hw3 - Error Correcting Codes Combinatorics Algorithms and...

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Error Correcting Codes: Combinatorics, Algorithms and Applications Spring 2011 Homework 3 Due Monday, April 4, 2011 in class You can collaborate in groups of up to 3. However, the write-ups must be done individually, that is, your group might have arrived at the solution of a problem together but everyone in the group has to write up the solution in their own words. Further, you must state at the beginning of your homework solution the names of your collaborators. Just to be sure that there is no confusion, the group that you pick has to be for all problems [i.e. you cannot pick different groups for different problems :-)] If you are not typesetting your homework, please make sure that your handwriting is legible. Illeg- ible handwriting will most probably lose you points. Unless stated otherwise, for all homeworks, you are only allowed to use notes from the course: this includes any notes that you might have taken in class or any scribed notes from Fall 07 or Spring 09 version or the current version of the course. Doing otherwise will be considered cheating. Note that if your collaborator cheats and you use his solution, then you have cheated too (ignorance is not a valid excuse). Please use the comment section of the post on HW 3 on the blog if you have any questions and/or you need any clarification. You might find Problem 2 in HW 0 useful for this homework. You can use any statement from HW 0 without proof. In total you can use at most seven pages for this homework. I encourage you to start thinking on the problems early . 1. ( Alternate definition of codes ) (10 + 5 = 15 points ) (a) We have defined Reed-Solomon in class. In this problem you will prove that a certain alternate definition also suffices. Consider the Reed-Solomon code over a field F of size q and block length n = q - 1 defined as C 1 = { ( p (1) , p ( α ) , . . . , p ( α n - 1 )) | p ( X ) F [ X ] has degree k - 1 } where α is the generator of the multiplicative group F * of F .

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