Find the minimum distance for each of the codes.
In coding theory, the minimum distance of a code is the smallest Hamming distance between any two distinct codewords in the code. The Hamming distance is the number of positions in which two codewords differ. To find the minimum ______ ____ ______ ___ ___.
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Let C1 and C2 be cyclic codes over , with generator polynomials g₁ (x) and g2(x), respectively. Prove that C1
C2 if and only if g2(x) | g1(x).
Let C be the [5, 2] ternary code generated by
Find the weight enumerator WC(x, y) of C.
Find the convolutional code for the message 11011. The convolutional encoder is given in
Fig. 1.
Which of the following binary codes are linear?
i)
ii)
Justify your answer.
The aim of this exercise is to show that every binary repetition code of odd length is perfect.
i) Find the value of t and d for a perfect code of length 2m + 1, m ∈ .
ii) Show that
(Hint: Start with the relation
iii) Deduce that every repetiition code of odd length is perfect.
Find the minimum distance for each of the codes.
Show that, if x ∈ ,
wt (x) = x.x (mod 3)
Deduce that, if C is a ternary self orthogonal code, the weight of each codeword is divisible by 3.
(Hint: Observe that x2 = 1 for all x ≠0 € )