4 votes
Accepted

Random codes in coding theory

The random coding technique is an application of the probabilistic method. Suppose you are working with some class of objects, and want to assert that at least one with a certain property exists. One ...
stochasticboy321's user avatar
4 votes

"Manin's theorem" reference, coding theory

My impression is that the intention of the authors is to introduce the function $\alpha_q$ and to state a few of its properties. But the way they do it ("there exists ... such that") looks a ...
azimut's user avatar
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3 votes
Accepted

Definition of smallest cyclic code in $\mathbb F_2$

Being C the smallest cyclic code refer to the dimension(number of word in the code). So the list of C's words, is given by the null vector, w, w' (first cyclic shifts of w), w+w'. At this point we don'...
francesco's user avatar
1 vote

A good decoder must be non-linear?

$H$ is $(n-k)\times n$ dimensional therefore the syndrome has dimension $n-k.$ Now each of the weight one errors span a space of dimension one and they're linearly independent. So a syndrome can ...
kodlu's user avatar
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1 vote
Accepted

"Manin's theorem" reference, coding theory

Here is the answer to the question as stated literally: Citation 2 is found here and the paper of Manin is here, see in particular Theorem 3 on p. 716. Still, I don't see the exact correlation to what ...
azimut's user avatar
  • 22.5k
1 vote
Accepted

Huffman Code with Weight

Given that the OP has now specified they created the tree to minimize the expected weight, I'm entering this answer: This is a valid Huffman tree, minimizing the expected path weight.
kodlu's user avatar
  • 9,243
1 vote

Prove that, if a code $C\subseteq \left \{ 0,1 \right \}^n$ has minimum distance $d$, then $C'=\left \{ (u,u):u\in C \right \}$ has distance $2d$

Your proof is almost correct; now you need to show the minimum distance is no more than $2d$. This doesn't take too much more work, though :) You know that the minimum distance the smallest pairwise ...
Aidan W. Murphy's user avatar

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