Toric Varieties Hirzebruch Surfaces and error-correcting Codes
Toric Varieties Hirzebruch Surfaces and error-correcting Codes Johan P. Hansen, [email protected] The Mathematical Institute, University of Aarhus, Denmark IV International Algebraic Conference in Ukraine Lviv, august 2003
0-0
Johan P. Hansen, Toric Varieties Hirzebruch Surfaces and Error-Correcting Codes, Applicable Algebra in Engineering, Communication and Computing, Springer-Verlag, Volume 13, Number 4/December 2002, Pages: 289 - 300 Resum´ e For any integral convex polytope in R2 there is an explicit construction of an error-correcting code of length (q − 1)2 over the finite field Fq , obtained by evaluation of rational functions on a toric surface associated to the polytope. The dimension of the code is equal to the number of integral points in the given polytope and the minimum distance is determined using the cohomology and intersection theory of the underlying surfaces. In detail we treat Hirzebruch surfaces.
1
Construction of Toric codes Let M ' Z2 be a Z-module af rank 2 over the integers Z. Let ¤ be a integral convex polytope in MR = M ⊗Z R. Example. Polytope with vertices (0, 0), (d, 0), (d, e + rd), (0, e). q-1
e
d
q-1
2
Toric Codes ξ ∈ Fq a primitive element. Pij = (ξ i , ξ j ) ∈ Fq ∗ × Fq ∗ ,
i = 0, . . . , q − 1; j = 0, . . . , q − 1.
m1 , m2 a Z-basis for M . For m = λ1 m1 + λ2 m2 ∈ M ∩ ¤: e(m)(Pij ) = (ξ i )λ1 (ξ j )λ2 . The toric code C¤ is the linear code of length n = (q − 1)2 generated by: {(e(m)(Pij ))i=0,...,q−1;j=0,...,q−1 | m ∈ M ∩ ¤}. The functions in the Fq -vectorspace L = Span{e(m)|m ∈ M ∩ ¤} are evaluated in the points Pij on the torus Fq ∗ × Fq ∗ : 2
φ : L = Span{e(m)|m ∈ M ∩ ¤}
→
F(q−1)
f
7→
(f (Pij )i=0,...,q−1;j=0,...,q−1 )
3
Theorem Let ¤ be the polytope in MR with vertices (0, 0), (d, 0), (d, e + rd), (0, e) as above. Assume d < q − 1, e < q − 1 and e + rd < q − 1. The toric code C¤ has • length (q − 1)2 • dimension #(M ∩ ¤) = (d + 1)(e + 1) + r d(d+1) (the number of lattice 2 points in ¤) • minimal distance (the minimal number of nonzero entries in a codeword different from zero) Min{(q − 1 − d)(q − 1 − e), (q − 1)(q − 1 − e − rd)}.
4
Rate and relative minimal distance (q = 32) 1
0.8
0.6
0.4
0.2
0
x=
0.2 0.4 0.6 0.8 dimension minimaldistance , y = , (q length length
5
1
= 32)
Toric varieties - support functions Let M be the lattice M ' Z2 . Let N = HomZ (M, Z) be the dual lattice with Z - bilinear parring < ,
>: M × N → Z.
Let ¤ be a 2-dimensional integral convex polytope in MR = M ⊗Z R. Then there is a support function h¤ : NR = N ⊗Z R → R h¤ (n) := inf{< m, n > | m ∈ ¤} such that ¤ can be reconstructed as: ¤h = {m ∈ M | < m, n > ≥ h(n) ∀n ∈ N }. The support function is piecewise linear: NR is the union of finitely many polyhedral cones in NR and h¤ is linear on each cone. 6
Hirzebruch surfaces - the toric surfaces asociated to the polyhedra in our example N as union of polyhedral cones in our example q-1
e
d
q-1
Generators for the 1-dimensional cones are: 1 0 −1 r n(ρ1 ) = , n(ρ2 ) = , n(ρ3 ) = , n(ρ4 ) = 0 1 0 −1 . 7
Toric variety - definition ∗
∗
∗
TN := HomZ (M, Fq ) w Fq × Fq is a 2-dimensional algebraic torus. ∗
e(m) : T → Fq , m ∈ M defined as e(m)(t) = t(m) for t ∈ TN is a multiplicative character. The toric surface X¤ associated to ¤ is X¤ = ∪σ∈∆ Uσ Uσ are the Fq -valued points on the affine scheme Spec(Fq [Sσ ]), that is Uσ = {u : Sσ → Fq |u(0) = 1, u(m + m0 ) = u(m)u(m0 ) ∀m, m0 ∈ Sσ }, where Sσ is the additive subsemigroup of M Sσ = {m ∈ M | < m, y >≥ 0∀y ∈ σ}. X¤ is irreducible, (smooth) and complete.
8
TN acts on X¤ . On u ∈ Uσ the element t ∈ TN acts in the following way: (tu)(m) := t(m)u(m) m ∈ Sσ For σ ∈ ∆ ∗
orb(σ) := {u : M ∩ σ → Fq |u is a group homomorphism} ia a TN orbit X¤ . V (σ) is defined to be the closure of orb(σ) in X¤ . A ∆-linear support function h gives rise to a Cartier divisor Dh : X Dh := − h(n(ρ)) V (ρ) ρ∈∆(1)
Dm = div(e(−m))
m ∈ M,
where ∆(1) are the 1-dimensional cones in ∆ and n(ρ) is a generator for the 1-dimensional cone ρ. Lemma 1. The vector space H0 (X, OX (Dh )) af globale sections of OX (Dh ) has dimension #(M ∩ ¤h ) and {e(m)|m ∈ M ∩ ¤h } is a basis. 9
q-1
e
d
q-1
Generators for the 1-dimensional cones are: 1 0 −1 r n(ρ1 ) = , n(ρ2 ) = , n(ρ3 ) = , n(ρ4 ) = 0 1 0 −1 . X Dh := − h(n(ρ)) V (ρ) = d V (ρ3 ) + e V (ρ4 ) ρ∈∆(1)
dim H0 (X, OX (Dh )) = (d + 1 )(e + 1 ) + r 10
d (d + 1 ) . 2
Toric surfaces - Intersection theory Let Dh be a Cartier divisor and let ¤h be the corresponding polytope. Then (Dh ; Dh ) = 2 vol2 (¤h ), where vol2 is the normalized Lesbesgue measure. I our example we get the intersection table V (ρ1 )
V (ρ2 )
V (ρ3 )
V (ρ4 )
V (ρ1 )
−r
1
0
1
V (ρ2 )
1
0
1
0
V (ρ3 )
0
1
r
1
V (ρ4 )
1
0
1
0
11
Result on Hirzebruch surfaces Sætning 1. Let ¤ be the polytope in MR with vertices (0, 0), (d, 0), (d, e + rd), (0, e). Assume d < q − 1, e < q − 1 and e + rd < q − 1. The toric code C¤ has • length (q − 1)2 • dimension #(M ∩ ¤) = (d + 1)(e + 1) + r d(d+1) (the number of lattice 2 points in ¤) • minimal distance (the minimal number of nonzero entries in a codeword different from zero) Min{(q − 1 − d)(q − 1 − e), (q − 1)(q − 1 − e − rd)}.
12
∗
∗
Bevis. For t ∈ T ' Fq × Fq , the rationale functions i H0 (X, OX (Dh )) are evaluated 0
H (X, OX (Dh )) → f
7→
Fq
∗
f (t).
Let H0 (X, OX (Dh ))Frob be the Frobenius invariante functions in H0 (X, OX (Dh )) (functions that are Fq − linear combinations of (e)(m)). Evaluating in all points in T (Fq ) gives the code C¤ : H0 (X, OX (Dh ))Frob
→
C¤ ⊂ (Fq ∗ )]T (Fq )
f
7→
(f (t))t∈T (Fq )
og generators for the code are the images of the basis functions e(m) 7→ (e(m)(t))t∈T (Fq ) . ∗
∗
Let m1 = (1, 0). The Fq -rationale points on T ' Fq × Fq are on the q − 1 Q lines on X¤ given by the equation η∈Fq (e(m1 ) − η) = 0. 13
Let 0 6= f ∈ H0 (X, OX (Dh )) and assume that f is identically zero on precisely a of these lines. As e(m1 ) − η and e(m1 ) have the same pole-divisor, they have equivalent divisors of zeros: (div(e(m1 ) − η))0 ∼ (div(e(m1 )))0 . Therefore div(f ) + Dh − a(div(e(m1 )))0 ≥ 0 or equivalently f ∈ H0 (X, OX (Dh − a(div(e(m1 )))0 ). This implies that a ≤ d according to Lemma 1 on cohomology.
14
On any of the q − 1 − a other lines the number of zeros for f is at most the intersection number: (Dh − a(div(e(m1 )))0 ; (div(e(m1 )))0 ). This is determined using the intersection table and the observation (div(e(m1 )))0 = V (ρ1 ) + rV (ρ4 ). We get (Dh − a(div(e(m1 )))0 ; (div(e(m1 )))0 ) = e + (d − a)r. As 0 ≤ a ≤ d, we conclude that the totale number of (rational) zeros for f is at most a(q − 1) + (q − 1 − a)(e + (d − a)r) ≤ max{d(q − 1) + (q − 1 − d)e, (q − 1)(e + dr)}. Therefore H0 (X, OX (Dh ))Frob
→
C¤ ⊂ (Fq ∗ )]T (Fq )
f
7→
(f (t))t∈T (Fq )
and the dimension and the lower bound for the minimal distance as claimed in the theorem is obtained. 15
Now we will see that we have determined the true minimal distance. Let b1 , . . . , be+rd ∈ Fq ∗ be pairwise distinct. The function xd (y − b1 ) · · · · · (y − be+rd ) ∈ H0 (X, OX (Dh ))Frob is zero in the (q − 1)(e + rd) points (x, bj ), x ∈ Fq ∗ ,
j = 1, . . . , e + rd
and gives a codeword of weight (q − 1)2 − (q − 1)(e + rd) = (q − 1)(q − 1 − (e + rd)). Let a1 , . . . , ad ∈ Fq ∗ be pairwise distinct and let b1 , . . . , be ∈ Fq ∗ be pairwise distinct. The function (x − a1 ) · · · · · (x − ad )(y − b1 ) · · · · · (y − be ) ∈ H0 (X, OX (Dh ))Frob is zero in the d(q − 1) + (q − 1)e − de points (ai , y), (x, bj ),
x, y ∈ Fq ∗ , i = 1, . . . e, j = 1, . . . , d
and gives a codeword of weight (q − 1 − d)(q − 1 − e). 16
Litteratur [1]
W. Fulton, “Introduction to Toric Varieties,”Annals of Mathematics Studies; no. 131, Princeton University Press, 1993.
[2] W. Fulton, “Intersection theory”Ergebnisse der Mathematik und uhrer Grenzgebiete, 3. Folge, Springer Verlag, 1998. [3]
J. P. Hansen, “Toric Surfaces and Error-correcting codes,”in Coding theory, cryptography and related areas (Guanajuato, 1998), 132-142, Springer, Berlin, 2000
[4]
J. P. Hansen, “Hirzebruch surfaces and Error-correcting Codes,”Preprint Series No. 6, 2000, University of Aarhus.
[5] J. P. Hansen, “Toric Varieties Hirzebruch Surfaces and Error-Correcting Codes,”Applicable Algebra in Engineering, Communication and Computing, Springer-Verlag, Volume 13, Number 4/December 2002, Pages: 289 300
16-1
[6]
Hansen, Søren Have, “Error-correcting codes from higher-dimensional varieties,”Finite Fields Appl.; no 7, 2001, 531–552
[7] Joyner, David, “Toric codes over finite fields,”Preprint, Aug. 2002, http://front.math.ucdavis.edu/math.AG/0208155 [8] T. Oda, “Convex Bodies and Algebraic Geometry, An Introduction to the Theory of Toric Varieties,”Ergebnisse der Mathematik und uhrer Grenzgebiete, 3. Folge, Band 15, Springer Verlag, 1985.
16-2
0-0
Johan P. Hansen, Toric Varieties Hirzebruch Surfaces and Error-Correcting Codes, Applicable Algebra in Engineering, Communication and Computing, Springer-Verlag, Volume 13, Number 4/December 2002, Pages: 289 - 300 Resum´ e For any integral convex polytope in R2 there is an explicit construction of an error-correcting code of length (q − 1)2 over the finite field Fq , obtained by evaluation of rational functions on a toric surface associated to the polytope. The dimension of the code is equal to the number of integral points in the given polytope and the minimum distance is determined using the cohomology and intersection theory of the underlying surfaces. In detail we treat Hirzebruch surfaces.
1
Construction of Toric codes Let M ' Z2 be a Z-module af rank 2 over the integers Z. Let ¤ be a integral convex polytope in MR = M ⊗Z R. Example. Polytope with vertices (0, 0), (d, 0), (d, e + rd), (0, e). q-1
e
d
q-1
2
Toric Codes ξ ∈ Fq a primitive element. Pij = (ξ i , ξ j ) ∈ Fq ∗ × Fq ∗ ,
i = 0, . . . , q − 1; j = 0, . . . , q − 1.
m1 , m2 a Z-basis for M . For m = λ1 m1 + λ2 m2 ∈ M ∩ ¤: e(m)(Pij ) = (ξ i )λ1 (ξ j )λ2 . The toric code C¤ is the linear code of length n = (q − 1)2 generated by: {(e(m)(Pij ))i=0,...,q−1;j=0,...,q−1 | m ∈ M ∩ ¤}. The functions in the Fq -vectorspace L = Span{e(m)|m ∈ M ∩ ¤} are evaluated in the points Pij on the torus Fq ∗ × Fq ∗ : 2
φ : L = Span{e(m)|m ∈ M ∩ ¤}
→
F(q−1)
f
7→
(f (Pij )i=0,...,q−1;j=0,...,q−1 )
3
Theorem Let ¤ be the polytope in MR with vertices (0, 0), (d, 0), (d, e + rd), (0, e) as above. Assume d < q − 1, e < q − 1 and e + rd < q − 1. The toric code C¤ has • length (q − 1)2 • dimension #(M ∩ ¤) = (d + 1)(e + 1) + r d(d+1) (the number of lattice 2 points in ¤) • minimal distance (the minimal number of nonzero entries in a codeword different from zero) Min{(q − 1 − d)(q − 1 − e), (q − 1)(q − 1 − e − rd)}.
4
Rate and relative minimal distance (q = 32) 1
0.8
0.6
0.4
0.2
0
x=
0.2 0.4 0.6 0.8 dimension minimaldistance , y = , (q length length
5
1
= 32)
Toric varieties - support functions Let M be the lattice M ' Z2 . Let N = HomZ (M, Z) be the dual lattice with Z - bilinear parring < ,
>: M × N → Z.
Let ¤ be a 2-dimensional integral convex polytope in MR = M ⊗Z R. Then there is a support function h¤ : NR = N ⊗Z R → R h¤ (n) := inf{< m, n > | m ∈ ¤} such that ¤ can be reconstructed as: ¤h = {m ∈ M | < m, n > ≥ h(n) ∀n ∈ N }. The support function is piecewise linear: NR is the union of finitely many polyhedral cones in NR and h¤ is linear on each cone. 6
Hirzebruch surfaces - the toric surfaces asociated to the polyhedra in our example N as union of polyhedral cones in our example q-1
e
d
q-1
Generators for the 1-dimensional cones are: 1 0 −1 r n(ρ1 ) = , n(ρ2 ) = , n(ρ3 ) = , n(ρ4 ) = 0 1 0 −1 . 7
Toric variety - definition ∗
∗
∗
TN := HomZ (M, Fq ) w Fq × Fq is a 2-dimensional algebraic torus. ∗
e(m) : T → Fq , m ∈ M defined as e(m)(t) = t(m) for t ∈ TN is a multiplicative character. The toric surface X¤ associated to ¤ is X¤ = ∪σ∈∆ Uσ Uσ are the Fq -valued points on the affine scheme Spec(Fq [Sσ ]), that is Uσ = {u : Sσ → Fq |u(0) = 1, u(m + m0 ) = u(m)u(m0 ) ∀m, m0 ∈ Sσ }, where Sσ is the additive subsemigroup of M Sσ = {m ∈ M | < m, y >≥ 0∀y ∈ σ}. X¤ is irreducible, (smooth) and complete.
8
TN acts on X¤ . On u ∈ Uσ the element t ∈ TN acts in the following way: (tu)(m) := t(m)u(m) m ∈ Sσ For σ ∈ ∆ ∗
orb(σ) := {u : M ∩ σ → Fq |u is a group homomorphism} ia a TN orbit X¤ . V (σ) is defined to be the closure of orb(σ) in X¤ . A ∆-linear support function h gives rise to a Cartier divisor Dh : X Dh := − h(n(ρ)) V (ρ) ρ∈∆(1)
Dm = div(e(−m))
m ∈ M,
where ∆(1) are the 1-dimensional cones in ∆ and n(ρ) is a generator for the 1-dimensional cone ρ. Lemma 1. The vector space H0 (X, OX (Dh )) af globale sections of OX (Dh ) has dimension #(M ∩ ¤h ) and {e(m)|m ∈ M ∩ ¤h } is a basis. 9
q-1
e
d
q-1
Generators for the 1-dimensional cones are: 1 0 −1 r n(ρ1 ) = , n(ρ2 ) = , n(ρ3 ) = , n(ρ4 ) = 0 1 0 −1 . X Dh := − h(n(ρ)) V (ρ) = d V (ρ3 ) + e V (ρ4 ) ρ∈∆(1)
dim H0 (X, OX (Dh )) = (d + 1 )(e + 1 ) + r 10
d (d + 1 ) . 2
Toric surfaces - Intersection theory Let Dh be a Cartier divisor and let ¤h be the corresponding polytope. Then (Dh ; Dh ) = 2 vol2 (¤h ), where vol2 is the normalized Lesbesgue measure. I our example we get the intersection table V (ρ1 )
V (ρ2 )
V (ρ3 )
V (ρ4 )
V (ρ1 )
−r
1
0
1
V (ρ2 )
1
0
1
0
V (ρ3 )
0
1
r
1
V (ρ4 )
1
0
1
0
11
Result on Hirzebruch surfaces Sætning 1. Let ¤ be the polytope in MR with vertices (0, 0), (d, 0), (d, e + rd), (0, e). Assume d < q − 1, e < q − 1 and e + rd < q − 1. The toric code C¤ has • length (q − 1)2 • dimension #(M ∩ ¤) = (d + 1)(e + 1) + r d(d+1) (the number of lattice 2 points in ¤) • minimal distance (the minimal number of nonzero entries in a codeword different from zero) Min{(q − 1 − d)(q − 1 − e), (q − 1)(q − 1 − e − rd)}.
12
∗
∗
Bevis. For t ∈ T ' Fq × Fq , the rationale functions i H0 (X, OX (Dh )) are evaluated 0
H (X, OX (Dh )) → f
7→
Fq
∗
f (t).
Let H0 (X, OX (Dh ))Frob be the Frobenius invariante functions in H0 (X, OX (Dh )) (functions that are Fq − linear combinations of (e)(m)). Evaluating in all points in T (Fq ) gives the code C¤ : H0 (X, OX (Dh ))Frob
→
C¤ ⊂ (Fq ∗ )]T (Fq )
f
7→
(f (t))t∈T (Fq )
og generators for the code are the images of the basis functions e(m) 7→ (e(m)(t))t∈T (Fq ) . ∗
∗
Let m1 = (1, 0). The Fq -rationale points on T ' Fq × Fq are on the q − 1 Q lines on X¤ given by the equation η∈Fq (e(m1 ) − η) = 0. 13
Let 0 6= f ∈ H0 (X, OX (Dh )) and assume that f is identically zero on precisely a of these lines. As e(m1 ) − η and e(m1 ) have the same pole-divisor, they have equivalent divisors of zeros: (div(e(m1 ) − η))0 ∼ (div(e(m1 )))0 . Therefore div(f ) + Dh − a(div(e(m1 )))0 ≥ 0 or equivalently f ∈ H0 (X, OX (Dh − a(div(e(m1 )))0 ). This implies that a ≤ d according to Lemma 1 on cohomology.
14
On any of the q − 1 − a other lines the number of zeros for f is at most the intersection number: (Dh − a(div(e(m1 )))0 ; (div(e(m1 )))0 ). This is determined using the intersection table and the observation (div(e(m1 )))0 = V (ρ1 ) + rV (ρ4 ). We get (Dh − a(div(e(m1 )))0 ; (div(e(m1 )))0 ) = e + (d − a)r. As 0 ≤ a ≤ d, we conclude that the totale number of (rational) zeros for f is at most a(q − 1) + (q − 1 − a)(e + (d − a)r) ≤ max{d(q − 1) + (q − 1 − d)e, (q − 1)(e + dr)}. Therefore H0 (X, OX (Dh ))Frob
→
C¤ ⊂ (Fq ∗ )]T (Fq )
f
7→
(f (t))t∈T (Fq )
and the dimension and the lower bound for the minimal distance as claimed in the theorem is obtained. 15
Now we will see that we have determined the true minimal distance. Let b1 , . . . , be+rd ∈ Fq ∗ be pairwise distinct. The function xd (y − b1 ) · · · · · (y − be+rd ) ∈ H0 (X, OX (Dh ))Frob is zero in the (q − 1)(e + rd) points (x, bj ), x ∈ Fq ∗ ,
j = 1, . . . , e + rd
and gives a codeword of weight (q − 1)2 − (q − 1)(e + rd) = (q − 1)(q − 1 − (e + rd)). Let a1 , . . . , ad ∈ Fq ∗ be pairwise distinct and let b1 , . . . , be ∈ Fq ∗ be pairwise distinct. The function (x − a1 ) · · · · · (x − ad )(y − b1 ) · · · · · (y − be ) ∈ H0 (X, OX (Dh ))Frob is zero in the d(q − 1) + (q − 1)e − de points (ai , y), (x, bj ),
x, y ∈ Fq ∗ , i = 1, . . . e, j = 1, . . . , d
and gives a codeword of weight (q − 1 − d)(q − 1 − e). 16
Litteratur [1]
W. Fulton, “Introduction to Toric Varieties,”Annals of Mathematics Studies; no. 131, Princeton University Press, 1993.
[2] W. Fulton, “Intersection theory”Ergebnisse der Mathematik und uhrer Grenzgebiete, 3. Folge, Springer Verlag, 1998. [3]
J. P. Hansen, “Toric Surfaces and Error-correcting codes,”in Coding theory, cryptography and related areas (Guanajuato, 1998), 132-142, Springer, Berlin, 2000
[4]
J. P. Hansen, “Hirzebruch surfaces and Error-correcting Codes,”Preprint Series No. 6, 2000, University of Aarhus.
[5] J. P. Hansen, “Toric Varieties Hirzebruch Surfaces and Error-Correcting Codes,”Applicable Algebra in Engineering, Communication and Computing, Springer-Verlag, Volume 13, Number 4/December 2002, Pages: 289 300
16-1
[6]
Hansen, Søren Have, “Error-correcting codes from higher-dimensional varieties,”Finite Fields Appl.; no 7, 2001, 531–552
[7] Joyner, David, “Toric codes over finite fields,”Preprint, Aug. 2002, http://front.math.ucdavis.edu/math.AG/0208155 [8] T. Oda, “Convex Bodies and Algebraic Geometry, An Introduction to the Theory of Toric Varieties,”Ergebnisse der Mathematik und uhrer Grenzgebiete, 3. Folge, Band 15, Springer Verlag, 1985.
16-2
