Does a code with length 6, size 32 and distance 2 exist?
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The problem is to prove or disprove the existence of $C$, s.t., $|c| = 6,forall cin C$; $|C| = 32$; $d(c_i,c_j)geq2,1leq i<jleq32$. ($d$ stands for hamming distance)
I tried to construct a satisfying code. The best I can get is to let $C = C'times C'$, a concatenation of $C' = {000,011,110,101}$, which is of size 16. 32 happens to be the theoretical upper bound of the size, now I don't know what to do next so as to solve the problem.
information-theory coding-theory encoding-scheme
asked 2 hours ago
MianguMiangu
614
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add a comment |
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The problem is to prove or disprove the existence of $C$, s.t., $|c| = 6,forall cin C$; $|C| = 32$; $d(c_i,c_j)geq2,1leq i<jleq32$. ($d$ stands for hamming distance)
I tried to construct a satisfying code. The best I can get is to let $C = C'times C'$, a concatenation of $C' = {000,011,110,101}$, which is of size 16. 32 happens to be the theoretical upper bound of the size, now I don't know what to do next so as to solve the problem.
information-theory coding-theory encoding-scheme
asked 2 hours ago
MianguMiangu
614
$endgroup$
add a comment |
$begingroup$
The problem is to prove or disprove the existence of $C$, s.t., $|c| = 6,forall cin C$; $|C| = 32$; $d(c_i,c_j)geq2,1leq i<jleq32$. ($d$ stands for hamming distance)
I tried to construct a satisfying code. The best I can get is to let $C = C'times C'$, a concatenation of $C' = {000,011,110,101}$, which is of size 16. 32 happens to be the theoretical upper bound of the size, now I don't know what to do next so as to solve the problem.
information-theory coding-theory encoding-scheme
asked 2 hours ago
MianguMiangu
614
$endgroup$
The problem is to prove or disprove the existence of $C$, s.t., $|c| = 6,forall cin C$; $|C| = 32$; $d(c_i,c_j)geq2,1leq i<jleq32$. ($d$ stands for hamming distance)
I tried to construct a satisfying code. The best I can get is to let $C = C'times C'$, a concatenation of $C' = {000,011,110,101}$, which is of size 16. 32 happens to be the theoretical upper bound of the size, now I don't know what to do next so as to solve the problem.
information-theory coding-theory encoding-scheme
information-theory coding-theory encoding-scheme
asked 2 hours ago
MianguMiangu
614
asked 2 hours ago
MianguMiangu
614
asked 2 hours ago
MianguMiangu
614
asked 2 hours ago
MianguMiangu
614
asked 2 hours ago
MianguMiangu
614
614
add a comment |
add a comment |
2 Answers
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All words of even parity from a linear code with $2^{n-1}$ codewords and minimum distance $2$.
More generally, if $A_2(n,d)$ is the maximum size of a code of length $n$ and minimum distance $d$, then $A_2(n,2d) = A_2(n-1,2d-1)$.
answered 1 hour ago
Yuval FilmusYuval Filmus
192k14180344
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Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
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– Apass.Jack
23 mins ago
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The subscript signifies the field $mathbb{F}_2$.
$endgroup$
– Yuval Filmus
23 mins ago
add a comment |
$begingroup$
Yes, there is such a set. You are actually on the right track to find the following example.
Let $C = {c : |c|=6 text{ and there are even number of 1's in c}}$. You can check the following.
$|C|=32$.
$d(u,v)geq2$ for all $u,vin C$, $unot=v$. (In fact, $d(u,v)=2$ or 4.)
Here are two related exercise.
Exercise 1. Give another example of a set of 32 words of length 6 and pairwise distance at least 2.
Exercise 2. Show that there are only two such sets, as given in the answer and in the exercise 1.
Exercise 3. Generalize the above to words of any given length.
answered 1 hour ago
Apass.JackApass.Jack
11k1939
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2 Answers
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2 Answers
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$begingroup$
All words of even parity from a linear code with $2^{n-1}$ codewords and minimum distance $2$.
More generally, if $A_2(n,d)$ is the maximum size of a code of length $n$ and minimum distance $d$, then $A_2(n,2d) = A_2(n-1,2d-1)$.
answered 1 hour ago
Yuval FilmusYuval Filmus
192k14180344
$endgroup$
$begingroup$
Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
$endgroup$
– Apass.Jack
23 mins ago
$begingroup$
The subscript signifies the field $mathbb{F}_2$.
$endgroup$
– Yuval Filmus
23 mins ago
add a comment |
$begingroup$
All words of even parity from a linear code with $2^{n-1}$ codewords and minimum distance $2$.
More generally, if $A_2(n,d)$ is the maximum size of a code of length $n$ and minimum distance $d$, then $A_2(n,2d) = A_2(n-1,2d-1)$.
answered 1 hour ago
Yuval FilmusYuval Filmus
192k14180344
$endgroup$
$begingroup$
Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
$endgroup$
– Apass.Jack
23 mins ago
$begingroup$
The subscript signifies the field $mathbb{F}_2$.
$endgroup$
– Yuval Filmus
23 mins ago
add a comment |
$begingroup$
All words of even parity from a linear code with $2^{n-1}$ codewords and minimum distance $2$.
More generally, if $A_2(n,d)$ is the maximum size of a code of length $n$ and minimum distance $d$, then $A_2(n,2d) = A_2(n-1,2d-1)$.
answered 1 hour ago
Yuval FilmusYuval Filmus
192k14180344
$endgroup$
All words of even parity from a linear code with $2^{n-1}$ codewords and minimum distance $2$.
More generally, if $A_2(n,d)$ is the maximum size of a code of length $n$ and minimum distance $d$, then $A_2(n,2d) = A_2(n-1,2d-1)$.
answered 1 hour ago
Yuval FilmusYuval Filmus
192k14180344
answered 1 hour ago
Yuval FilmusYuval Filmus
192k14180344
answered 1 hour ago
Yuval FilmusYuval Filmus
192k14180344
answered 1 hour ago
Yuval FilmusYuval Filmus
192k14180344
192k14180344
$begingroup$
Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
$endgroup$
– Apass.Jack
23 mins ago
$begingroup$
The subscript signifies the field $mathbb{F}_2$.
$endgroup$
– Yuval Filmus
23 mins ago
add a comment |
$begingroup$
Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
$endgroup$
– Apass.Jack
23 mins ago
$begingroup$
The subscript signifies the field $mathbb{F}_2$.
$endgroup$
– Yuval Filmus
23 mins ago
$begingroup$
Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
$endgroup$
– Apass.Jack
23 mins ago
$begingroup$
Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
$endgroup$
– Apass.Jack
23 mins ago
$begingroup$
The subscript signifies the field $mathbb{F}_2$.
$endgroup$
– Yuval Filmus
23 mins ago
$begingroup$
The subscript signifies the field $mathbb{F}_2$.
$endgroup$
– Yuval Filmus
23 mins ago
add a comment |
$begingroup$
Yes, there is such a set. You are actually on the right track to find the following example.
Let $C = {c : |c|=6 text{ and there are even number of 1's in c}}$. You can check the following.
$|C|=32$.
$d(u,v)geq2$ for all $u,vin C$, $unot=v$. (In fact, $d(u,v)=2$ or 4.)
Here are two related exercise.
Exercise 1. Give another example of a set of 32 words of length 6 and pairwise distance at least 2.
Exercise 2. Show that there are only two such sets, as given in the answer and in the exercise 1.
Exercise 3. Generalize the above to words of any given length.
answered 1 hour ago
Apass.JackApass.Jack
11k1939
$endgroup$
add a comment |
$begingroup$
Yes, there is such a set. You are actually on the right track to find the following example.
Let $C = {c : |c|=6 text{ and there are even number of 1's in c}}$. You can check the following.
$|C|=32$.
$d(u,v)geq2$ for all $u,vin C$, $unot=v$. (In fact, $d(u,v)=2$ or 4.)
Here are two related exercise.
Exercise 1. Give another example of a set of 32 words of length 6 and pairwise distance at least 2.
Exercise 2. Show that there are only two such sets, as given in the answer and in the exercise 1.
Exercise 3. Generalize the above to words of any given length.
answered 1 hour ago
Apass.JackApass.Jack
11k1939
$endgroup$
add a comment |
$begingroup$
Yes, there is such a set. You are actually on the right track to find the following example.
Let $C = {c : |c|=6 text{ and there are even number of 1's in c}}$. You can check the following.
$|C|=32$.
$d(u,v)geq2$ for all $u,vin C$, $unot=v$. (In fact, $d(u,v)=2$ or 4.)
Here are two related exercise.
Exercise 1. Give another example of a set of 32 words of length 6 and pairwise distance at least 2.
Exercise 2. Show that there are only two such sets, as given in the answer and in the exercise 1.
Exercise 3. Generalize the above to words of any given length.
answered 1 hour ago
Apass.JackApass.Jack
11k1939
$endgroup$
Yes, there is such a set. You are actually on the right track to find the following example.
Let $C = {c : |c|=6 text{ and there are even number of 1's in c}}$. You can check the following.
$|C|=32$.
$d(u,v)geq2$ for all $u,vin C$, $unot=v$. (In fact, $d(u,v)=2$ or 4.)
Here are two related exercise.
Exercise 1. Give another example of a set of 32 words of length 6 and pairwise distance at least 2.
Exercise 2. Show that there are only two such sets, as given in the answer and in the exercise 1.
Exercise 3. Generalize the above to words of any given length.
answered 1 hour ago
Apass.JackApass.Jack
11k1939
edited 34 mins ago
answered 1 hour ago
Apass.JackApass.Jack
11k1939
answered 1 hour ago
Apass.JackApass.Jack
11k1939
answered 1 hour ago
Apass.JackApass.Jack
11k1939
11k1939
add a comment |
add a comment |
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