By N. Boccara

ISBN-10: 2729896058

ISBN-13: 9782729896058

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**Additional resources for Analyse Fonctionnelle. Une introduction pour physiciens**

**Example text**

If the entry in the corresponding coordinate is 1, we include that element in the subset; if the entry is 0 we will not include it. For example, 0011 corresponds to the subset {3, 4}, 1001 to the subset {1, 4}, 1111 to the total set {1, 2, 3, 4} and 0000 to the empty set. We see that the bitstrings of length n and weight m correspond precisely to the subsets of m elements of a fixed set with n elements. 4. Consider paths starting at the top of the triangle, where in each step the choice is between going southeast or southwest.

It indicates that this tiny structure of only two elements has something in common with the field of real numbers and the complex number field: each of these structures satisfies the same set of axioms. Each of them is a field. We will come back to this topic later. The idea behind the addition in F2 is to model the errors in information transmission: a coordinate of a message is in error if and only if 1 is added to the entry in this coordinate. This explains why we must have 1 + 1 = 0 : if a bit is flipped twice, then no error occurs, or: if 1 is flipped, then 0 is received.

5, it suffices to check that each of the 15 nonzero codewords has weight ≥ 3 to check that indeed d = 3. Here is a generator matrix: 10 0 1 0 0 00 01 01 10 01 10 01 11 11 1 1 1 0 How can we be sure? At first we check that each row of the matrix is a nonzero codeword. One method of controlling that we have a generator matrix is to make sure that each codeword is indeed a linear combination of rows, equivalently that different linear combinations of rows yield different codewords. We can speed up by using basic facts of linear algebra.

### Analyse Fonctionnelle. Une introduction pour physiciens by N. Boccara

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