (AJ ) + rkFq (BJ ) ≤ |J | = wt(v). 43 In particular, we have the van-Lint-Wilson bound: If for some d ∈ N and all subsets ∅ = I ⊆ {1, . . , n} such that |I| ≤ d − 1 we have rkFq (AI ) + rkFq (BI ) > |I|, then C has minimum distance at least d.

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Coding Theory by Jürgen Müller


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