Does a code with length 6, size 32 and distance 2 exist?












7












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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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$endgroup$

















    7












    $begingroup$


    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









    $endgroup$















      7












      7








      7





      $begingroup$


      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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      share|cite|improve this question










      asked 21 hours ago









      MianguMiangu

      865




      865






















          2 Answers
          2






          active

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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 $C = {c : |c|=6 text{ and there are even number of 1's in c}}$. 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 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
            14 hours ago










          • $begingroup$
            @siegi, thanks. Updated.
            $endgroup$
            – Apass.Jack
            11 hours ago



















          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
            19 hours ago








          • 1




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











          Your Answer





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          2 Answers
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          2 Answers
          2






          active

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          active

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          active

          oldest

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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 $C = {c : |c|=6 text{ and there are even number of 1's in c}}$. 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 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
            14 hours ago










          • $begingroup$
            @siegi, thanks. Updated.
            $endgroup$
            – Apass.Jack
            11 hours ago
















          8












          $begingroup$

          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 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
            14 hours ago










          • $begingroup$
            @siegi, thanks. Updated.
            $endgroup$
            – Apass.Jack
            11 hours ago














          8












          8








          8





          $begingroup$

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



          Let $C = {c : |c|=6 text{ and there are even number of 1's in c}}$. 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 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 $C = {c : |c|=6 text{ and there are even number of 1's in c}}$. 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 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 11 hours ago

























          answered 20 hours ago









          Apass.JackApass.Jack

          11.1k1939




          11.1k1939








          • 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
            14 hours ago










          • $begingroup$
            @siegi, thanks. Updated.
            $endgroup$
            – Apass.Jack
            11 hours ago














          • 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
            14 hours ago










          • $begingroup$
            @siegi, thanks. Updated.
            $endgroup$
            – Apass.Jack
            11 hours ago








          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
          14 hours ago




          $begingroup$
          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
          14 hours ago












          $begingroup$
          @siegi, thanks. Updated.
          $endgroup$
          – Apass.Jack
          11 hours ago




          $begingroup$
          @siegi, thanks. Updated.
          $endgroup$
          – Apass.Jack
          11 hours ago











          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
            19 hours ago








          • 1




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
















          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
            19 hours ago








          • 1




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














          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 20 hours ago









          Yuval FilmusYuval Filmus

          193k14180345




          193k14180345








          • 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
            19 hours ago








          • 1




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














          • 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
            19 hours ago








          • 1




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








          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
          19 hours 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
          19 hours ago






          1




          1




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




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


















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