Gilbert–Varshamov bound for linear codes: Difference between revisions

Revision as of 19:28, 12 July 2024

Latest revision as of 16:25, 16 August 2025

(One intermediate revision by one other user not shown)

Line 52:

:<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 [httphttps://www.cse.buffalo.edu/~faculty/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>