Finding a binary prefix code provided lengths












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Firstly, I am relatively new to this particular forum, and I usually use Stack exchange (maths). I do not know if this is the right place to post so please be aware in case, I should ask this question on a different stack exchange.



For the following numbers: 1,2,3,3,3. Find a binary prefix code for these lengths.



In my working out I realise by the sum of 3^x, where x is each of the respective numbers is {/frac 5/9}



Then with 3 symbols I have the following prefix code:



With length 1, I have the code 0. With length 2, I have the code 10. With 3 words that have length 3 I have the codes: 110, 111, 112.



I am asking this question because my tutor said that 112 was a possible code. I have only thought of this now, but since I am required to find a binary code how can there be a code 112? Should it not be something like 011? From my understanding, the code which is binary should only contain 0's and 1's.



I have attempted to contact my tutor but to no avail.



Any help would be welcome.



Thnx






computability information-theory coding-theory binary







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    0














    Firstly, I am relatively new to this particular forum, and I usually use Stack exchange (maths). I do not know if this is the right place to post so please be aware in case, I should ask this question on a different stack exchange.



    For the following numbers: 1,2,3,3,3. Find a binary prefix code for these lengths.



    In my working out I realise by the sum of 3^x, where x is each of the respective numbers is {/frac 5/9}



    Then with 3 symbols I have the following prefix code:



    With length 1, I have the code 0. With length 2, I have the code 10. With 3 words that have length 3 I have the codes: 110, 111, 112.



    I am asking this question because my tutor said that 112 was a possible code. I have only thought of this now, but since I am required to find a binary code how can there be a code 112? Should it not be something like 011? From my understanding, the code which is binary should only contain 0's and 1's.



    I have attempted to contact my tutor but to no avail.



    Any help would be welcome.



    Thnx






    computability information-theory coding-theory binary







    share|cite|improve this question

























      0












      0








      0


      1





      Firstly, I am relatively new to this particular forum, and I usually use Stack exchange (maths). I do not know if this is the right place to post so please be aware in case, I should ask this question on a different stack exchange.



      For the following numbers: 1,2,3,3,3. Find a binary prefix code for these lengths.



      In my working out I realise by the sum of 3^x, where x is each of the respective numbers is {/frac 5/9}



      Then with 3 symbols I have the following prefix code:



      With length 1, I have the code 0. With length 2, I have the code 10. With 3 words that have length 3 I have the codes: 110, 111, 112.



      I am asking this question because my tutor said that 112 was a possible code. I have only thought of this now, but since I am required to find a binary code how can there be a code 112? Should it not be something like 011? From my understanding, the code which is binary should only contain 0's and 1's.



      I have attempted to contact my tutor but to no avail.



      Any help would be welcome.



      Thnx






      computability information-theory coding-theory binary







      share|cite|improve this question













      Firstly, I am relatively new to this particular forum, and I usually use Stack exchange (maths). I do not know if this is the right place to post so please be aware in case, I should ask this question on a different stack exchange.



      For the following numbers: 1,2,3,3,3. Find a binary prefix code for these lengths.



      In my working out I realise by the sum of 3^x, where x is each of the respective numbers is {/frac 5/9}



      Then with 3 symbols I have the following prefix code:



      With length 1, I have the code 0. With length 2, I have the code 10. With 3 words that have length 3 I have the codes: 110, 111, 112.



      I am asking this question because my tutor said that 112 was a possible code. I have only thought of this now, but since I am required to find a binary code how can there be a code 112? Should it not be something like 011? From my understanding, the code which is binary should only contain 0's and 1's.



      I have attempted to contact my tutor but to no avail.



      Any help would be welcome.



      Thnx






      computability information-theory coding-theory binary




      computability information-theory coding-theory binary






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      share|cite|improve this question











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      asked Jan 4 at 14:10









      princetongirl818princetongirl818

      828




      828






















          2 Answers
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          1















          With 3 words that have length 3 I have the codes: 110, 111, 112




          Your try:



          L C
          
1 0
          
2 10
          
3 110
          
3 111
          
3 112


          If you are looking for a binary prefix-free code , then 112 does not make sense.



          Before looking for such a code, with prescribed lengths, you might test if it's possible. A necessary and sufficient criterion is Kraft's inequality: $sum 2^{-l_i} le 1$.
          In this case we have $frac12 + frac 14 + 3 frac18=frac98>1$
          Hence, it's not possible to find a binary prefix-free code with those lengths.



          BTW: I have no idea why you've you've computed $sum 3^{-l_i} $ (I guess that's you meant). That would make sense if you wanted a ternary code. In that case, yes, the Kraft's inequality is verified, and you can build a ternary prefix-free code (your try is fine).






          share|cite|improve this answer























          • Thank you sir, for your help. This makes a lot more sense. I have marked your solution as the verified answer.
            – princetongirl818
            Jan 5 at 21:35



















          1














          I would think but cannot be certain that 112 is not a binary code because 2 is not a binary number, and that it should be 0's and 1's only. However I am not completely sure.






          share|cite|improve this answer





















          • Thank you for your help I appreciate it.
            – princetongirl818
            Jan 5 at 21:21











          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









          1















          With 3 words that have length 3 I have the codes: 110, 111, 112




          Your try:



          L C
          
1 0
          
2 10
          
3 110
          
3 111
          
3 112


          If you are looking for a binary prefix-free code , then 112 does not make sense.



          Before looking for such a code, with prescribed lengths, you might test if it's possible. A necessary and sufficient criterion is Kraft's inequality: $sum 2^{-l_i} le 1$.
          In this case we have $frac12 + frac 14 + 3 frac18=frac98>1$
          Hence, it's not possible to find a binary prefix-free code with those lengths.



          BTW: I have no idea why you've you've computed $sum 3^{-l_i} $ (I guess that's you meant). That would make sense if you wanted a ternary code. In that case, yes, the Kraft's inequality is verified, and you can build a ternary prefix-free code (your try is fine).






          share|cite|improve this answer























          • Thank you sir, for your help. This makes a lot more sense. I have marked your solution as the verified answer.
            – princetongirl818
            Jan 5 at 21:35
















          1















          With 3 words that have length 3 I have the codes: 110, 111, 112




          Your try:



          L C
          
1 0
          
2 10
          
3 110
          
3 111
          
3 112


          If you are looking for a binary prefix-free code , then 112 does not make sense.



          Before looking for such a code, with prescribed lengths, you might test if it's possible. A necessary and sufficient criterion is Kraft's inequality: $sum 2^{-l_i} le 1$.
          In this case we have $frac12 + frac 14 + 3 frac18=frac98>1$
          Hence, it's not possible to find a binary prefix-free code with those lengths.



          BTW: I have no idea why you've you've computed $sum 3^{-l_i} $ (I guess that's you meant). That would make sense if you wanted a ternary code. In that case, yes, the Kraft's inequality is verified, and you can build a ternary prefix-free code (your try is fine).






          share|cite|improve this answer























          • Thank you sir, for your help. This makes a lot more sense. I have marked your solution as the verified answer.
            – princetongirl818
            Jan 5 at 21:35














          1












          1








          1







          With 3 words that have length 3 I have the codes: 110, 111, 112




          Your try:



          L C
          
1 0
          
2 10
          
3 110
          
3 111
          
3 112


          If you are looking for a binary prefix-free code , then 112 does not make sense.



          Before looking for such a code, with prescribed lengths, you might test if it's possible. A necessary and sufficient criterion is Kraft's inequality: $sum 2^{-l_i} le 1$.
          In this case we have $frac12 + frac 14 + 3 frac18=frac98>1$
          Hence, it's not possible to find a binary prefix-free code with those lengths.



          BTW: I have no idea why you've you've computed $sum 3^{-l_i} $ (I guess that's you meant). That would make sense if you wanted a ternary code. In that case, yes, the Kraft's inequality is verified, and you can build a ternary prefix-free code (your try is fine).






          share|cite|improve this answer















          With 3 words that have length 3 I have the codes: 110, 111, 112




          Your try:



          L C
          
1 0
          
2 10
          
3 110
          
3 111
          
3 112


          If you are looking for a binary prefix-free code , then 112 does not make sense.



          Before looking for such a code, with prescribed lengths, you might test if it's possible. A necessary and sufficient criterion is Kraft's inequality: $sum 2^{-l_i} le 1$.
          In this case we have $frac12 + frac 14 + 3 frac18=frac98>1$
          Hence, it's not possible to find a binary prefix-free code with those lengths.



          BTW: I have no idea why you've you've computed $sum 3^{-l_i} $ (I guess that's you meant). That would make sense if you wanted a ternary code. In that case, yes, the Kraft's inequality is verified, and you can build a ternary prefix-free code (your try is fine).







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 5 at 16:13

























          answered Jan 5 at 15:17









          leonbloyleonbloy

          40.3k645107




          40.3k645107












          • Thank you sir, for your help. This makes a lot more sense. I have marked your solution as the verified answer.
            – princetongirl818
            Jan 5 at 21:35


















          • Thank you sir, for your help. This makes a lot more sense. I have marked your solution as the verified answer.
            – princetongirl818
            Jan 5 at 21:35
















          Thank you sir, for your help. This makes a lot more sense. I have marked your solution as the verified answer.
          – princetongirl818
          Jan 5 at 21:35




          Thank you sir, for your help. This makes a lot more sense. I have marked your solution as the verified answer.
          – princetongirl818
          Jan 5 at 21:35











          1














          I would think but cannot be certain that 112 is not a binary code because 2 is not a binary number, and that it should be 0's and 1's only. However I am not completely sure.






          share|cite|improve this answer





















          • Thank you for your help I appreciate it.
            – princetongirl818
            Jan 5 at 21:21
















          1














          I would think but cannot be certain that 112 is not a binary code because 2 is not a binary number, and that it should be 0's and 1's only. However I am not completely sure.






          share|cite|improve this answer





















          • Thank you for your help I appreciate it.
            – princetongirl818
            Jan 5 at 21:21














          1












          1








          1






          I would think but cannot be certain that 112 is not a binary code because 2 is not a binary number, and that it should be 0's and 1's only. However I am not completely sure.






          share|cite|improve this answer












          I would think but cannot be certain that 112 is not a binary code because 2 is not a binary number, and that it should be 0's and 1's only. However I am not completely sure.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 4 at 23:18









          Teddy MontgomeryTeddy Montgomery

          132




          132












          • Thank you for your help I appreciate it.
            – princetongirl818
            Jan 5 at 21:21


















          • Thank you for your help I appreciate it.
            – princetongirl818
            Jan 5 at 21:21
















          Thank you for your help I appreciate it.
          – princetongirl818
          Jan 5 at 21:21




          Thank you for your help I appreciate it.
          – princetongirl818
          Jan 5 at 21:21


















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