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







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    3












    $begingroup$


    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







    share|cite|improve this question









    $endgroup$















      3












      3








      3





      $begingroup$


      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







      share|cite|improve this question









      $endgroup$




      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






      share|cite|improve this question













      share|cite|improve this question











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      asked 3 hours ago









      MianguMiangu

      664




      664






















          2 Answers
          2






          active

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          3












          $begingroup$

          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.










          share|cite|improve this answer











          $endgroup$





















            2












            $begingroup$

            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 $A_2(n,2d) = A_2(n-1,2d-1)$.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
              $endgroup$
              – Apass.Jack
              1 hour ago












            • $begingroup$
              The subscript signifies the field $mathbb{F}_2$.
              $endgroup$
              – Yuval Filmus
              1 hour ago











            Your Answer





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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            3












            $begingroup$

            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.










            share|cite|improve this answer











            $endgroup$


















              3












              $begingroup$

              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.










              share|cite|improve this answer











              $endgroup$
















                3












                3








                3





                $begingroup$

                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.










                share|cite|improve this answer











                $endgroup$



                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.











                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 1 hour ago

























                answered 2 hours ago









                Apass.JackApass.Jack

                11k1939




                11k1939























                    2












                    $begingroup$

                    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 $A_2(n,2d) = A_2(n-1,2d-1)$.






                    share|cite|improve this answer









                    $endgroup$













                    • $begingroup$
                      Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
                      $endgroup$
                      – Apass.Jack
                      1 hour ago












                    • $begingroup$
                      The subscript signifies the field $mathbb{F}_2$.
                      $endgroup$
                      – Yuval Filmus
                      1 hour ago
















                    2












                    $begingroup$

                    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 $A_2(n,2d) = A_2(n-1,2d-1)$.






                    share|cite|improve this answer









                    $endgroup$













                    • $begingroup$
                      Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
                      $endgroup$
                      – Apass.Jack
                      1 hour ago












                    • $begingroup$
                      The subscript signifies the field $mathbb{F}_2$.
                      $endgroup$
                      – Yuval Filmus
                      1 hour ago














                    2












                    2








                    2





                    $begingroup$

                    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 $A_2(n,2d) = A_2(n-1,2d-1)$.






                    share|cite|improve this answer









                    $endgroup$



                    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 $A_2(n,2d) = A_2(n-1,2d-1)$.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 2 hours 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
                      1 hour ago












                    • $begingroup$
                      The subscript signifies the field $mathbb{F}_2$.
                      $endgroup$
                      – Yuval Filmus
                      1 hour 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
                      1 hour ago












                    • $begingroup$
                      The subscript signifies the field $mathbb{F}_2$.
                      $endgroup$
                      – Yuval Filmus
                      1 hour 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
                    1 hour 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
                    1 hour ago














                    $begingroup$
                    The subscript signifies the field $mathbb{F}_2$.
                    $endgroup$
                    – Yuval Filmus
                    1 hour ago




                    $begingroup$
                    The subscript signifies the field $mathbb{F}_2$.
                    $endgroup$
                    – Yuval Filmus
                    1 hour ago


















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