How to prove that the dual code of a punctured code C is equal to the shortened dual code of C. [closed]

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Let $C$ be a binary linear [n,k,d] code



The shortened code $C_S$ of $C$ with respect to the first position is the linear code
with length n − 1, obtained by taking the codewords of $C$ that are zero in the
first position, and by removing the first entry.



The punctured code $C_P$ with respect to the first position is the linear code with
length n − 1, obtained removing the first entry from every codeword of $C$.



Show that



$$(C_P)^perp=(C^perp)_S$$



Hint: show that $(C_P)^perpsubseteq(C^perp)_S$ and $(C^perp)_Ssubseteq(C_P)^perp$




I just have no idea how to prove that one code is the subset of the other. All I can figure out is that each side of the equation has the same length and dimension.




coding-theory




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closed as off-topic by Jack D'Aurizio♦ Aug 10 at 17:14


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jack D'Aurizio
If this question can be reworded to fit the rules in the help center, please edit the question.












  • I do not have enough reputation to comment, so here's my answer: This is an assignment question taken from MATH324. I would appreciate it if you take this question down.
    – YourLecturer
    Aug 10 at 2:18














up vote
0
down vote

favorite













Let $C$ be a binary linear [n,k,d] code



The shortened code $C_S$ of $C$ with respect to the first position is the linear code
with length n − 1, obtained by taking the codewords of $C$ that are zero in the
first position, and by removing the first entry.



The punctured code $C_P$ with respect to the first position is the linear code with
length n − 1, obtained removing the first entry from every codeword of $C$.



Show that



$$(C_P)^perp=(C^perp)_S$$



Hint: show that $(C_P)^perpsubseteq(C^perp)_S$ and $(C^perp)_Ssubseteq(C_P)^perp$




I just have no idea how to prove that one code is the subset of the other. All I can figure out is that each side of the equation has the same length and dimension.




coding-theory




share|cite|improve this question













closed as off-topic by Jack D'Aurizio♦ Aug 10 at 17:14


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jack D'Aurizio
If this question can be reworded to fit the rules in the help center, please edit the question.












  • I do not have enough reputation to comment, so here's my answer: This is an assignment question taken from MATH324. I would appreciate it if you take this question down.
    – YourLecturer
    Aug 10 at 2:18












up vote
0
down vote

favorite









up vote
0
down vote

favorite












Let $C$ be a binary linear [n,k,d] code



The shortened code $C_S$ of $C$ with respect to the first position is the linear code
with length n − 1, obtained by taking the codewords of $C$ that are zero in the
first position, and by removing the first entry.



The punctured code $C_P$ with respect to the first position is the linear code with
length n − 1, obtained removing the first entry from every codeword of $C$.



Show that



$$(C_P)^perp=(C^perp)_S$$



Hint: show that $(C_P)^perpsubseteq(C^perp)_S$ and $(C^perp)_Ssubseteq(C_P)^perp$




I just have no idea how to prove that one code is the subset of the other. All I can figure out is that each side of the equation has the same length and dimension.




coding-theory




share|cite|improve this question














Let $C$ be a binary linear [n,k,d] code



The shortened code $C_S$ of $C$ with respect to the first position is the linear code
with length n − 1, obtained by taking the codewords of $C$ that are zero in the
first position, and by removing the first entry.



The punctured code $C_P$ with respect to the first position is the linear code with
length n − 1, obtained removing the first entry from every codeword of $C$.



Show that



$$(C_P)^perp=(C^perp)_S$$



Hint: show that $(C_P)^perpsubseteq(C^perp)_S$ and $(C^perp)_Ssubseteq(C_P)^perp$




I just have no idea how to prove that one code is the subset of the other. All I can figure out is that each side of the equation has the same length and dimension.





coding-theory





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 6 at 9:17
























asked Aug 6 at 9:06









MeLaan

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closed as off-topic by Jack D'Aurizio♦ Aug 10 at 17:14


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jack D'Aurizio
If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by Jack D'Aurizio♦ Aug 10 at 17:14


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jack D'Aurizio
If this question can be reworded to fit the rules in the help center, please edit the question.











  • I do not have enough reputation to comment, so here's my answer: This is an assignment question taken from MATH324. I would appreciate it if you take this question down.
    – YourLecturer
    Aug 10 at 2:18
















  • I do not have enough reputation to comment, so here's my answer: This is an assignment question taken from MATH324. I would appreciate it if you take this question down.
    – YourLecturer
    Aug 10 at 2:18















I do not have enough reputation to comment, so here's my answer: This is an assignment question taken from MATH324. I would appreciate it if you take this question down.
– YourLecturer
Aug 10 at 2:18




I do not have enough reputation to comment, so here's my answer: This is an assignment question taken from MATH324. I would appreciate it if you take this question down.
– YourLecturer
Aug 10 at 2:18















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