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Symplectic group over F2

  1. link
  2. notation
    • \(\mathbb{F}_n\): finite field
    • \(\mathbb{R}\): real field
    • \(\mathbb{C}\): complex field
    • \(U(n)=\lbrace x\in\mathbb{C}^{n\times n}:x^\dagger x=I_n \rbrace\): unitary group
    • \(\Lambda_n=\sigma_x\otimes I_n\)

Symplectic vector space

\[x,y\in \mathbb{F}_2^{2n}\Rightarrow \langle x,y\rangle=\sum_{i=1}^n x_iy_{i+n}+x_{i+n}y_i\pmod{2}\]
  1. \(x\in\mathbb{F}_2^{2n},\langle x,x\rangle=0\)

Symplectic pair vector set

\[ S_{n}=\lbrace \left(x,y\right)\in\mathbb{F}_{2}^{2n}\times\mathbb{F}_{2}^{2n}:\langle x,y\rangle=x^{T}\Lambda_{n}y=1\rbrace \]

  1. order of the set

    \[ \left|S_{n}\right|=10\left|S_{n-1}\right|+6\left(16^{n-1}-\left|S_{n-1}\right|\right)=2^{2n-1}\times\left(4^{n}-1\right) \]

Symplectic group

\[ Sp\left(2n,\mathbb{F}_{2}\right)=\lbrace x\in\mathbb{F}_{2}^{2n\times2n}:x\Lambda_{n}x^{T}=\Lambda_{n}\rbrace \]
  1. \(x\in Sp\left(2n,\mathbb{F}_{2}\right)\Rightarrow x^{T}\in Sp\left(2n,\mathbb{F}_{2}\right)\)
  2. \(x,y\in Sp\left(2n,\mathbb{F}_{2}\right)\Rightarrow xy\in Sp\left(2n,\mathbb{F}_{2}\right)\)
  3. \(x\in Sp\left(2n,\mathbb{F}_{2}\right)\Rightarrow x^{-1}=\Lambda_{n}x^{T}\Lambda_{n}\)

  4. order of the group

    \[ \left|Sp\left(2n,\mathbb{F}_{2}\right)\right|=\prod_{i=1}^{n}\left|S_{i}\right|=\prod_{i=1}^{n}(4^{i}-1)2^{2n-1} \]

  5. there exists a one-to-one mapping: \(x\in Sp\left(2n,\mathbb{F}_{2}\right),x\mapsto\left(a_{1},b_{1},a_{2},b_{2},\cdots,a_{n},b_{n}\right)\)

    • \(0\leq a_{i}<4^{i}-1\)
    • \(0\leq b_{i}<2^{2i-1}\)
    • the mapping is constructed using transvection, details see paper doi-link

transvection

\[h\in\mathbb{F}_{2}^{2n},T_{h}\left(x\right)=T_{h}x=x+\langle x,h\rangle h:\mathbb{F}_{2}^{2n} \mapsto\mathbb{F}_{2}^{2n}\]
  1. \(\forall h\in\mathbb{F}_{2}^{2n},T_{h}T_{h}x=x=T_{0}x\)
  2. \(\forall x,y\in\mathbb{F}_{2}^{2n}\setminus\lbrace 0\rbrace \Rightarrow\exists a,b\in\mathbb{F}_{2}^{2n},y=T_{b}T_{a}x\)