EXERCISES CODING THEORY SHEET 02
(1) Using Magma determine for which values of m, n, e the Ball B
e(∗) defined over an alphabet with m elements and of length n, has size a power of m, with 1 ≤ n, m ≤ 10 and 1 ≤ e ≤ n.
(2) Let H be the 7 by 3 matrix over F
2whose i-th row is (a
0, a
1, a
2), where i = a
0+2a
1+4a
3. Let σ ∈Sym(7) and define σ(H) as the 7×3 matrix whose σ(i)-th row is the i-th row of H. Let C
σ= ker σ(H).
Using Magma determine |{C
σ: σ ∈Sym(7)}|. Can you interpret the result analysing the action of Sym(7) on the above set?
(3) Given an alphabet A and C ⊆ A
n, show that d
min(C) = n implies
|C| ≤ | A|.
(4) Given 1 ≤ s ≤ m, m = | A|, exhibit codes as above with |C| = s.
(5) Let F be a field of size q, C =Rep
n(F ). Prove that for any v ∈ F
n, d
min(C + v) = n, where C + v = {c + v : c ∈ C}.
E-mail address: andrea.previtali@unimib.it Webpage: http://www.matapp.unimib/~prevital
Date: November 14, 2016.
Andrea Previtali. c
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