Simulation d`un code de Hamming (4,7)

(4,7)
7{0,1}
4
𝔽4
2
𝛼
→𝔽7
2
𝑎𝑏𝑐𝑑 7→ 𝑎𝑏𝑐𝑑𝑥𝑦𝑧
⎧
⎨
⎩
𝑥=𝑏+𝑐+𝑑
𝑦=𝑎+𝑐+𝑑
𝑧=𝑎+𝑏+𝑑
𝑥, 𝑦, 𝑧
(4,7) 𝐺 𝛼
𝔽4
2𝔽7 2
𝐺
3
𝐻(3,7)
𝐻=⎛
⎝
1010101
0110011
0001111
⎞
⎠.
𝐺𝑡 𝐻= 0
𝑡
𝐻𝔽7
2
1𝔽3
2
𝑚𝔽4
2𝛼(𝑚) = 𝑚𝐺 𝑚𝐺 +𝑒
𝑒(𝑚𝐺+𝑒)𝑡 𝐻=𝑒𝑡 𝐻
1
𝑒
𝑝
𝑘
𝑏0𝑏1...𝑏𝑘−1∈𝐶⇔𝑏1𝑏2...𝑏𝑘−1𝑏0∈𝐶
𝑎, 𝑏, 𝑐, 𝑑
𝑏0𝑏1𝑏2...𝑏6𝑏0+𝑏1𝑋+⋅ ⋅ ⋅ +𝑏6𝑋6
𝑋7+ 1
𝔽2[𝑋]/(𝑋7+ 1)