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Someone please explain what is the meaning of the words in shade

From Proof from the Book, 4th edition page 96:

Let $q$ be a prime power, set $n=4q-2$, and let

$$Q = \{x \in \{+1,-1\}^n: x_1 = 1, \#\{i:x_i=-1\} \text{ is even}\}.$$

This $Q$ is a set of $2^{n-2}$ vectors in $\Bbb R^n$.

We will see that $\langle x,y \rangle \equiv 2 \pmod 4$ holds for all vectors $x,y \in Q$.

Remark: These 4 sentences came together.

And I need to understand what is $|\langle x,y \rangle|$ is in.

We will call $x,y$ nearly orthogonal if $|\langle x,y \rangle|=2$.

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"the book"? I guess there is only one... – Arturo Magidin Jul 20 '12 at 22:08
Do you mean, "Proofs from the Book", 4th Edition? – Arturo Magidin Jul 20 '12 at 22:13
@ArturoMagidin - Yes, and thank you again for your answer below – Victor Jul 20 '12 at 22:24
up vote 3 down vote accepted

$\langle x,y\rangle$ denotes the result of applying the inner product of $\mathbb{R}^n$ to the vectors $x$ and $y$ (which happen to be in $Q$); in this case, it is the usual "dot product". $|\langle x,y\rangle|$ denotes the absolute value of that operation.

E.g., $q=2$, $n=6$, $x=(1,-1,1,1,-1,1)$, $y = (1,1,-1,1,1,-1)$, then $$\langle x,y\rangle = (1)(1) + (-1)(1) + (1)(-1) + (1)(1) + (-1)(1) + (1)(-1) = -2$$ so $|\langle x,y\rangle| = |-2| = 2$.

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What is $\#\{i:x_i=-1\} \text{ is even}\$ mean? – Victor Jul 20 '12 at 22:11
@Victor: If you have multiple questions about the notation, why not put them all in the question in the first place? It means "the cardinality" ($\#$) "of the set of indices corresponding to entries equal to $-1$". In other words, you are looking at vectors in $\mathbb{R}^n$, all of whose entries are either $1$ or $-1$, and such that the number of entries that are equal to $-1$ is even. – Arturo Magidin Jul 20 '12 at 22:13
Appreciate to Arturo Magidin – Victor Jul 20 '12 at 22:21

$Q = \{x \in \{+1,-1\}^n: x_1 = 1, \#\{i:x_i=-1\} \text{ is even}\}.$

$Q$ is the set of all vector $x$ with all of the following conditions:

  1. $x$ is a $n$-dimensional vector with entries $+1$ or $-1.$

  2. The first entry in, $x_1 = 1.$

  3. The number of indices $i$ where $x_i = -1$ is even. The cardinality of such set is even. i.e., the number of $-1$ component in the vector is even.

This $Q$ is a set of $2^{n-2}$ vectors in $\Bbb R^n$.

$Q \subset \Bbb R^2,$ and the number of vectors in $Q$ is $2^{n-2}.$

We will see that $\langle x,y \rangle \equiv 2 \pmod 4$ holds for all vectors $x,y \in Q$.

The inner product between any two vectors in $Q$ is congruent to $2$ modulo 4.

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