Question: Prove that, for a linear code C, either all the code vectors have even weight or exactly half of them do. [Let E be the

Prove that, for a linear code C, either all the code vectors have even weight or exactly half of them do. [Let E be the set of vectors in C with even weight and O the set of vectors in C with odd weight. If O is not empty, let c_o be in O and consider O' = {c_o + e : e in E}. Show that O' = O.]

Step by Step Solution

★★★★★

3.32 Rating (167 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

As suggested We will show O c 0 e e E O vectors in C with odd weight wc 0 e wc 0 we 2 o... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (2 attachments)

1623_606abeddab84e_684562.pdf

180 KBs PDF File

1623_606abeddab84e_684562.docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

For a linear code C, show that (C ) without using matrices.

If C is an (n, k) linear code that is self dual, prove that n must be even. [Use the analogue in Z n 2 of Theorem 5.13.]

Prove that for a random variable X with mean X , Where n is any positive integer. Pr(|X- u > e)

Coding Theory (14 points) Let n be a positive integer and let C En, ie., let C be the binary code consisting of all length n binary vectors of even weight: (a) Find a parity check check matrix H for...

Set Student Name: 1. Describe the relationship between two variables that have a correlation coefficient value: a. Near -1 b. Near 0 c. Near 1 2. Data was collected where a weightlifter was asked to...

Let i and j be positive integers. (i) Prove that there exist natural numbers a and b such that ai = bj+gcd(i, j). You may use standard results provided that you state them clearly. [4 marks] (ii) Let...

seventh pages Chapter 3 Curve Sketching How much metal would be required to make a 400-mL soup can? What is the least amount of cardboard needed to build a box that holds 3000 cm3 of cereal? The...

Instuctor's Annotated Edition TENTH EDITION Understandable Statistics Concepts and Methods Charles Henry Brase Regis University Corrinne Pellillo Brase Arapahoe Community College Australia Brazil...

4 This assignment, based on the content of Unit 4 of the Study Guide, is worth 3% of your grade. We recommend that you hand it in after you complete Unit 4. You must show all of your work in order to...

\fCOMPLEX VARIABLES AND APPLICATIONS Eighth Edition James Ward Brown Professor of Mathematics The University of Michigan-Dearborn Ruel V. Churchill Late Professor of Mathematics The University of...

The last census of the entire community from which the sample in Exercise 7 was taken was 2010 when the mean loss was $10,200. From the sample in Exercise 7, use a t-test to test whether the mean...

What are some criteria for defining report problems?

Onnt Item 2 , Todd Company acquired 4 0 % of the outstanding stock of MCGuire Company for On January 2 , Todd Company acquirember 3 1 , McGuire earned income of $ 4 8 , 0 0 0 and paid dividends of $...

Task 1: A student was interested in whether there was a positive relationship between the time spent doing an essay and the mark received. He got 45 of his friends and timed how long they spent...

A square matrix is strictly diagonally dominant if the absolute value of each diagonal entry is greater than the sum of the absolute values of the remaining entries in that row. Use Gerschgorin's...

If A is an n Ã n matrix, let ||A|| denote the maximum of the sums of the absolute values of the rows of A; that is, Allmax

Let be an eigenvalue of a stochastic matrix A (see section 3.7). Prove that | | 1.

Task (2): Consider that the geometry of the arc for billboard for the sport club is following the hyperbolic equation: y= 10 - 0. 658 (x - 4. 025). It is required to attach two torches for each...

Hickock Mining is evaluating when to open a gold mine. The mine has 39,000 ounces of gold left that can be mined, and mining operations will produce 6,500 ounces per year. The required return on the...

Question 39 Here is a sample of amounts of weight change(kg) of college students in their freshmanyear: 12, 6, 3, 4, where 4 represents a loss of 4 kg and positive values represent weight gained....