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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 := Fq(x, y) where h(x, y) = 0 | Fq 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 : 9f 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 := Fq(x, y) where h(x, y) = 0 | Fq 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 : 9f 2 F with (f )1 = nP} . The Weierstrass semigroup of an m-tuple of rational places (P1, . . . , Pm) is H(P1, . . . , Pm) = {(n1, . . . , nm) 2 Nm : 9f 2 F with (f )1 = n1P1 + · · · + nmPm} .

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 := Fq(x, y) where h(x, y) = 0 | Fq 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 : 9f 2 F with (f )1 = nP} . The Weierstrass semigroup of an m-tuple of rational places (P1, . . . , Pm) is H(P1, . . . , Pm) = {(n1, . . . , nm) 2 Nm : 9f 2 F with (f )1 = n1P1 + · · · + nmPm} .

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Examples of Weierstrass semigroups

1

Consider the Hermitian function field Fq2(x, y) given by y q + y = xq+1. Here, (x) = X

β:βq+β=0

P0β qP1 (y) = (q + 1)P00 (q + 1)P1 Hence H(P1) = hq, q + 1i, because |N \ H(P1)| = g = q(q + 1) 2 = |N \ hq, q + 1i |.

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Examples of Weierstrass semigroups

1

Consider the Hermitian function field Fq2(x, y) given by y q + y = xq+1. Here, (x) = X

β:βq+β=0

P0β qP1 (y) = (q + 1)P00 (q + 1)P1 Hence H(P1) = hq, q + 1i, because |N \ H(P1)| = g = q(q + 1) 2 = |N \ hq, q + 1i |.

2

More generally, consider Fqr (x, y) where L(y) = xu with L(y) a linearized polynomial and u| qr 1

q1 . Here,

(x)1 = qdP1 and (y)1 = uP1 where d = logq deg L. Then H(P1) = ⌦ qd, u ↵ .

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Examples of Weierstrass semigroups

3

Consider the Suzuki function field F := Fq(x, y)/Fq defined by y q y = xq0(xq x) where q0 = 2r and q = 22r+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

3

Consider the Suzuki function field F := Fq(x, y)/Fq defined by y q y = xq0(xq x) where q0 = 2r and q = 22r+1 and r is a positive integer. According to [Hansen & Stichtenoth, 1990] the functions x, y, v := y

q q0 x q q0 +1, w := y q q0 x q q2

1 + v

q q0 2 F

have pole divisors (x)1 = qP1 (y)1 = (q + q0)P1 (v)1 = (q + q

q0 )P1

(w)1 = (q + q

q0 + 1)P1.

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Examples of Weierstrass semigroups

3

Consider the Suzuki function field F := Fq(x, y)/Fq defined by y q y = xq0(xq x) where q0 = 2r and q = 22r+1 and r is a positive integer. According to [Hansen & Stichtenoth, 1990] the functions x, y, v := y

q q0 x q q0 +1, w := y q q0 x q q2

1 + v

q q0 2 F

have pole divisors (x)1 = qP1 (y)1 = (q + q0)P1 (v)1 = (q + q

q0 )P1

(w)1 = (q + q

q0 + 1)P1.

The Weierstrass semigroup H(P) = D q, q + q0, q + q

q0 , q + q q0 + 1

E 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 9f 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

1

For the Hermitian function field Fq2(x, y)/Fq2, L(aP1) = Span

• xiy j : iq + j(q + 1)  a

.

2

For the function field Fqr (x, y)/Fqr defined by L(y) = xu, L(aP1) = Span

• xiy j : iqd + ju  a

.

3

For the Suzuki function field Fq(x, y)/Fq, L(aP1) = Span ⇢ xiy jv kw l : iq + j(q + q0) + k(q + q q0 ) + l(q + q q0 + 1)  a

• .

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 Fm⇥n

2

and y 2 Fm⇥1

2

where y = Ax. 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 Fm⇥n

2

and y 2 Fm⇥1

2

where y = Ax. 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 efficient 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

Application to compressed sensing

Set up: Given a matrix A 2 Fm⇥n

2

and y 2 Fm⇥1

2

where y = Ax. 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 efficient algorithms. (see [Donoho, 2006], [Candes, Romberg, & Tao, 2006]) Question: How do we construct/find such matrices A?

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 Fm⇥n

2

and y 2 Fm⇥1

2

where y = Ax. 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 efficient algorithms. (see [Donoho, 2006], [Candes, Romberg, & Tao, 2006]) Question: How do we construct/find such matrices A? The coherence of a matrix A with unit column vectors ui is µ(A) = max

i6=j |hui, uji|.

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to compressed sensing

Theorem If a matrix A has coherence µ, then a unique and exact recovery of x is guaranteed for all vectors x of weight k < 1 + 1

µ.

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to compressed sensing

Fix a 2 N and distinct rational places P, P1, . . . , Pn of F/Fq. Set A = 2 6 6 6 6 6 4 ⇥ ef1(P1) ⇤ ⇥ ef2(P1) ⇤ · · · ⇥ eft(P1) ⇤ ⇥ ef1(P2) ⇤ ⇥ ef2(P2) ⇤ · · · ⇥ eft(P2) ⇤ . . . . . . . . . ⇥ ef1(Pn) ⇤ ⇥ ef2(Pn) ⇤ · · · ⇥ eft(Pn) ⇤ 3 7 7 7 7 7 5 2 Fnq⇥qdim L(aP)

2

where L(aP) = {f1, . . . , ft}.

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to compressed sensing

Fix a 2 N and distinct rational places P, P1, . . . , Pn of F/Fq. Set A = 2 6 6 6 6 6 4 ⇥ ef1(P1) ⇤ ⇥ ef2(P1) ⇤ · · · ⇥ eft(P1) ⇤ ⇥ ef1(P2) ⇤ ⇥ ef2(P2) ⇤ · · · ⇥ eft(P2) ⇤ . . . . . . . . . ⇥ ef1(Pn) ⇤ ⇥ ef2(Pn) ⇤ · · · ⇥ eft(Pn) ⇤ 3 7 7 7 7 7 5 2 Fnq⇥qdim L(aP)

2

where L(aP) = {f1, . . . , ft}. Lemma (Li, Gao, Ge, & Zhang, 2012) The matrix

1 pnA is a sensing matrix with coherence µ  a n.

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to compressed sensing

Take Fqr (x, y)/Fqr where y qr1 + y qr2 + · · · + y = x

qr 1 q1 .

Theorem (M & Peachey) Let s 2 Z+. Then there exists a sensing matrix A of size q3r1 ⇥ qr dim L(sP1) with coherence bounded by µ(A) 

s q2r1 .

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to compressed sensing

Take Fqr (x, y)/Fqr where y qr1 + y qr2 + · · · + y = x

qr 1 q1 .

Theorem (M & Peachey) Let s 2 Z+. Then there exists a sensing matrix A of size q3r1 ⇥ qr dim L(sP1) with coherence bounded by µ(A) 

s q2r1 .

Example Consider the norm-trace function field F over F27 and the Hermitian function field over F24. The corresponding sensing matrix constructed over the norm-trace function field has dimensions 220 ⇥ 216352, and µnt  6064

213 < 4093 26 .

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Definition Let F/F be an algebraic function field, where F is a finite field. Let G and D = Q1 + · · · + Qn be divisors of F/F where Q1, . . . , Qn are distinct places of degree one and Qi / 2 suppG for all 1  i  n. The AG code C(D, G) is C (D, G) := {(f (Q1) , f (Q2) , . . . , f (Qn)) : f 2 L(G)} ✓ Fn where L (G) is the Riemann-Roch space of G.

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Definition Let F/F be an algebraic function field, where F is a finite field. Let G and D = Q1 + · · · + Qn be divisors of F/F where Q1, . . . , Qn are distinct places of degree one and Qi / 2 suppG for all 1  i  n. The AG code C(D, G) is C (D, G) := {(f (Q1) , f (Q2) , . . . , f (Qn)) : f 2 L(G)} ✓ Fn where L (G) is the Riemann-Roch space of G. Code parameters The code C(D, G) has length n,

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Definition Let F/F be an algebraic function field, where F is a finite field. Let G and D = Q1 + · · · + Qn be divisors of F/F where Q1, . . . , Qn are distinct places of degree one and Qi / 2 suppG for all 1  i  n. The AG code C(D, G) is C (D, G) := {(f (Q1) , f (Q2) , . . . , f (Qn)) : f 2 L(G)} ✓ Fn where L (G) is the Riemann-Roch space of G. Code parameters The code C(D, G) has length n, dimension dim L (G) dim L(G D),

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Definition Let F/F be an algebraic function field, where F is a finite field. Let G and D = Q1 + · · · + Qn be divisors of F/F where Q1, . . . , Qn are distinct places of degree one and Qi / 2 suppG for all 1  i  n. The AG code C(D, G) is C (D, G) := {(f (Q1) , f (Q2) , . . . , f (Qn)) : f 2 L(G)} ✓ Fn where L (G) is the Riemann-Roch space of G. Code parameters The code C(D, G) has length n, dimension dim L (G) dim L(G D), which is dim L (G) if deg G < n and

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Definition Let F/F be an algebraic function field, where F is a finite field. Let G and D = Q1 + · · · + Qn be divisors of F/F where Q1, . . . , Qn are distinct places of degree one and Qi / 2 suppG for all 1  i  n. The AG code C(D, G) is C (D, G) := {(f (Q1) , f (Q2) , . . . , f (Qn)) : f 2 L(G)} ✓ Fn where L (G) is the Riemann-Roch space of G. Code parameters The code C(D, G) has length n, dimension dim L (G) dim L(G D), which is dim L (G) if deg G < n and minimum distance at least n deg G.

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Recall that C (D, aP) := {(f (Q1) , f (Q2) , . . . , f (Qn)) : f 2 L(aP)} ✓ Fn Hence, a 2 H(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

Application to polar coding

Recall that C (D, aP) := {(f (Q1) , f (Q2) , . . . , f (Qn)) : f 2 L(aP)} ✓ Fn Hence, a 2 H(P) if and only if L((a 1)P) $ L(aP) if and only if C(D, (a 1)P) $ C(D, aP) for a < n If 0 = α1 < α2 < · · · < αng are the first n g elements of H(P), then C(D, α1P) $ C(D, α2P) $ · · · $ C(D, αngP).

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Recall that C (D, aP) := {(f (Q1) , f (Q2) , . . . , f (Qn)) : f 2 L(aP)} ✓ Fn Hence, a 2 H(P) if and only if L((a 1)P) $ L(aP) if and only if C(D, (a 1)P) $ C(D, aP) for a < n If 0 = α1 < α2 < · · · < αng are the first n g elements of H(P), then C(D, α1P) $ C(D, α2P) $ · · · $ C(D, αngP). We will consider the kernel matrix G = 2 6 6 6 4 fn(P1) fn(P2) · · · fn(Pn) fn1(P1) fn1(P2) · · · fn1(Pn) . . . . . . . . . f1(P1) f1(P2) · · · f1(Pn) 3 7 7 7 5 , where for each i, 1  i  n, {f1, . . . , fi} is a basis for L(αiP).

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Example

Let F = F8(x, y) be the function field of the Suzuki curve with defining equation y 8 y = x4(x8 x), and let α be a primitive element of F8. The Weierstrass semigroup at P1 is H(P1) = h8, 10, 12, 13i . Using this, we see that C(D, P1) $ C(D, 8P1) $ C(D, 10P1) $ C(D, 12P1) $ C(D, 13P1) $ C(D, 16P1) $ C(D, 18P1) $ C(D, 20P1) $ C(D, 21P1) $ C(D, 22P1) $ C(D, 23P1) $ C(D, 24P1) $ C(D, 25P1) $ C(D, 26P1) $ C(D, 28P1) $ C(D, 29P1) $ · · · $ C(D, 63P1) $ C(D, 65P1) $ C(D, 66P1) $ C(D, 67P1) $ C(D, 68P1) $ C(D, 69P1) $ C(D, 70P1) $ C(D, 71P1) $ C(D, 73P1) $ C(D, 75P1) $ C(D, 78P1) $ C(D, 79P1) $ C(D, 81P1) $ C(D, 83P1) $ C(D, 90P1) $ C(D, 91P1) = F64

8

.

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Polar codes were developed in [Arikan, 2009] as an explicit construction of symmetric capacity achieving codes for binary discrete memoryless channels with low encoding and decoding complexity. Definition The symmetric capacity of a discrete memoryless channel (DMC) W : X ! Y where | X |= q is I(W ) = X

y2Y

X

x2X

1 q W (y|x) logq W (y|x)

1 q

P

x02X W (y|x0)

! . Chain rule for mutual information Suppose that G 2 Fn⇥n

2

is nonsingular and un

1 2 Fn

• 2. Let xn

1 := un 1G and y n 1 be

the output of W n : X n ! Yn given xn

1 . Then

nI(W ) = I(X n

1 ; Y n 1 ) = I(Un 1 ; Y n 1 ) = n

X

i=1

I(Ui; Y n

1 , Ui1 1

) Transform n independent uses of W : X ! Y into uses of channels

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Copies of the channel are synthesized to obtain an upgraded channel and a degraded channel. Given a binary DMC W : X ! Y, form the vector channel W2 : X 2 ! Y2 and then split W2 to obtain channels W (1)

2

: X ! Y2 W (2)

2

: X ! Y2 ⇥ X

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Copies of the channel are synthesized to obtain an upgraded channel and a degraded channel. Given a binary DMC W : X ! Y, form the vector channel W2 : X 2 ! Y2 and then split W2 to obtain channels W (1)

2

: X ! Y2 W (2)

2

: X ! Y2 ⇥ X Recall that I ⇣ W (1)

2

⌘ + I ⇣ W (2)

2

⌘ = 2I (W ) .

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Copies of the channel are synthesized to obtain an upgraded channel and a degraded channel. Given a binary DMC W : X ! Y, form the vector channel W2 : X 2 ! Y2 and then split W2 to obtain channels W (1)

2

: X ! Y2 W (2)

2

: X ! Y2 ⇥ X Recall that I ⇣ W (1)

2

⌘ + I ⇣ W (2)

2

⌘ = 2I (W ) . The idea is to combine and split so that I ⇣ W (1)

2

⌘  I (W )  I ⇣ W (2)

2

⌘ .

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Suppose G 2 Fl⇥l

q , N := ln, and W : X ! Y where X = Fq. Consider

W (i)

N : X ! YN ⇥ X i1 defined by

W (i)

N

• y N

1 , ui1 1

| ui

• = P

uN

i+12X Ni

1 2N1 W N

y N

1 | uBNG ⌦n

.

. . . . . . . . .

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Suppose G 2 Fl⇥l

q , N := ln, and W : X ! Y where X = Fq. Consider

W (i)

N : X ! YN ⇥ X i1 defined by

W (i)

N

• y N

1 , ui1 1

| ui

• = P

uN

i+12X Ni

1 2N1 W N

y N

1 | uBNG ⌦n

.

. . . . . . . . .

Associated with the kernel G is a value called the exponent E(G) which provides an approximation to the rate of decay error probability for sufficiently large blocklengths.

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Consider a function field F/Fq with rational places P, P1, . . . , Pn and 0 = α1 < α2 < · · · < αng < αng+1 < · · · < αn. Find a basis for L(αnP) so that for each i {f1, . . . , fi} is a basis for L(αiP). Set G = 2 6 6 6 4 fn(P1) fn(P2) · · · fn(Pn) fn1(P1) fn1(P2) · · · fn1(Pn) . . . . . . . . . f1(P1) f1(P2) · · · f1(Pn) 3 7 7 7 5 ,

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Application to polar coding

Consider a function field F/Fq with rational places P, P1, . . . , Pn and 0 = α1 < α2 < · · · < αng < αng+1 < · · · < αn. Find a basis for L(αnP) so that for each i {f1, . . . , fi} is a basis for L(αiP). Set G = 2 6 6 6 4 fn(P1) fn(P2) · · · fn(Pn) fn1(P1) fn1(P2) · · · fn1(Pn) . . . . . . . . . f1(P1) f1(P2) · · · f1(Pn) 3 7 7 7 5 , Theorem (Anderson & M) The exponent of the polar code with kernel G satisfies E(G) 1 n 2 4logn((n g)!) +

n

X

i=ng+1

logn(di) 3 5 , where di denotes the minimum distance of C(D, αiP).

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding

Thank you!

Gretchen L. Matthews This work is supported by the NSF and NSA. Applications of algebraic geometry codes beyond classical coding