Limit of Gilbert-Varshamov

The limits of Gilbert-Varshamov is a decrease of the distance minimal of the codes. One usually supposes, although that does not have ever proven, that binary linear codes generated by one random Matrice satisfies this terminal. It has a value close to $0,11n$ when $n=2k$, which makes it possible to say that there are strong chances that there are no nonnull words of the code of weight lower than $0,11n.$

For an unspecified linear code on $\backslash \; mathbb\; \{F\}\; \_\; \{Q\},$ one has shown that the median number of words of weight $w$ of a code was close of: $\{N\; \backslash \; choose\; W\}\; (q-1)\; q^\; \{n-k\}$

but this formula was not proven for the binary codes (case $q=2$), although it has chances not to be too distant from the truth. Indeed, for $x$ random, the events ($Hx=i$) are equiprobable, and by supposing that the words of the code are distributed by chance, according to a Binomial distribution of elementary probability $p=1/2$ (what is far from being proven), one a: $card\; \backslash \; \{X,\; Hx=0\; \backslash \}\; =card\; \backslash \; \{Ker\; H\; \backslash \}\; =2^\; \{n-k\}$ $P\; (|X|=w)\; =\; \backslash \; frac\; \{2^\; \{N\}\}.$ It is noticed, in experiments, that, for a random binary code, this formula give a number not no one of words of weight $w$ if $w$ is higher on the terminal of Gilbert-Varshamov (this number grows then extremely quickly with $w$), and no one if $w$ is lower than this one.

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