Does a binary 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.










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    8












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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.










    share|cite|improve this question











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      8












      8








      8





      $begingroup$


      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.










      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






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      edited Feb 27 at 13:57







      Miangu

















      asked Feb 20 at 3:53









      MianguMiangu

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          2 Answers
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          8












          $begingroup$

          Yes, there is such a set. You are actually on the right track to find the following example.



          Let $C = c$. 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 or 6.)


          Here are four related exercise, listed in the order of increasing difficulty. As in the question, only binary code is concerned.



          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 and pairwise distance at least 2. (Hint, $32=2^6-1$.)





          Exercise 4. (further generalization stated in Yuval's answer) If $A(n,d)$ is the maximum size of a code of length $n$ and minimum pairwise distance $d$, then $A(d,2d)=A(n-1,2d-1)$.








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          • 1




            $begingroup$
            I think $d(u,v)$ may also be 6, specifically for $u=000000$ and $v=111111$, as both $uin C$ and $vin C$ because both have even number of 1's. Or am I missing something?
            $endgroup$
            – siegi
            Feb 20 at 11:26










          • $begingroup$
            @siegi, thanks. Updated.
            $endgroup$
            – Apass.Jack
            Feb 20 at 14:06


















          6












          $begingroup$

          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









          $endgroup$








          • 1




            $begingroup$
            Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
            $endgroup$
            – Apass.Jack
            Feb 20 at 6:22







          • 1




            $begingroup$
            The subscript signifies the field $mathbbF_2$.
            $endgroup$
            – Yuval Filmus
            Feb 20 at 6:22










          Your Answer





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          2 Answers
          2






          active

          oldest

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          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          8












          $begingroup$

          Yes, there is such a set. You are actually on the right track to find the following example.



          Let $C = c$. 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 or 6.)


          Here are four related exercise, listed in the order of increasing difficulty. As in the question, only binary code is concerned.



          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 and pairwise distance at least 2. (Hint, $32=2^6-1$.)





          Exercise 4. (further generalization stated in Yuval's answer) If $A(n,d)$ is the maximum size of a code of length $n$ and minimum pairwise distance $d$, then $A(d,2d)=A(n-1,2d-1)$.








          share|cite|improve this answer











          $endgroup$








          • 1




            $begingroup$
            I think $d(u,v)$ may also be 6, specifically for $u=000000$ and $v=111111$, as both $uin C$ and $vin C$ because both have even number of 1's. Or am I missing something?
            $endgroup$
            – siegi
            Feb 20 at 11:26










          • $begingroup$
            @siegi, thanks. Updated.
            $endgroup$
            – Apass.Jack
            Feb 20 at 14:06















          8












          $begingroup$

          Yes, there is such a set. You are actually on the right track to find the following example.



          Let $C = c$. 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 or 6.)


          Here are four related exercise, listed in the order of increasing difficulty. As in the question, only binary code is concerned.



          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 and pairwise distance at least 2. (Hint, $32=2^6-1$.)





          Exercise 4. (further generalization stated in Yuval's answer) If $A(n,d)$ is the maximum size of a code of length $n$ and minimum pairwise distance $d$, then $A(d,2d)=A(n-1,2d-1)$.








          share|cite|improve this answer











          $endgroup$








          • 1




            $begingroup$
            I think $d(u,v)$ may also be 6, specifically for $u=000000$ and $v=111111$, as both $uin C$ and $vin C$ because both have even number of 1's. Or am I missing something?
            $endgroup$
            – siegi
            Feb 20 at 11:26










          • $begingroup$
            @siegi, thanks. Updated.
            $endgroup$
            – Apass.Jack
            Feb 20 at 14:06













          8












          8








          8





          $begingroup$

          Yes, there is such a set. You are actually on the right track to find the following example.



          Let $C = c$. 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 or 6.)


          Here are four related exercise, listed in the order of increasing difficulty. As in the question, only binary code is concerned.



          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 and pairwise distance at least 2. (Hint, $32=2^6-1$.)





          Exercise 4. (further generalization stated in Yuval's answer) If $A(n,d)$ is the maximum size of a code of length $n$ and minimum pairwise distance $d$, then $A(d,2d)=A(n-1,2d-1)$.








          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 $C = c$. 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 or 6.)


          Here are four related exercise, listed in the order of increasing difficulty. As in the question, only binary code is concerned.



          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 and pairwise distance at least 2. (Hint, $32=2^6-1$.)





          Exercise 4. (further generalization stated in Yuval's answer) If $A(n,d)$ is the maximum size of a code of length $n$ and minimum pairwise distance $d$, then $A(d,2d)=A(n-1,2d-1)$.









          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Feb 20 at 14:05

























          answered Feb 20 at 5:06









          Apass.JackApass.Jack

          12.8k1939




          12.8k1939







          • 1




            $begingroup$
            I think $d(u,v)$ may also be 6, specifically for $u=000000$ and $v=111111$, as both $uin C$ and $vin C$ because both have even number of 1's. Or am I missing something?
            $endgroup$
            – siegi
            Feb 20 at 11:26










          • $begingroup$
            @siegi, thanks. Updated.
            $endgroup$
            – Apass.Jack
            Feb 20 at 14:06












          • 1




            $begingroup$
            I think $d(u,v)$ may also be 6, specifically for $u=000000$ and $v=111111$, as both $uin C$ and $vin C$ because both have even number of 1's. Or am I missing something?
            $endgroup$
            – siegi
            Feb 20 at 11:26










          • $begingroup$
            @siegi, thanks. Updated.
            $endgroup$
            – Apass.Jack
            Feb 20 at 14:06







          1




          1




          $begingroup$
          I think $d(u,v)$ may also be 6, specifically for $u=000000$ and $v=111111$, as both $uin C$ and $vin C$ because both have even number of 1's. Or am I missing something?
          $endgroup$
          – siegi
          Feb 20 at 11:26




          $begingroup$
          I think $d(u,v)$ may also be 6, specifically for $u=000000$ and $v=111111$, as both $uin C$ and $vin C$ because both have even number of 1's. Or am I missing something?
          $endgroup$
          – siegi
          Feb 20 at 11:26












          $begingroup$
          @siegi, thanks. Updated.
          $endgroup$
          – Apass.Jack
          Feb 20 at 14:06




          $begingroup$
          @siegi, thanks. Updated.
          $endgroup$
          – Apass.Jack
          Feb 20 at 14:06











          6












          $begingroup$

          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









          $endgroup$








          • 1




            $begingroup$
            Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
            $endgroup$
            – Apass.Jack
            Feb 20 at 6:22







          • 1




            $begingroup$
            The subscript signifies the field $mathbbF_2$.
            $endgroup$
            – Yuval Filmus
            Feb 20 at 6:22















          6












          $begingroup$

          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









          $endgroup$








          • 1




            $begingroup$
            Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
            $endgroup$
            – Apass.Jack
            Feb 20 at 6:22







          • 1




            $begingroup$
            The subscript signifies the field $mathbbF_2$.
            $endgroup$
            – Yuval Filmus
            Feb 20 at 6:22













          6












          6








          6





          $begingroup$

          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 Feb 20 at 5:04









          Yuval FilmusYuval Filmus

          195k14183347




          195k14183347







          • 1




            $begingroup$
            Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
            $endgroup$
            – Apass.Jack
            Feb 20 at 6:22







          • 1




            $begingroup$
            The subscript signifies the field $mathbbF_2$.
            $endgroup$
            – Yuval Filmus
            Feb 20 at 6:22












          • 1




            $begingroup$
            Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
            $endgroup$
            – Apass.Jack
            Feb 20 at 6:22







          • 1




            $begingroup$
            The subscript signifies the field $mathbbF_2$.
            $endgroup$
            – Yuval Filmus
            Feb 20 at 6:22







          1




          1




          $begingroup$
          Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
          $endgroup$
          – Apass.Jack
          Feb 20 at 6:22





          $begingroup$
          Nice fact, upvoted. By the way, why not just $A(n,d)$ instead of $A_2(n,d)$? Oh, two letters.
          $endgroup$
          – Apass.Jack
          Feb 20 at 6:22





          1




          1




          $begingroup$
          The subscript signifies the field $mathbbF_2$.
          $endgroup$
          – Yuval Filmus
          Feb 20 at 6:22




          $begingroup$
          The subscript signifies the field $mathbbF_2$.
          $endgroup$
          – Yuval Filmus
          Feb 20 at 6:22

















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