Gilbert–Varshamov bound for linear codes: Difference between revisions

Revision as of 02:58, 28 May 2011

Revision as of 00:42, 29 May 2011

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Denote <math>wt(y)</math> be the weight of the codeword <math>y</math>.

So

So <math>P = \Pr_{\text{Random }G} [\text{linear code generated by }G\text{ has distance} < d] = \Pr_{\text{Random }G}</math> [there exists a codeword <math>y \ne 0</math> in a linear code generated by <math>G</math> such that <math>wt(y) < d</math>]

: <math>

\begin{align}

P & = {\Pr}_{\text{random }G} [\text{linear code generated by }G\text{ has distance} < d] \\

So <math>P& = {\Pr_Pr}_{\text{Randomrandom }G} [\text{linear code generated by }G\text{ has distance} < d] = \Pr_{\text{Random }G}</math> [there exists a codeword <math>}y \ne 0</math>\text{ in a linear code generated by <math>}G</math>\text{ such that <math>}\mathrm{wt}(y) < d</math>]

\end{align}

</math>

Also if codeword <math>y</math> belongs to a linear code generated by <math>G</math>, then <math>y = mG</math> for some vector <math>m \in \mathbb{F}_q^k</math>.