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Asymptotic for binary linear code
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Let $C=[n,k,d]$ is a binary linear code of length $n$, dimension $k$, and minimum distance $d$. Let us consider all possible binary linear codes with $k=d$ and $nin mathbb N$. Is it true that
$$A=mathoplim limits_kto infty frackn=0 ?$$
I look codetables.de and this assumption looks plausible. Are suitable boundaries or hypotheses known?
reference-request coding-theory
asked yesterday
grizzly
25829
add a comment |Â
up vote
1
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Let $C=[n,k,d]$ is a binary linear code of length $n$, dimension $k$, and minimum distance $d$. Let us consider all possible binary linear codes with $k=d$ and $nin mathbb N$. Is it true that
$$A=mathoplim limits_kto infty frackn=0 ?$$
I look codetables.de and this assumption looks plausible. Are suitable boundaries or hypotheses known?
reference-request coding-theory
asked yesterday
grizzly
25829
1
I assume you want the number of codes, not just $frac kn$, in your limit. For fixed $n$, once $k gt frac n2$ there are no codes
– Ross Millikan
yesterday
@RossMillikan $n=n(k)$ and $n→infty $ as $k→infty $, of course. From your comments it follows that $A=mathoplim sup _k→infty k/n le 1/2$. áan there exist codes $[10k,k,k]$ for arbitrarily large $k$? If such codes exist only a finite number, no more, then $A le 1/10$ etc.
– grizzly
yesterday
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $C=[n,k,d]$ is a binary linear code of length $n$, dimension $k$, and minimum distance $d$. Let us consider all possible binary linear codes with $k=d$ and $nin mathbb N$. Is it true that
$$A=mathoplim limits_kto infty frackn=0 ?$$
I look codetables.de and this assumption looks plausible. Are suitable boundaries or hypotheses known?
reference-request coding-theory
asked yesterday
grizzly
25829
Let $C=[n,k,d]$ is a binary linear code of length $n$, dimension $k$, and minimum distance $d$. Let us consider all possible binary linear codes with $k=d$ and $nin mathbb N$. Is it true that
$$A=mathoplim limits_kto infty frackn=0 ?$$
I look codetables.de and this assumption looks plausible. Are suitable boundaries or hypotheses known?
reference-request coding-theory
asked yesterday
grizzly
25829
edited yesterday
asked yesterday
grizzly
25829
asked yesterday
grizzly
25829
asked yesterday
grizzly
25829
25829
1
I assume you want the number of codes, not just $frac kn$, in your limit. For fixed $n$, once $k gt frac n2$ there are no codes
– Ross Millikan
yesterday
@RossMillikan $n=n(k)$ and $n→infty $ as $k→infty $, of course. From your comments it follows that $A=mathoplim sup _k→infty k/n le 1/2$. áan there exist codes $[10k,k,k]$ for arbitrarily large $k$? If such codes exist only a finite number, no more, then $A le 1/10$ etc.
– grizzly
yesterday
add a comment |Â
1
I assume you want the number of codes, not just $frac kn$, in your limit. For fixed $n$, once $k gt frac n2$ there are no codes
– Ross Millikan
yesterday
@RossMillikan $n=n(k)$ and $n→infty $ as $k→infty $, of course. From your comments it follows that $A=mathoplim sup _k→infty k/n le 1/2$. áan there exist codes $[10k,k,k]$ for arbitrarily large $k$? If such codes exist only a finite number, no more, then $A le 1/10$ etc.
– grizzly
yesterday
1
1
I assume you want the number of codes, not just $frac kn$, in your limit. For fixed $n$, once $k gt frac n2$ there are no codes
– Ross Millikan
yesterday
I assume you want the number of codes, not just $frac kn$, in your limit. For fixed $n$, once $k gt frac n2$ there are no codes
– Ross Millikan
yesterday
@RossMillikan $n=n(k)$ and $n→infty $ as $k→infty $, of course. From your comments it follows that $A=mathoplim sup _k→infty k/n le 1/2$. áan there exist codes $[10k,k,k]$ for arbitrarily large $k$? If such codes exist only a finite number, no more, then $A le 1/10$ etc.
– grizzly
yesterday
@RossMillikan $n=n(k)$ and $n→infty $ as $k→infty $, of course. From your comments it follows that $A=mathoplim sup _k→infty k/n le 1/2$. áan there exist codes $[10k,k,k]$ for arbitrarily large $k$? If such codes exist only a finite number, no more, then $A le 1/10$ etc.
– grizzly
yesterday
add a comment |Â
1 Answer
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up vote
1
down vote
accepted
The Gilbert Varshamov bound shows the existence of a binary linear code of dimension $$kgeq n-lfloor log_2 V(n,d-1)rfloor,$$
where $V(n,r)$ is the volume of the hamming sphere in the $n$ dimensional hypercube.
Let $d=k$, and find the maximal $k$ satisfying this equation, for each $n$. Along a subsequence of integers, something like the existence of codes with parameters
$$
[Ak_i,k_i,k_i]
$$
with $A=5,$ should hold.
More generally, let $nrightarrow infty$ and use the fact that for each $n$ the binomial coefficients $binomnk,$ are superincreasing in $k$ up to about $k<n/3,$ and the approximation
$$
V(n,k-1)approx binomnk approx 2^n (h(k/n)+o(1)),
$$
where $h$ is binary Shannon entropy (see here) to conclude that the solution $theta=0.227cdots$ to
$$
1-theta=h(theta)
$$
where $theta=k/n,$ means that asymptotically there is a sequence of codes with parameters $[Ak,k,k]$ and $A=1/theta.$
answered 14 hours ago
kodlu
2,871615
This is very useful for me. Many thanks!
– grizzly
7 hours ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
The Gilbert Varshamov bound shows the existence of a binary linear code of dimension $$kgeq n-lfloor log_2 V(n,d-1)rfloor,$$
where $V(n,r)$ is the volume of the hamming sphere in the $n$ dimensional hypercube.
Let $d=k$, and find the maximal $k$ satisfying this equation, for each $n$. Along a subsequence of integers, something like the existence of codes with parameters
$$
[Ak_i,k_i,k_i]
$$
with $A=5,$ should hold.
More generally, let $nrightarrow infty$ and use the fact that for each $n$ the binomial coefficients $binomnk,$ are superincreasing in $k$ up to about $k<n/3,$ and the approximation
$$
V(n,k-1)approx binomnk approx 2^n (h(k/n)+o(1)),
$$
where $h$ is binary Shannon entropy (see here) to conclude that the solution $theta=0.227cdots$ to
$$
1-theta=h(theta)
$$
where $theta=k/n,$ means that asymptotically there is a sequence of codes with parameters $[Ak,k,k]$ and $A=1/theta.$
answered 14 hours ago
kodlu
2,871615
This is very useful for me. Many thanks!
– grizzly
7 hours ago
add a comment |Â
up vote
1
down vote
accepted
The Gilbert Varshamov bound shows the existence of a binary linear code of dimension $$kgeq n-lfloor log_2 V(n,d-1)rfloor,$$
where $V(n,r)$ is the volume of the hamming sphere in the $n$ dimensional hypercube.
Let $d=k$, and find the maximal $k$ satisfying this equation, for each $n$. Along a subsequence of integers, something like the existence of codes with parameters
$$
[Ak_i,k_i,k_i]
$$
with $A=5,$ should hold.
More generally, let $nrightarrow infty$ and use the fact that for each $n$ the binomial coefficients $binomnk,$ are superincreasing in $k$ up to about $k<n/3,$ and the approximation
$$
V(n,k-1)approx binomnk approx 2^n (h(k/n)+o(1)),
$$
where $h$ is binary Shannon entropy (see here) to conclude that the solution $theta=0.227cdots$ to
$$
1-theta=h(theta)
$$
where $theta=k/n,$ means that asymptotically there is a sequence of codes with parameters $[Ak,k,k]$ and $A=1/theta.$
answered 14 hours ago
kodlu
2,871615
This is very useful for me. Many thanks!
– grizzly
7 hours ago
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
The Gilbert Varshamov bound shows the existence of a binary linear code of dimension $$kgeq n-lfloor log_2 V(n,d-1)rfloor,$$
where $V(n,r)$ is the volume of the hamming sphere in the $n$ dimensional hypercube.
Let $d=k$, and find the maximal $k$ satisfying this equation, for each $n$. Along a subsequence of integers, something like the existence of codes with parameters
$$
[Ak_i,k_i,k_i]
$$
with $A=5,$ should hold.
More generally, let $nrightarrow infty$ and use the fact that for each $n$ the binomial coefficients $binomnk,$ are superincreasing in $k$ up to about $k<n/3,$ and the approximation
$$
V(n,k-1)approx binomnk approx 2^n (h(k/n)+o(1)),
$$
where $h$ is binary Shannon entropy (see here) to conclude that the solution $theta=0.227cdots$ to
$$
1-theta=h(theta)
$$
where $theta=k/n,$ means that asymptotically there is a sequence of codes with parameters $[Ak,k,k]$ and $A=1/theta.$
answered 14 hours ago
kodlu
2,871615
The Gilbert Varshamov bound shows the existence of a binary linear code of dimension $$kgeq n-lfloor log_2 V(n,d-1)rfloor,$$
where $V(n,r)$ is the volume of the hamming sphere in the $n$ dimensional hypercube.
Let $d=k$, and find the maximal $k$ satisfying this equation, for each $n$. Along a subsequence of integers, something like the existence of codes with parameters
$$
[Ak_i,k_i,k_i]
$$
with $A=5,$ should hold.
More generally, let $nrightarrow infty$ and use the fact that for each $n$ the binomial coefficients $binomnk,$ are superincreasing in $k$ up to about $k<n/3,$ and the approximation
$$
V(n,k-1)approx binomnk approx 2^n (h(k/n)+o(1)),
$$
where $h$ is binary Shannon entropy (see here) to conclude that the solution $theta=0.227cdots$ to
$$
1-theta=h(theta)
$$
where $theta=k/n,$ means that asymptotically there is a sequence of codes with parameters $[Ak,k,k]$ and $A=1/theta.$
answered 14 hours ago
kodlu
2,871615
answered 14 hours ago
kodlu
2,871615
answered 14 hours ago
kodlu
2,871615
answered 14 hours ago
kodlu
2,871615
2,871615
This is very useful for me. Many thanks!
– grizzly
7 hours ago
add a comment |Â
This is very useful for me. Many thanks!
– grizzly
7 hours ago
This is very useful for me. Many thanks!
– grizzly
7 hours ago
This is very useful for me. Many thanks!
– grizzly
7 hours ago
add a comment |Â
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1
I assume you want the number of codes, not just $frac kn$, in your limit. For fixed $n$, once $k gt frac n2$ there are no codes
– Ross Millikan
yesterday
@RossMillikan $n=n(k)$ and $n→infty $ as $k→infty $, of course. From your comments it follows that $A=mathoplim sup _k→infty k/n le 1/2$. áan there exist codes $[10k,k,k]$ for arbitrarily large $k$? If such codes exist only a finite number, no more, then $A le 1/10$ etc.
– grizzly
yesterday