EXERCISES CODING THEORY SHEET 02
(1) Using Magma determine for which values of m, n, e the Ball Be (∗)
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 F2 whose i-th row is (a0 , a1 , a2 ),
where i = a0 +2a1 +4a3 . 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 Cσ = ker σ(H).
Using Magma determine |{Cσ : 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 ⊆ An , show that dmin (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 =Repn (F ). Prove that for any v ∈ F n ,
dmin (C + v) = n, where C + v = {c + v : c ∈ C}.
E-mail address: [email protected]
Webpage: http://www.matapp.unimib/~prevital
Date: November 14, 2016.
c
Andrea
Previtali.
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