Proving that the union of a linear code with its left coset is also linear

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
1
down vote

favorite












Suppose $C$ is a linear code, with elements $textbfc$ and $textbfv$ is a code word not in $C$, but it is an element of We form the coset of $C$, given by
$$
C + textbfv = textbfc + textbfv : textbfc in C, textbfv not in X.
$$
I want to prove that $C cup C + textbfv$. is also a linear code, but am struggling a bit.



I considered breaking this into cases by considering a partition $C cup (C + textbfv)$, and then establishing closure for each of the partitions. But, there seems to be a lapse in information about $textbfv$, save for the fact that it's a field element and thus clearly is closed under the group operation.



Any comments or insights would be greatly appreciated.




coding-theory




share|cite|improve this question























    up vote
    1
    down vote

    favorite












    Suppose $C$ is a linear code, with elements $textbfc$ and $textbfv$ is a code word not in $C$, but it is an element of We form the coset of $C$, given by
    $$
    C + textbfv = textbfc + textbfv : textbfc in C, textbfv not in X.
    $$
    I want to prove that $C cup C + textbfv$. is also a linear code, but am struggling a bit.



    I considered breaking this into cases by considering a partition $C cup (C + textbfv)$, and then establishing closure for each of the partitions. But, there seems to be a lapse in information about $textbfv$, save for the fact that it's a field element and thus clearly is closed under the group operation.



    Any comments or insights would be greatly appreciated.




    coding-theory




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Suppose $C$ is a linear code, with elements $textbfc$ and $textbfv$ is a code word not in $C$, but it is an element of We form the coset of $C$, given by
      $$
      C + textbfv = textbfc + textbfv : textbfc in C, textbfv not in X.
      $$
      I want to prove that $C cup C + textbfv$. is also a linear code, but am struggling a bit.



      I considered breaking this into cases by considering a partition $C cup (C + textbfv)$, and then establishing closure for each of the partitions. But, there seems to be a lapse in information about $textbfv$, save for the fact that it's a field element and thus clearly is closed under the group operation.



      Any comments or insights would be greatly appreciated.




      coding-theory




      share|cite|improve this question











      Suppose $C$ is a linear code, with elements $textbfc$ and $textbfv$ is a code word not in $C$, but it is an element of We form the coset of $C$, given by
      $$
      C + textbfv = textbfc + textbfv : textbfc in C, textbfv not in X.
      $$
      I want to prove that $C cup C + textbfv$. is also a linear code, but am struggling a bit.



      I considered breaking this into cases by considering a partition $C cup (C + textbfv)$, and then establishing closure for each of the partitions. But, there seems to be a lapse in information about $textbfv$, save for the fact that it's a field element and thus clearly is closed under the group operation.



      Any comments or insights would be greatly appreciated.





      coding-theory





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Aug 5 at 23:21









      Matt.P

      768313




      768313




















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          1
          down vote













          That's true iff $C$ is a binary linear code, that is it's linear
          over $Bbb F_2$, the field of two elements.



          The difficult case is showing the sum of two elements of $C+v$
          lies in $Ccup(C+v)$. If those elements are $c_1+v$ and $c_2+v$ then
          they add to
          $$(c_1+v)+(c_2+v)=c_1+c_2+2v=c_1+c_2in C$$
          as $2=0$ within $Bbb F_2$.






          share|cite|improve this answer





















            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',
            convertImagesToLinks: true,
            noModals: false,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            );



            );








             

            draft saved


            draft discarded


















            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2873427%2fproving-that-the-union-of-a-linear-code-with-its-left-coset-is-also-linear%23new-answer', 'question_page');

            );

            Post as a guest

















            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            1
            down vote













            That's true iff $C$ is a binary linear code, that is it's linear
            over $Bbb F_2$, the field of two elements.



            The difficult case is showing the sum of two elements of $C+v$
            lies in $Ccup(C+v)$. If those elements are $c_1+v$ and $c_2+v$ then
            they add to
            $$(c_1+v)+(c_2+v)=c_1+c_2+2v=c_1+c_2in C$$
            as $2=0$ within $Bbb F_2$.






            share|cite|improve this answer

























              up vote
              1
              down vote













              That's true iff $C$ is a binary linear code, that is it's linear
              over $Bbb F_2$, the field of two elements.



              The difficult case is showing the sum of two elements of $C+v$
              lies in $Ccup(C+v)$. If those elements are $c_1+v$ and $c_2+v$ then
              they add to
              $$(c_1+v)+(c_2+v)=c_1+c_2+2v=c_1+c_2in C$$
              as $2=0$ within $Bbb F_2$.






              share|cite|improve this answer























                up vote
                1
                down vote










                up vote
                1
                down vote









                That's true iff $C$ is a binary linear code, that is it's linear
                over $Bbb F_2$, the field of two elements.



                The difficult case is showing the sum of two elements of $C+v$
                lies in $Ccup(C+v)$. If those elements are $c_1+v$ and $c_2+v$ then
                they add to
                $$(c_1+v)+(c_2+v)=c_1+c_2+2v=c_1+c_2in C$$
                as $2=0$ within $Bbb F_2$.






                share|cite|improve this answer













                That's true iff $C$ is a binary linear code, that is it's linear
                over $Bbb F_2$, the field of two elements.



                The difficult case is showing the sum of two elements of $C+v$
                lies in $Ccup(C+v)$. If those elements are $c_1+v$ and $c_2+v$ then
                they add to
                $$(c_1+v)+(c_2+v)=c_1+c_2+2v=c_1+c_2in C$$
                as $2=0$ within $Bbb F_2$.







                share|cite|improve this answer













                share|cite|improve this answer



                share|cite|improve this answer











                answered Aug 6 at 3:11









                Lord Shark the Unknown

                86k951112




                86k951112






















                     

                    draft saved


                    draft discarded


























                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2873427%2fproving-that-the-union-of-a-linear-code-with-its-left-coset-is-also-linear%23new-answer', 'question_page');

                    );

                    Post as a guest
































































                    Comments

                    Post a Comment

                    Popular posts from this blog

                    Color the edges and diagonals of a regular polygon

                    Image
                    Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne
                    Read more

                    Relationship between determinant of matrix and determinant of adjoint?

                    Image
                    Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/… – Hector Blandin Nov 17 &
                    Read more

                    What is the equation of a 3D cone with generalised tilt?

                    Image
                    Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an
                    Read more