CODES ET RÉSEAUX

Rn
R C
K
2

nK
2X1, . . . , Xn
f(X1, . . . , Xn) = P1≤i≤j≤nαi,j XiXj
i < j
αi,j XiXj=αi,j
2(XiXj+XjXi).
f(X1, . . . , Xn) =
Pn
i,j=1 ai,j XiXji6=j ai,j =aj,i f
Mf= (ai,j )
f(X1, . . . , Xn) = (X1, . . . , Xn)Mf
X1
Xn
.
Mff(X1, X2, X3) = X2
1+
2X2
2+ 4X2 3+X1X2+ 2X1X3+X2X3+ 3X3X2
f
˜
f:Kn→Kx= (x1, . . . , xn)7→ f(x1, . . . , xn)
∀λ∈K˜ f(λx) = λ2˜ f(x)
b˜
f:Kn×Kn→Kb˜ f(x, y) = 1
2(˜
f(x+y)−˜ f(x)−˜ f(y))
f2
b˜
f(x, y) = 1
2(x1, . . . , xn)Mf
y1
yn
+ (y1, . . . , yn)Mf
x1
xn
=Pn
i,j=1 ai,j xiyj= (x1, . . . , xn)Mf
y1
yn
.
b˜
fx y
f b ˜
f
b˜
f(x, y) = x1y1+ 2x2y2+ 4x3y3+1
2(x1y2+x2y1)+(x1y3+x3y1) + 2(x2y3+x3y2).
FqXq−X
0

f
˜
f˜ f b ˜
f
Mf= (ai,j )f(e1, . . . , en)
Knai,j =b˜
f(ei, ej)
K(V, q)
KV q :V→K
∀x∈V λ ∈Kq(λx) = λ2q(x)
bq:V×V→Kbq(x, y) = 1
2(q(x+y)−q(x)−q(y))
f2n
(Kn,˜ f)K
(e1, . . . , en)V bq
x=x1e1+···+xnenq(x) = bq(x1e1+···+xnen, x1e1+···+xnen) =
Pn
i,j=1 xixjbq(ei, ej)q
2
(x, y)7→ bq(x, y)
V q
q bqq(x) = bq(x, x)
K
2
(V, q) (V0, q0)K
(V, q) (V0, q0)f:V→V0
q0◦f=q
bq0(f(x), f(y)) = bq(x, y)
q q0
(V, q) (V0, q0)

C R Q
(e1, . . . , en)V q
B= (bq(ei, ej))
˜
f2
(Kn,˜ f)Mf§
B q (e1, . . . , en)X
Y x y ∈V
q(x) = tXBX bq(x, y) = tXBY.
bqB x =
x1e1+··· +xneny=y1e1+··· +ynenbq(x, y) = Pn
i,j=1 xiyjbq(ei, ej)
§
˜
f
E E0V
PE E0E
(e0
1, . . . , e0 n)E0x∈V
X X0E E0X=P X0
B0qE0
B q EB0=tP BP.
bq(x, y) = tX0B0Y0=tXBY =tX0tP BP Y 0
X0Y0tX0B0Y0=tX0tP BP Y 0
P Q Mn(K)
X Y tXP Y =tXQY
tXP Y =tXQY Kn
P Q
BtP BP

(V, q)KB
EV(V0, q0)B0E0
f:V→V0
E E0A f
B=tAB0A
f x
y V b0(f(x), f(y)) = b(x, y)
X Y t(AX)B0(AY ) = tXBY tX(tAB0A)Y=tXBY
tAB0A=B
(V, q) (V0, q0)
q q0
q q0
f:V→V0
E= (e1, . . . , en)V(f(e1), . . . , f(en))
V0bq0(f(ei), f(ej)) = bq(ei, ej)
⇒
§B B0B0=tP BP
P f :V→V0
E E0P−1
q q0P−1f
(V, q)
(V, q)GL(V)
V
EV B b E
GL(V)GLn(K)Mn(K)
O(V, q){P∈GLn(K),tP BP =B}
x y ∈V x ⊥y
bq(x, y) = 0
x U ⊂V U
U W ⊂V U ⊥W
U⊂V V U
U⊥={x∈V, ∀u∈U bq(x, u) = 0}.