Applications of algebraic geometry codes beyond classical coding Gretchen L. Matthews September 10, 2014 C L E M S O N M A T H E M A T I C A L S C I E N C E S Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Two applications of Weierstrass semigroups 1 compressed sensing (joint work with Justin Peachey) Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Two applications of Weierstrass semigroups 1 compressed sensing (joint work with Justin Peachey) 2 polar coding (joint work with Sarah Anderson) Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Two applications of Weierstrass semigroups 0 background 1 compressed sensing (joint work with Justin Peachey) 2 polar coding (joint work with Sarah Anderson) Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Weierstrass semigroups contain information about poles of functions. Consider the algebraic function field F := F q ( x , y ) where h ( x , y ) = 0 | F q g the genus of F ( f ) the divisor of f 2 F \ { 0 } ( f ) 1 the pole divisor of f 2 F \ { 0 } The Weierstrass semigroup of a rational place P of F is H ( P ) = { n 2 N : 9 f 2 F with ( f ) 1 = nP } . Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Weierstrass semigroups contain information about poles of functions. Consider the algebraic function field F := F q ( x , y ) where h ( x , y ) = 0 | F q g the genus of F ( f ) the divisor of f 2 F \ { 0 } ( f ) 1 the pole divisor of f 2 F \ { 0 } The Weierstrass semigroup of a rational place P of F is H ( P ) = { n 2 N : 9 f 2 F with ( f ) 1 = nP } . The Weierstrass semigroup of an m -tuple of rational places ( P 1 , . . . , P m ) is H ( P 1 , . . . , P m ) = { ( n 1 , . . . , n m ) 2 N m : 9 f 2 F with ( f ) 1 = n 1 P 1 + · · · + n m P m } . Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Weierstrass semigroups contain information about poles of functions. Consider the algebraic function field F := F q ( x , y ) where h ( x , y ) = 0 | F q g the genus of F ( f ) the divisor of f 2 F \ { 0 } ( f ) 1 the pole divisor of f 2 F \ { 0 } The Weierstrass semigroup of a rational place P of F is H ( P ) = { n 2 N : 9 f 2 F with ( f ) 1 = nP } . The Weierstrass semigroup of an m -tuple of rational places ( P 1 , . . . , P m ) is H ( P 1 , . . . , P m ) = { ( n 1 , . . . , n m ) 2 N m : 9 f 2 F with ( f ) 1 = n 1 P 1 + · · · + n m P m } . Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Examples of Weierstrass semigroups Consider the Hermitian function field F q 2 ( x , y ) given by y q + y = x q +1 . 1 Here, X ( x ) = P 0 β � qP 1 β : β q + β =0 ( y ) = ( q + 1) P 00 � ( q + 1) P 1 Hence H ( P 1 ) = h q , q + 1 i , because | N \ H ( P 1 ) | = g = q ( q + 1) = | N \ h q , q + 1 i | . 2 Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Examples of Weierstrass semigroups Consider the Hermitian function field F q 2 ( x , y ) given by y q + y = x q +1 . 1 Here, X ( x ) = P 0 β � qP 1 β : β q + β =0 ( y ) = ( q + 1) P 00 � ( q + 1) P 1 Hence H ( P 1 ) = h q , q + 1 i , because | N \ H ( P 1 ) | = g = q ( q + 1) = | N \ h q , q + 1 i | . 2 More generally, consider F q r ( x , y ) where 2 L ( y ) = x u with L ( y ) a linearized polynomial and u | q r � 1 q � 1 . Here, ( x ) 1 = q d P 1 and ( y ) 1 = uP 1 ⌦ q d , u ↵ where d = log q deg L . Then H ( P 1 ) = . Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Examples of Weierstrass semigroups Consider the Suzuki function field F := F q ( x , y ) / F q defined by 3 y q � y = x q 0 ( x q � x ) where q 0 = 2 r and q = 2 2 r +1 and r is a positive integer. Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Examples of Weierstrass semigroups Consider the Suzuki function field F := F q ( x , y ) / F q defined by 3 y q � y = x q 0 ( x q � x ) where q 0 = 2 r and q = 2 2 r +1 and r is a positive integer. According to [Hansen & Stichtenoth, 1990] the functions � 1 + v q q q q q q 0 +1 , w := y q 2 q 0 � x q 0 x q 0 2 F x , y , v := y 0 have pole divisors ( x ) 1 = qP 1 ( y ) 1 = ( q + q 0 ) P 1 ( q + q ( v ) 1 = q 0 ) P 1 ( q + q ( w ) 1 = q 0 + 1) P 1 . Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Examples of Weierstrass semigroups Consider the Suzuki function field F := F q ( x , y ) / F q defined by 3 y q � y = x q 0 ( x q � x ) where q 0 = 2 r and q = 2 2 r +1 and r is a positive integer. According to [Hansen & Stichtenoth, 1990] the functions � 1 + v q q q q q q 0 +1 , w := y q 2 q 0 � x q 0 x q 0 2 F x , y , v := y 0 have pole divisors ( x ) 1 = qP 1 ( y ) 1 = ( q + q 0 ) P 1 ( q + q ( v ) 1 = q 0 ) P 1 ( q + q ( w ) 1 = q 0 + 1) P 1 . D E q , q + q 0 , q + q q 0 , q + q The Weierstrass semigroup H ( P ) = q 0 + 1 for any rational place P of F . Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Weierstrass semigroups and Riemann-Roch spaces Definition The Riemann-Roch space of a divisor G is L ( G ) := { f 2 F : ( f ) � � G } [ { 0 } . Notice that A B = ) L ( A ) ✓ L ( B ) , Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Weierstrass semigroups and Riemann-Roch spaces Definition The Riemann-Roch space of a divisor G is L ( G ) := { f 2 F : ( f ) � � G } [ { 0 } . Notice that A B = ) L ( A ) ✓ L ( B ) , as f 2 L ( A ) = ) ( f ) � � A Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Weierstrass semigroups and Riemann-Roch spaces Definition The Riemann-Roch space of a divisor G is L ( G ) := { f 2 F : ( f ) � � G } [ { 0 } . Notice that A B = ) L ( A ) ✓ L ( B ) , as f 2 L ( A ) = ) ( f ) � � A � � B Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Weierstrass semigroups and Riemann-Roch spaces Definition The Riemann-Roch space of a divisor G is L ( G ) := { f 2 F : ( f ) � � G } [ { 0 } . Notice that A B = ) L ( A ) ✓ L ( B ) , as f 2 L ( A ) = ) ( f ) � � A � � B = ) f 2 L ( B ). Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Weierstrass semigroups and Riemann-Roch spaces Definition The Riemann-Roch space of a divisor G is L ( G ) := { f 2 F : ( f ) � � G } [ { 0 } . Notice that A B = ) L ( A ) ✓ L ( B ) , as f 2 L ( A ) = ) ( f ) � � A � � B = ) f 2 L ( B ). In particular, L (( a � 1) P ) ✓ L ( aP ). Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Weierstrass semigroups and Riemann-Roch spaces Definition The Riemann-Roch space of a divisor G is L ( G ) := { f 2 F : ( f ) � � G } [ { 0 } . Notice that A B = ) L ( A ) ✓ L ( B ) , as f 2 L ( A ) = ) ( f ) � � A � � B = ) f 2 L ( B ). In particular, L (( a � 1) P ) ✓ L ( aP ). Moreover, a 2 H ( P ) if and only if 9 f 2 L ( aP ) \ L (( a � 1) P ) if and only if L (( a � 1) P ) $ L ( aP ) . Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Weierstrass semigroups and Riemann-Roch spaces For the Hermitian function field F q 2 ( x , y ) / F q 2 , 1 x i y j : iq + j ( q + 1) a � L ( aP 1 ) = Span . For the function field F q r ( x , y ) / F q r defined by L ( y ) = x u , 2 x i y j : iq d + ju a � L ( aP 1 ) = Span . For the Suzuki function field F q ( x , y ) / F q , L ( aP 1 ) = 3 ⇢ � x i y j v k w l : iq + j ( q + q 0 ) + k ( q + q ) + l ( q + q Span + 1) a . q 0 q 0 Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Application to compressed sensing Set up: Given a matrix A 2 F m ⇥ n and y 2 F m ⇥ 1 where y = Ax . 2 2 Goal: Reconstruct x from y . Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
Application to compressed sensing Set up: Given a matrix A 2 F m ⇥ n and y 2 F m ⇥ 1 where y = Ax . 2 2 Goal: Reconstruct x from y . For appropriately chosen matrices A , the reconstruction of a sparse signal reduces to an optimization problem for which there are e ffi cient algorithms. (see [Donoho, 2006], [Candes, Romberg, & Tao, 2006]) Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding
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