Gilbert–Varshamov bound for linear codes: Difference between revisions

Revision as of 19:28, 12 July 2024

Revision as of 21:34, 28 February 2025

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:<math>\Pr_{\text{random }G} (mG = y) = q^{-n}</math>

Let <math>\operatorname{Vol}_q(r,n)</math> be the volume of a [[Hamming ball]] with the radius <math>r</math>. Then:<ref>The later inequality comes from [http://www.cse.buffalo.edu/~atri/courses/coding-theory/lectures/lect9.pdf the upper bound of the Volume of Hamming ball] {{Webarchive|url=https://web.archive.org/web/20131108081414/http://www.cse.buffalo.edu/~atri/courses/coding-theory/lectures/lect9.pdf |date=2013-11-08 }}</ref>

: <math> P \leqslant q^k W = q^k \left ( \frac{\operatorname{Vol}_q(d-1,n)}{q^n} \right ) \leqslant q^k \left ( \frac{q^{nH_q(\delta)}}{q^n} \right ) = q^k q^{-n(1-H_q(\delta))}</math>