Good covering codes from algebraic curves Massimo Giulietti - PowerPoint PPT Presentation

applying Segre’s criterion (Segre, 1962) if there exists a point P ∈ C and a tangent ℓ of C at P such that ℓ counts once among the tangents of C at P , the intersection multiplicity of C and ℓ at P equals deg( C ), C has no linear components through P , then C is irreducible.

applying Segre’s criterion (Segre, 1962) if there exists a point P ∈ C and a tangent ℓ of C at P such that ℓ counts once among the tangents of C at P , the intersection multiplicity of C and ℓ at P equals deg( C ), C has no linear components through P , then C is irreducible. a + ( X p − X + ¯ t )( Y p − Y + ¯ t ) 2 + F a , b ( X , Y ) := ( X p − X + ¯ t ) 2 ( Y p − Y + ¯ t ) − b (( X p − X + ¯ t ) 2 +( X p − X + ¯ t )( Y p − Y + ¯ t ) + ( Y p − Y + ¯ t ) 2 ) = 0

applying Segre’s criterion (Segre, 1962) if there exists a point P ∈ C and a tangent ℓ of C at P such that ℓ counts once among the tangents of C at P , the intersection multiplicity of C and ℓ at P equals deg( C ), C has no linear components through P , then C is irreducible. a + ( X p − X + ¯ t )( Y p − Y + ¯ t ) 2 + F a , b ( X , Y ) := ( X p − X + ¯ t ) 2 ( Y p − Y + ¯ t ) − b (( X p − X + ¯ t ) 2 +( X p − X + ¯ t )( Y p − Y + ¯ t ) + ( Y p − Y + ¯ t ) 2 ) = 0 at P = X ∞ the tangents are ℓ : Y = β with β p − β + ¯ t = b

applying Segre’s criterion (Segre, 1962) if there exists a point P ∈ C and a tangent ℓ of C at P such that ℓ counts once among the tangents of C at P , the intersection multiplicity of C and ℓ at P equals deg( C ), C has no linear components through P , then C is irreducible. a + ( X p − X + ¯ t )( Y p − Y + ¯ t ) 2 + F a , b ( X , Y ) := ( X p − X + ¯ t ) 2 ( Y p − Y + ¯ t ) − b (( X p − X + ¯ t ) 2 +( X p − X + ¯ t )( Y p − Y + ¯ t ) + ( Y p − Y + ¯ t ) 2 ) = 0 at P = X ∞ the tangents are ℓ : Y = β with β p − β + ¯ t = b F a , b ( X , β ) = a − b 3

applying Segre’s criterion (Segre, 1962) if there exists a point P ∈ C and a tangent ℓ of C at P such that ℓ counts once among the tangents of C at P , the intersection multiplicity of C and ℓ at P equals deg( C ), C has no linear components through P , then C is irreducible. a + ( X p − X + ¯ t )( Y p − Y + ¯ t ) 2 + F a , b ( X , Y ) := ( X p − X + ¯ t ) 2 ( Y p − Y + ¯ t ) − b (( X p − X + ¯ t ) 2 +( X p − X + ¯ t )( Y p − Y + ¯ t ) + ( Y p − Y + ¯ t ) 2 ) = 0 at P = X ∞ the tangents are ℓ : Y = β with β p − β + ¯ t = b F a , b ( X , β ) = a − b 3 if P / ∈ X C P is irreducible of genus g ≤ 3 p 2 − 3 p + 1

applying Segre’s criterion (Segre, 1962) if there exists a point P ∈ C and a tangent ℓ of C at P such that ℓ counts once among the tangents of C at P , the intersection multiplicity of C and ℓ at P equals deg( C ), C has no linear components through P , then C is irreducible. a + ( X p − X + ¯ t )( Y p − Y + ¯ t ) 2 + F a , b ( X , Y ) := ( X p − X + ¯ t ) 2 ( Y p − Y + ¯ t ) − b (( X p − X + ¯ t ) 2 +( X p − X + ¯ t )( Y p − Y + ¯ t ) + ( Y p − Y + ¯ t ) 2 ) = 0 at P = X ∞ the tangents are ℓ : Y = β with β p − β + ¯ t = b F a , b ( X , β ) = a − b 3 if P / ∈ X C P is irreducible of genus g ≤ 3 p 2 − 3 p + 1 C P has at least q + 1 − (6 p 2 − 6 p + 2) √ q points

cuspidal case: Y = X 3 G is elementary abelian, isomorphic to ( F q , +)

cuspidal case: Y = X 3 G is elementary abelian, isomorphic to ( F q , +) S = { ( L ( t ) + ¯ t , ( L ( t ) + ¯ t ) 3 ) | t ∈ F q } � �� � P t � L ( T ) = ( T − α ) , M < ( F q , +) , # M = m α ∈ M

cuspidal case: Y = X 3 G is elementary abelian, isomorphic to ( F q , +) S = { ( L ( t ) + ¯ t , ( L ( t ) + ¯ t ) 3 ) | t ∈ F q } � �� � P t � L ( T ) = ( T − α ) , M < ( F q , +) , # M = m α ∈ M P = ( a , b ) is collinear with P x and P y if and only if t ) 2 + a + ( L ( x ) + ¯ t )( L ( y ) + ¯ F a , b ( x , y ) := ( L ( x ) + ¯ t ) 2 ( L ( y ) + ¯ t ) − b (( L ( x ) + ¯ t ) 2 +( L ( x ) + ¯ t )( L ( y ) + ¯ t ) + ( L ( y ) + ¯ t ) 2 ) = 0

cuspidal case: Y = X 3 G is elementary abelian, isomorphic to ( F q , +) S = { ( L ( t ) + ¯ t , ( L ( t ) + ¯ t ) 3 ) | t ∈ F q } � �� � P t � L ( T ) = ( T − α ) , M < ( F q , +) , # M = m α ∈ M P = ( a , b ) is collinear with P x and P y if and only if t ) 2 + a + ( L ( x ) + ¯ t )( L ( y ) + ¯ F a , b ( x , y ) := ( L ( x ) + ¯ t ) 2 ( L ( y ) + ¯ t ) − b (( L ( x ) + ¯ t ) 2 +( L ( x ) + ¯ t )( L ( y ) + ¯ t ) + ( L ( y ) + ¯ t ) 2 ) = 0 if P / ∈ X C P is irreducible of genus g ≤ 3 m 2 − 3 m + 1

cuspidal case: Y = X 3 G is elementary abelian, isomorphic to ( F q , +) S = { ( L ( t ) + ¯ t , ( L ( t ) + ¯ t ) 3 ) | t ∈ F q } � �� � P t � L ( T ) = ( T − α ) , M < ( F q , +) , # M = m α ∈ M P = ( a , b ) is collinear with P x and P y if and only if t ) 2 + a + ( L ( x ) + ¯ t )( L ( y ) + ¯ F a , b ( x , y ) := ( L ( x ) + ¯ t ) 2 ( L ( y ) + ¯ t ) − b (( L ( x ) + ¯ t ) 2 +( L ( x ) + ¯ t )( L ( y ) + ¯ t ) + ( L ( y ) + ¯ t ) 2 ) = 0 if P / ∈ X C P is irreducible of genus g ≤ 3 m 2 − 3 m + 1 C P has at least q + 1 − (6 m 2 − 6 m + 2) √ q points

(Sz˝ onyi, 1985 - Anbar, Bartoli, G., Platoni, 2013) let P = ( a , b ) be a point in A 2 ( F q ) \ X ; if � m < 4 q / 36 then there is a secant of S passing through P .

(Sz˝ onyi, 1985 - Anbar, Bartoli, G., Platoni, 2013) let P = ( a , b ) be a point in A 2 ( F q ) \ X ; if � m < 4 q / 36 then there is a secant of S passing through P . m is a power of p

(Sz˝ onyi, 1985 - Anbar, Bartoli, G., Platoni, 2013) let P = ( a , b ) be a point in A 2 ( F q ) \ X ; if � m < 4 q / 36 then there is a secant of S passing through P . m is a power of p the points in X \ S need to be dealt with

(Sz˝ onyi, 1985 - Anbar, Bartoli, G., Platoni, 2013) let P = ( a , b ) be a point in A 2 ( F q ) \ X ; if � m < 4 q / 36 then there is a secant of S passing through P . m is a power of p the points in X \ S need to be dealt with theorem � q / 36 , then there exists a complete cap in A 2 ( F q ) with size if m < 4 m + q m − 3

(Sz˝ onyi, 1985 - Anbar, Bartoli, G., Platoni, 2013) let P = ( a , b ) be a point in A 2 ( F q ) \ X ; if � m < 4 q / 36 then there is a secant of S passing through P . m is a power of p the points in X \ S need to be dealt with theorem � q / 36 , then there exists a complete cap in A 2 ( F q ) with size if m < 4 m + q m − 3 ∼ p 1 / 4 · q 3 / 4

nodal case: XY = ( X − 1) 3 G is isomorphic to ( F ∗ q , · ) v , ( v − 1) 3 � � G → F ∗ �→ v q v

nodal case: XY = ( X − 1) 3 G is isomorphic to ( F ∗ q , · ) v , ( v − 1) 3 � � G → F ∗ �→ v q v the subgroup of index m ( m a divisor of q − 1): t m , ( t m − 1) 3 �� � � | t ∈ F ∗ K = q t m

nodal case: XY = ( X − 1) 3 G is isomorphic to ( F ∗ q , · ) v , ( v − 1) 3 � � G → F ∗ �→ v q v the subgroup of index m ( m a divisor of q − 1): t m , ( t m − 1) 3 �� � � | t ∈ F ∗ K = q t m a coset: tt m − 1) 3 tt m , (¯ �� � � ¯ | t ∈ F ∗ S = ¯ q tt m

nodal case: XY = ( X − 1) 3 G is isomorphic to ( F ∗ q , · ) v , ( v − 1) 3 � � G → F ∗ �→ v q v the subgroup of index m ( m a divisor of q − 1): t m , ( t m − 1) 3 �� � � | t ∈ F ∗ K = q t m a coset: g ( t ) f ( t ) � �� � tt m − 1) 3 (¯ � � � � ���� tt m , | t ∈ F ∗ ¯ S = ¯ q tt m � �� � P t

nodal case: XY = ( X − 1) 3 G is isomorphic to ( F ∗ q , · ) v , ( v − 1) 3 � � G → F ∗ �→ v q v the subgroup of index m ( m a divisor of q − 1): t m , ( t m − 1) 3 �� � � | t ∈ F ∗ K = q t m a coset: g ( t ) f ( t ) � �� � tt m − 1) 3 (¯ � � � � ���� tt m , | t ∈ F ∗ ¯ S = ¯ q tt m � �� � P t the curve C P : t 3 X 2 m Y m + ¯ t 3 X m Y 2 m − 3¯ t 2 X m Y m + 1) a (¯ F a , b ( X , Y ) = t 2 X m Y m − ¯ t 4 X 2 m Y 2 m + 3¯ − b ¯ t 2 X m Y m tX m − ¯ tY m = 0 − ¯

(Anbar-Bartoli-G.-Platoni, 2013) let P be a point in A 2 ( F q ) \ X ; if � 4 m < q / 36 then there is a secant of S passing through P

(Anbar-Bartoli-G.-Platoni, 2013) let P be a point in A 2 ( F q ) \ X ; if � 4 m < q / 36 then there is a secant of S passing through P m is a divisor of q − 1

(Anbar-Bartoli-G.-Platoni, 2013) let P be a point in A 2 ( F q ) \ X ; if � 4 m < q / 36 then there is a secant of S passing through P m is a divisor of q − 1 some points from X \ S need to be added to S

(Anbar-Bartoli-G.-Platoni, 2013) let P be a point in A 2 ( F q ) \ X ; if � 4 m < q / 36 then there is a secant of S passing through P m is a divisor of q − 1 some points from X \ S need to be added to S theorem � q / 36 , and in addition ( m , q − 1 if m is a divisor of q − 1 with m < 4 m ) = 1 , then there exists a complete cap in A 2 ( F q ) with size m + q − 1 − 3 m

(Anbar-Bartoli-G.-Platoni, 2013) let P be a point in A 2 ( F q ) \ X ; if � 4 m < q / 36 then there is a secant of S passing through P m is a divisor of q − 1 some points from X \ S need to be added to S theorem � q / 36 , and in addition ( m , q − 1 if m is a divisor of q − 1 with m < 4 m ) = 1 , then there exists a complete cap in A 2 ( F q ) with size m + q − 1 ∼ q 3 / 4 − 3 m

isolated double point case: Y ( X 2 − β ) = 1 G cyclic of order q + 1

isolated double point case: Y ( X 2 − β ) = 1 G cyclic of order q + 1 (Anbar-Bartoli-G.-Platoni, 2013) � q / 36, and in addition ( m , q +1 if m is a divisor of q + 1 with m < 4 m ) = 1, then there exists a complete cap in A 2 ( F q ) with size at most m + q + 1 m

isolated double point case: Y ( X 2 − β ) = 1 G cyclic of order q + 1 (Anbar-Bartoli-G.-Platoni, 2013) � q / 36, and in addition ( m , q +1 if m is a divisor of q + 1 with m < 4 m ) = 1, then there exists a complete cap in A 2 ( F q ) with size at most m + q + 1 ∼ q 3 / 4 m

elliptic case: Y 2 = X 3 + AX + B if n ∈ [ q + 1 − 2 √ q , q + 1 + 2 √ q ] n �≡ q + 1 (mod p ) there exists an elliptic cubic curve X over F q with # G = n

elliptic case: Y 2 = X 3 + AX + B if n ∈ [ q + 1 − 2 √ q , q + 1 + 2 √ q ] n �≡ q + 1 (mod p ) there exists an elliptic cubic curve X over F q with # G = n (Voloch, 1988) if p does not divide # G − 1, then G can be assumed to be cyclic

elliptic case: Y 2 = X 3 + AX + B if n ∈ [ q + 1 − 2 √ q , q + 1 + 2 √ q ] n �≡ q + 1 (mod p ) there exists an elliptic cubic curve X over F q with # G = n (Voloch, 1988) if p does not divide # G − 1, then G can be assumed to be cyclic problem: no polynomial or rational parametrization of the points of S is possible

elliptic case: Y 2 = X 3 + AX + B if n ∈ [ q + 1 − 2 √ q , q + 1 + 2 √ q ] n �≡ q + 1 (mod p ) there exists an elliptic cubic curve X over F q with # G = n (Voloch, 1988) if p does not divide # G − 1, then G can be assumed to be cyclic problem: no polynomial or rational parametrization of the points of S is possible Voloch’s solution (1990): implicit description of C P

elliptic case: Y 2 = X 3 + AX + B if n ∈ [ q + 1 − 2 √ q , q + 1 + 2 √ q ] n �≡ q + 1 (mod p ) there exists an elliptic cubic curve X over F q with # G = n (Voloch, 1988) if p does not divide # G − 1, then G can be assumed to be cyclic problem: no polynomial or rational parametrization of the points of S is possible Voloch’s solution (1990): implicit description of C P Voloch’s result would provide complete caps of size ∼ q 3 / 4 for every q large enough

elliptic case: Y 2 = X 3 + AX + B if n ∈ [ q + 1 − 2 √ q , q + 1 + 2 √ q ] n �≡ q + 1 (mod p ) there exists an elliptic cubic curve X over F q with # G = n (Voloch, 1988) if p does not divide # G − 1, then G can be assumed to be cyclic problem: no polynomial or rational parametrization of the points of S is possible Voloch’s solution (1990): implicit description of C P Voloch’s result would provide complete caps of size ∼ q 3 / 4 for every q large enough ?

elliptic case G cyclic m | q − 1 m prime

elliptic case G cyclic m | q − 1 m prime Tate-Lichtenbaum Pairing F ∗ q / ( F ∗ q ) m < · , · > : G [ m ] × G / K →

elliptic case G cyclic m | q − 1 m prime Tate-Lichtenbaum Pairing F ∗ q / ( F ∗ q ) m < · , · > : G [ m ] × G / K → if m 2 does not divide # G , then for some T in G [ m ] F ∗ q / ( F ∗ q ) m < T , · > : G / K → is an isomorphism such that K ⊕ Q �→ [ α T ( Q )] where α T is a rational function on X

elliptic case G cyclic m | q − 1 m prime Tate-Lichtenbaum Pairing F ∗ q / ( F ∗ q ) m < · , · > : G [ m ] × G / K → if m 2 does not divide # G , then for some T in G [ m ] F ∗ q / ( F ∗ q ) m < T , · > : G / K → is an isomorphism such that K ⊕ Q �→ [ α T ( Q )] where α T is a rational function on X S = { R ∈ G | α T ( R ) = dt m for some t ∈ F q }

elliptic case S = { R ∈ X | α ( R ) = dt m for some t ∈ F q }

elliptic case S = { R ∈ X | α ( R ) = dt m for some t ∈ F q }

elliptic case S = { R ∈ X | α ( R ) = dt m for some t ∈ F q } P = ( a , b ) collinear with two points ( x , y ) , ( u , v ) ∈ S if there exist x , y , u , v , t , z ∈ F q with

b b b elliptic case S = { R ∈ X | α ( R ) = dt m for some t ∈ F q } P = ( a , b ) collinear with two points ( x , y ) , ( u , v ) ∈ S if there exist x , y , u , v , t , z ∈ F q with ( a , b )  y 2 = x 3 + Ax + B ( x , y )   v 2 = u 3 + Au + B        ( u , v )         

b b b elliptic case S = { R ∈ X | α ( R ) = dt m for some t ∈ F q } P = ( a , b ) collinear with two points ( x , y ) , ( u , v ) ∈ S if there exist x , y , u , v , t , z ∈ F q with ( a , b )  y 2 = x 3 + Ax + B ( x , y )   v 2 = u 3 + Au + B      α ( x , y ) = dt m   α ( u , v ) = dz m ( u , v )         

b b b elliptic case S = { R ∈ X | α ( R ) = dt m for some t ∈ F q } P = ( a , b ) collinear with two points ( x , y ) , ( u , v ) ∈ S if there exist x , y , u , v , t , z ∈ F q with ( a , b )  y 2 = x 3 + Ax + B ( x , y )   v 2 = u 3 + Au + B      α ( x , y ) = dt m   α ( u , v ) = dz m ( u , v )    1 a b     = 0  det 1 x y      1 u v

b b b elliptic case S = { R ∈ X | α ( R ) = dt m for some t ∈ F q } P = ( a , b ) collinear with two points ( x , y ) , ( u , v ) ∈ S if there exist x , y , u , v , t , z ∈ F q with ( a , b )  y 2 = x 3 + Ax + B ( x , y )   v 2 = u 3 + Au + B      α ( x , y ) = dt m   α ( u , v ) = dz m C P : ( u , v )    1 a b     = 0  det 1 x y      1 u v

(Anbar-G., 2012) if A � = 0, then C P is irreducible or admits an irreducible F q -rational component

(Anbar-G., 2012) if A � = 0, then C P is irreducible or admits an irreducible F q -rational component � if m is a prime divisor of q − 1 with m < 4 q / 64, then there exists a complete cap in A 2 ( F q ) with size at most m + ⌊ q − 2 √ q + 1 ⌋ + 31 m

(Anbar-G., 2012) if A � = 0, then C P is irreducible or admits an irreducible F q -rational component � if m is a prime divisor of q − 1 with m < 4 q / 64, then there exists a complete cap in A 2 ( F q ) with size at most m + ⌊ q − 2 √ q + 1 ∼ q 3 / 4 ⌋ + 31 m

ℓ ( r , q ) 2 , 4

in geometrical terms... proposition ℓ ( r , q ) 2 , 4 = minimum size of a complete cap in P r − 1 ( F q )

in geometrical terms... proposition ℓ ( r , q ) 2 , 4 = minimum size of a complete cap in P r − 1 ( F q ) trivial lower bound √ 2 q ( N − 1) / 2 in P N ( F q ) # S ≥