I can't understand the notation $\mathbb{Z}_{q}^{n} \times \mathbb{T}$ as defined below. As far as I know $\mathbb{Z_{q}}$ comprises all integers modulo $q$. But with $n$ as a power symbol I can't understand it. Also: $\mathbb{R/Z}$, what does it denote?

"... $ \mathbb{T} = \mathbb{R}/\mathbb{Z} $ the additive group on reals modulo one. Denote by $ A_{s,\phi} $ the distribution on $ \mathbb{Z}^n_q \times \mathbb{T}$ ..."


2 Answers 2

  • $\mathbb{Z}_q^n$ means the vector space of lenght $n$ over $\mathbb{Z}_q$
  • $\times$ is for cartesian product
  • $\mathbb{R}/\mathbb{Z}$ means the set $\mathbb{R}$ modulus the set $\mathbb{Z}$ so in this specific case it means the set $\{ x | x \in \mathbb{R}\wedge 0\leq x < 1\}$
  • 4
    $\begingroup$ Technically, $\mathbb R / \mathbb Z$ is the set of equivalence classes of the form $[x] = x + \mathbb Z = \{\dotsc, x-2, x-1, x, x+1, x+2, \dotsc\}$ for all $x \in \mathbb R$. But it's true that identifying each equivalence class with its smallest positive (or non-negative) representative is a commonly adopted convention, since it makes formulas look a lot nicer. $\endgroup$ Jun 27, 2012 at 14:32
  • $\begingroup$ Modulo $\Bbb Z$... $\endgroup$
    – anon
    Jun 27, 2012 at 15:59
  • 5
    $\begingroup$ To add a little more technical trivia to Ilmari Karonen's comment, $\mathbb Z_q$ is not a field unless $q$ is a prime, and so $\mathbb Z_q^n$ would in general be called a module and not a vector space. However, regardless of whether it is a vector space or a module, $\mathbb Z_q^n$ consists of the $q^n$ $n$-tuples with elements from $\mathbb Z_q$. $\endgroup$ Jun 27, 2012 at 16:32

You write "afaik $\mathbb Z_q$ comprises..." You have to be careful here what is meant by this notation. There are two common options:

1) $\mathbb Z_q$ is the ring of integers module $q$. Many people think this should be better written as $\mathbb Z/q$, to avoid confusion wth 2). However it is not uncommon to write $\mathbb Z_q$.

2) The ring of $q$-adic integers, i.e. formal powerseries in $q$ with coefficients in $\mathbb Z\cap [0,q-1]$.


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