The Three Trivial Perfect Codes
On Coding Theory, there's three trivial perfect codes.
They are:
- Binary codes of odd length
- Codes with contains only one codeword
- Codes that are the whole $A^n_q$
So, the $A^n_q$ case, considering the definition that says "A perfect code is a code such $A_q^n$ is the disjoint union of balls of some fixed radius centered on the codewords" is trivial by taking the radius to be 0.
But the other two cases I can't see why. Any hints to demonstrate that?
(Could it be that codes with one codeword are analogue to the $A_q^n$ case? Taking the radius to be n or M=|C|)
coding-theory
asked Dec 27 '18 at 13:53
Rodrigo Geaquinto Gonçalves
62
add a comment |
On Coding Theory, there's three trivial perfect codes.
They are:
- Binary codes of odd length
- Codes with contains only one codeword
- Codes that are the whole $A^n_q$
So, the $A^n_q$ case, considering the definition that says "A perfect code is a code such $A_q^n$ is the disjoint union of balls of some fixed radius centered on the codewords" is trivial by taking the radius to be 0.
But the other two cases I can't see why. Any hints to demonstrate that?
(Could it be that codes with one codeword are analogue to the $A_q^n$ case? Taking the radius to be n or M=|C|)
coding-theory
asked Dec 27 '18 at 13:53
Rodrigo Geaquinto Gonçalves
62
For the second case, yes, just take your radius to be suitably large, and you're done. For the first, this should help.
– user3482749
Dec 27 '18 at 13:59
Oh, I see it now! Thanks!
– Rodrigo Geaquinto Gonçalves
Dec 27 '18 at 15:01
add a comment |
On Coding Theory, there's three trivial perfect codes.
They are:
- Binary codes of odd length
- Codes with contains only one codeword
- Codes that are the whole $A^n_q$
So, the $A^n_q$ case, considering the definition that says "A perfect code is a code such $A_q^n$ is the disjoint union of balls of some fixed radius centered on the codewords" is trivial by taking the radius to be 0.
But the other two cases I can't see why. Any hints to demonstrate that?
(Could it be that codes with one codeword are analogue to the $A_q^n$ case? Taking the radius to be n or M=|C|)
coding-theory
asked Dec 27 '18 at 13:53
Rodrigo Geaquinto Gonçalves
62
On Coding Theory, there's three trivial perfect codes.
They are:
- Binary codes of odd length
- Codes with contains only one codeword
- Codes that are the whole $A^n_q$
So, the $A^n_q$ case, considering the definition that says "A perfect code is a code such $A_q^n$ is the disjoint union of balls of some fixed radius centered on the codewords" is trivial by taking the radius to be 0.
But the other two cases I can't see why. Any hints to demonstrate that?
(Could it be that codes with one codeword are analogue to the $A_q^n$ case? Taking the radius to be n or M=|C|)
coding-theory
coding-theory
asked Dec 27 '18 at 13:53
Rodrigo Geaquinto Gonçalves
62
asked Dec 27 '18 at 13:53
Rodrigo Geaquinto Gonçalves
62
asked Dec 27 '18 at 13:53
Rodrigo Geaquinto Gonçalves
62
asked Dec 27 '18 at 13:53
Rodrigo Geaquinto Gonçalves
62
asked Dec 27 '18 at 13:53
Rodrigo Geaquinto Gonçalves
62
62
For the second case, yes, just take your radius to be suitably large, and you're done. For the first, this should help.
– user3482749
Dec 27 '18 at 13:59
Oh, I see it now! Thanks!
– Rodrigo Geaquinto Gonçalves
Dec 27 '18 at 15:01
add a comment |
For the second case, yes, just take your radius to be suitably large, and you're done. For the first, this should help.
– user3482749
Dec 27 '18 at 13:59
Oh, I see it now! Thanks!
– Rodrigo Geaquinto Gonçalves
Dec 27 '18 at 15:01
For the second case, yes, just take your radius to be suitably large, and you're done. For the first, this should help.
– user3482749
Dec 27 '18 at 13:59
For the second case, yes, just take your radius to be suitably large, and you're done. For the first, this should help.
– user3482749
Dec 27 '18 at 13:59
Oh, I see it now! Thanks!
– Rodrigo Geaquinto Gonçalves
Dec 27 '18 at 15:01
Oh, I see it now! Thanks!
– Rodrigo Geaquinto Gonçalves
Dec 27 '18 at 15:01
add a comment |
0
active
oldest
votes
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: "69"
};
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: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
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
},
noCode: 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%2fmath.stackexchange.com%2fquestions%2f3053947%2fthe-three-trivial-perfect-codes%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics 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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2fmath.stackexchange.com%2fquestions%2f3053947%2fthe-three-trivial-perfect-codes%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
For the second case, yes, just take your radius to be suitably large, and you're done. For the first, this should help.
– user3482749
Dec 27 '18 at 13:59
Oh, I see it now! Thanks!
– Rodrigo Geaquinto Gonçalves
Dec 27 '18 at 15:01