Finding a binary prefix code provided lengths
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
asked Jan 4 at 14:10
princetongirl818princetongirl818
828
add a comment |
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
asked Jan 4 at 14:10
princetongirl818princetongirl818
828
add a comment |
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
asked Jan 4 at 14:10
princetongirl818princetongirl818
828
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
asked Jan 4 at 14:10
princetongirl818princetongirl818
828
asked Jan 4 at 14:10
princetongirl818princetongirl818
828
asked Jan 4 at 14:10
princetongirl818princetongirl818
828
asked Jan 4 at 14:10
princetongirl818princetongirl818
828
asked Jan 4 at 14:10
princetongirl818princetongirl818
828
828
add a comment |
add a comment |
2 Answers
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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).
answered Jan 5 at 15:17
leonbloyleonbloy
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
add a comment |
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.
answered Jan 4 at 23:18
Teddy MontgomeryTeddy Montgomery
132
Thank you for your help I appreciate it.
– princetongirl818
Jan 5 at 21:21
add a comment |
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2 Answers
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2 Answers
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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).
answered Jan 5 at 15:17
leonbloyleonbloy
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
add a comment |
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).
answered Jan 5 at 15:17
leonbloyleonbloy
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
add a comment |
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).
answered Jan 5 at 15:17
leonbloyleonbloy
40.3k645107
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).
answered Jan 5 at 15:17
leonbloyleonbloy
40.3k645107
edited Jan 5 at 16:13
answered Jan 5 at 15:17
leonbloyleonbloy
40.3k645107
answered Jan 5 at 15:17
leonbloyleonbloy
40.3k645107
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
add a comment |
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
add a comment |
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.
answered Jan 4 at 23:18
Teddy MontgomeryTeddy Montgomery
132
Thank you for your help I appreciate it.
– princetongirl818
Jan 5 at 21:21
add a comment |
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.
answered Jan 4 at 23:18
Teddy MontgomeryTeddy Montgomery
132
Thank you for your help I appreciate it.
– princetongirl818
Jan 5 at 21:21
add a comment |
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.
answered Jan 4 at 23:18
Teddy MontgomeryTeddy Montgomery
132
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.
answered Jan 4 at 23:18
Teddy MontgomeryTeddy Montgomery
132
answered Jan 4 at 23:18
Teddy MontgomeryTeddy Montgomery
132
answered Jan 4 at 23:18
Teddy MontgomeryTeddy Montgomery
132
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
add a comment |
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
add a comment |
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