Does a code with length 6, size 32 and distance 2 exist?
$begingroup$
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
$endgroup$
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
$endgroup$
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
2
active
oldest
votes
$begingroup$
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
29 mins ago
$begingroup$
The subscript signifies the field $mathbb{F}_2$.
$endgroup$
– Yuval Filmus
28 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 |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "419"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcs.stackexchange.com%2fquestions%2f104581%2fdoes-a-code-with-length-6-size-32-and-distance-2-exist%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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
29 mins ago
$begingroup$
The subscript signifies the field $mathbb{F}_2$.
$endgroup$
– Yuval Filmus
28 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
29 mins ago
$begingroup$
The subscript signifies the field $mathbb{F}_2$.
$endgroup$
– Yuval Filmus
28 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
29 mins ago
$begingroup$
The subscript signifies the field $mathbb{F}_2$.
$endgroup$
– Yuval Filmus
28 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
29 mins ago
$begingroup$
The subscript signifies the field $mathbb{F}_2$.
$endgroup$
– Yuval Filmus
28 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
29 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
29 mins ago
$begingroup$
The subscript signifies the field $mathbb{F}_2$.
$endgroup$
– Yuval Filmus
28 mins ago
$begingroup$
The subscript signifies the field $mathbb{F}_2$.
$endgroup$
– Yuval Filmus
28 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 40 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 |
Thanks for contributing an answer to Computer Science Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcs.stackexchange.com%2fquestions%2f104581%2fdoes-a-code-with-length-6-size-32-and-distance-2-exist%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
