Section II.5.3

Projective spaces

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Problem II.5.3.1

The famous Hamming code HH of length 7 is based on a configuration of seven points and seven lines of the projective plane F2P2(7=22+2+1)\mathbb {F}_{2} \mathbb {P}^{2}\left(7=2^{2}+2+1\right). This configuration is depicted schematically in Fig. 21.

In each row of the incidence matrix JJ there stand three 1's: they correspond to three points on one line. Each row also serves as a codeword in HH of weight 3. The codewords of weight 4 make up the matrix J~\widetilde{J};

J=∥0110100001101000011011000110010001110100011101000∥,J~=∥1001011110010111100100111001101110001011100010111∥ J=\left\Vert \begin{smallmatrix} 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 \\ 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 \end{smallmatrix}\right\Vert , \quad \widetilde{J}=\left\Vert \begin{smallmatrix} 1 & 0 & 0 & 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 1 & 1 \end{smallmatrix}\right\Vert

Fig. 21 and Fig. 22

The matrix J~\tilde{J} is obtained from JJ by replacing 0 with 1 and 1 with 0. In all, HH has 16 codewords. We have written out 14. Here are the remaining two: (0000000), (1111111).

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(a)

Verify that the code HH is precisely a four-dimensional vector subspace of F27\mathbb {F}_{2}^{7}, defined by the linear system

x1+x3+x4+x5=0,x1+x2+x3+x6=0,x2+x3+x4+x7=0x_{1}+x_{3}+x_{4}+x_{5}=0, \quad x_{1}+x_{2}+x_{3}+x_{6}=0, \quad x_{2}+x_{3}+x_{4}+x_{7}=0

(each equation involves the variables lying in one of the three circles in Fig. 22).

(b)

Prove that PGL(F2P2)P G L\left(\mathbb {F}_{2} \mathbb {P}^{2}\right) is a nonabelian group of order 168, which can be realized by permutations of degree 7, acting on the set of points of the projective plane.

(c)

Prove that

PGL(F2P2)=Aut⁡(H):={σ∈S7∣σ(H)=H} P G L\left(\mathbb {F}_{2} \mathbb {P}^{2}\right)=\operatorname {Aut}(H):=\left\{ \sigma \in S_{7} \mid \sigma (H)=H\right\}
Problem II.5.3.2

Let Π=F2P2\Pi =\mathbb {F}_{2} \mathbb {P}^{2} be the projective plane consisting of seven points, and let p~,q~\widetilde{p}, \widetilde{q} be two distinct points of the plane Π\Pi. How many automorphisms are there taking p~\widetilde{p} to q~\widetilde{q}?

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Problem II.5.3.3

Prove that over the field C\mathbb {C} every projective transformation has at least one fixed point.

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