Algebraic Geometry Codes. Shelly Manber. Linear Codes. Algebraic Geometry Codes. Example: Hermitian. Shelly Manber. Codes. Decoding.

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1 Linear December 2, 2011

2 References Linear Main Source: Stichtenoth, Henning. Function Fields and. Springer, Other Sources: Høholdt, Lint and Pellikaan. geometry codes. Handbook of Coding Theory, vol 1, p , Amsterdam, Bartley and Walker. Geometric over Rings. World Scientific Review, June Guruswami,Venkatesan. Notes 2: Gilbert-Varshamov bound. Jan 2010.

3 Error Correcting Linear

4 Error Correcting Linear Natural way of sending information is a stream of bits:

5 Error Correcting Linear Natural way of sending information is a stream of bits: If one bit is off, the entire message may be 100% different

6 Error Correcting Linear Natural way of sending information is a stream of bits: If one bit is off, the entire message may be 100% different Error Correcting : a way to send information along a noisy channel so that the original message can be recovered with high probability

7 Definitions Definition: A linear code C is a linear subpace of F n q. Linear

8 Definitions Definition: A linear code C is a linear subpace of F n q. Linear Definition: The dimension k of a linear code is its dimension as a vector space

9 Definitions Definition: A linear code C is a linear subpace of F n q. Linear Definition: The dimension k of a linear code is its dimension as a vector space Definition: The Hamming distance between two elements of F n q is the number of coefficients on which the two elements differ. The weight of an element, wt(e), is the number of nonzero coefficients.

10 Definitions Definition: A linear code C is a linear subpace of F n q. Linear Definition: The dimension k of a linear code is its dimension as a vector space Definition: The Hamming distance between two elements of F n q is the number of coefficients on which the two elements differ. The weight of an element, wt(e), is the number of nonzero coefficients. Definition: The minimum distance of a code is the minimum Hamming distance between any two elements of the linear code (equivalently the minimum weight of a codeword)

11 Encoding An encoding is a map from a message into the code C Linear

12 Encoding An encoding is a map from a message into the code C Linear A generator matrix is a k n matrix whose rows are a basis for C

13 Encoding An encoding is a map from a message into the code C Linear A generator matrix is a k n matrix whose rows are a basis for C A generator matrix gives an encoding: ( x1 x 2 ) x 3 a 1 a 2 a 3 a 4 a 5 b 1 b 2 b 3 b 4 b 5 c 1 c 2 c 3 c 4 c 5

14 Error Checking Definition: The dual C of a code C is its dual as a vector space under the canonical inner product. Linear

15 Error Checking Linear Definition: The dual C of a code C is its dual as a vector space under the canonical inner product. Definition: A parity check matrix M for C is an (n k) n matrix whose rows are a basis for C

16 Error Checking Linear Definition: The dual C of a code C is its dual as a vector space under the canonical inner product. Definition: A parity check matrix M for C is an (n k) n matrix whose rows are a basis for C Claim: A vector x := (x 1,..., x n ) F n q is in C if and only if Mx = 0.

17 Error Checking Linear Definition: The dual C of a code C is its dual as a vector space under the canonical inner product. Definition: A parity check matrix M for C is an (n k) n matrix whose rows are a basis for C Claim: A vector x := (x 1,..., x n ) F n q is in C if and only if Mx = 0. Proof. (C ) = C, so

18 Error Checking Linear Definition: The dual C of a code C is its dual as a vector space under the canonical inner product. Definition: A parity check matrix M for C is an (n k) n matrix whose rows are a basis for C Claim: A vector x := (x 1,..., x n ) F n q is in C if and only if Mx = 0. Proof. (C ) = C, so x C x, c = 0 c C

19 Error Checking Linear Definition: The dual C of a code C is its dual as a vector space under the canonical inner product. Definition: A parity check matrix M for C is an (n k) n matrix whose rows are a basis for C Claim: A vector x := (x 1,..., x n ) F n q is in C if and only if Mx = 0. Proof. (C ) = C, so x C x, c = 0 c C Mx = 0

20 Linear Let C be an [n, k, d] code. If x F n q has Hamming distance (d 1)/2 from a codeword c C then c is the unique codeword with minimal distance to x.

21 Linear Let C be an [n, k, d] code. If x F n q has Hamming distance (d 1)/2 from a codeword c C then c is the unique codeword with minimal distance to x. Definition: A decoding is an algorithm, given a F n q and the guarantee that a = c + e for some c C and e F n q with weight (d 1)/2, to recover c.

22 Bounds To maximize code efficacy: Should d be higher or lower with respect to n? Linear

23 Bounds To maximize code efficacy: Should d be higher or lower with respect to n? Should k be higher or lower with respect to n? Linear

24 Bounds Linear To maximize code efficacy: Should d be higher or lower with respect to n? Should k be higher or lower with respect to n? Theorem (Singleton Bound): for all linear codes k + d n + 1

25 Bounds Linear To maximize code efficacy: Should d be higher or lower with respect to n? Should k be higher or lower with respect to n? Theorem (Singleton Bound): for all linear codes k + d n + 1 Proof. Let V := {(a 1,..., a n ) F n q a i = 0 i d}

26 Bounds Linear To maximize code efficacy: Should d be higher or lower with respect to n? Should k be higher or lower with respect to n? Theorem (Singleton Bound): for all linear codes Proof. Let k + d n + 1 V := {(a 1,..., a n ) F n q a i = 0 i d} So dim(v ) = d 1 and V C =, so

27 Bounds Linear To maximize code efficacy: Should d be higher or lower with respect to n? Should k be higher or lower with respect to n? Theorem (Singleton Bound): for all linear codes Proof. Let k + d n + 1 V := {(a 1,..., a n ) F n q a i = 0 i d} So dim(v ) = d 1 and V C =, so n = dim(f n q) dim(v + C) = dim(v ) + dim(c) = (d 1) + k

28 Bounds Linear To maximize code efficacy: Should d be higher or lower with respect to n? Should k be higher or lower with respect to n? Theorem (Singleton Bound): for all linear codes Proof. Let k + d n + 1 V := {(a 1,..., a n ) F n q a i = 0 i d} So dim(v ) = d 1 and V C =, so n = dim(f n q) dim(v + C) = dim(v ) + dim(c) = (d 1) + k

29 Reed-Solomon Linear Let n = q 1 and F q = {0, 1, α,..., α n 1 }. Choose k n, and define L k = {f F q [X ] deg f k 1}

30 Reed-Solomon Linear Let n = q 1 and F q = {0, 1, α,..., α n 1 }. Choose k n, and define Define L k = {f F q [X ] deg f k 1} C RS = {(f (1), f (α),..., f (α n 1 ) f L k }

31 Reed-Solomon Linear Let n = q 1 and F q = {0, 1, α,..., α n 1 }. Choose k n, and define Define L k = {f F q [X ] deg f k 1} C RS = {(f (1), f (α),..., f (α n 1 ) f L k } Claim: C RS is an [n, k, n k + 1] code. Proof. 1 L k C RS is an injective F q -linear map

32 Reed-Solomon Linear Let n = q 1 and F q = {0, 1, α,..., α n 1 }. Choose k n, and define Define L k = {f F q [X ] deg f k 1} C RS = {(f (1), f (α),..., f (α n 1 ) f L k } Claim: C RS is an [n, k, n k + 1] code. Proof. 1 L k C RS is an injective F q -linear map 2 A polynomial of degree k 1 has at most k 1 zeros, so d n k + 1

33 Reed-Solomon Linear Let n = q 1 and F q = {0, 1, α,..., α n 1 }. Choose k n, and define Define L k = {f F q [X ] deg f k 1} C RS = {(f (1), f (α),..., f (α n 1 ) f L k } Claim: C RS is an [n, k, n k + 1] code. Proof. 1 L k C RS is an injective F q -linear map 2 A polynomial of degree k 1 has at most k 1 zeros, so d n k By the Singleton bound, d n k + 1

34 Definition Linear Given: X a curve of genus g over F q with function field F

35 Definition Linear Given: X a curve of genus g over F q with function field F P 1,..., P n distinct places of F of degree one

36 Definition Linear Given: X a curve of genus g over F q with function field F P 1,..., P n distinct places of F of degree one D := P P n a divisor of X

37 Definition Linear Given: X a curve of genus g over F q with function field F P 1,..., P n distinct places of F of degree one D := P P n a divisor of X G a divisor of X such that Supp G Supp D = Define C L (D, G) := {(x(p 1 ),..., x(p n )) x L (G)} F n q

38 Definition Linear Given: X a curve of genus g over F q with function field F P 1,..., P n distinct places of F of degree one D := P P n a divisor of X G a divisor of X such that Supp G Supp D = Define C L (D, G) := {(x(p 1 ),..., x(p n )) x L (G)} F n q Claim: k = l(g) l(g D) and d n deg G.

39 Some Nice Properties Linear If deg G < n then 1 k = l(g) deg(g) + 1 g (Riemann-Roch)

40 Some Nice Properties Linear If deg G < n then 1 k = l(g) deg(g) + 1 g (Riemann-Roch) 2 L (G) C L (D, G) is injective

41 Some Nice Properties Linear If deg G < n then 1 k = l(g) deg(g) + 1 g (Riemann-Roch) 2 L (G) C L (D, G) is injective 3 For a basis {x 1,..., x k } of L (G), the matrix: x 1 (P 1 ) x 1 (P 2 )... x 1 (P n ) x 2 (P 1 ) x 2 (P 2 )... x 2 (P n ) x n (P 1 ) x n (P 2 )... x n (P n ) is a generator matrix for C L.

42 Bounds If deg G < n then k deg(g) + 1 g and d n deg G so Linear

43 Bounds Linear If deg G < n then k deg(g) + 1 g and d n deg G so k + d n + 1 g

44 Bounds If deg G < n then k deg(g) + 1 g and d n deg G so Linear But recall that k + d n + 1 g k + d n + 1

45 Bounds Linear If deg G < n then k deg(g) + 1 g and d n deg G so k + d n + 1 g But recall that k + d n + 1 So for genus 0 curves, k + d = n + 1

46 Bounds Linear If deg G < n then k deg(g) + 1 g and d n deg G so k + d n + 1 g But recall that k + d n + 1 So for genus 0 curves, k + d = n + 1 Unfortunately, for genus zero curves, n q + 1, so over a fixed alphabet, we can t send very large messages.

47 Bounds Linear If deg G < n then k deg(g) + 1 g and d n deg G so k + d n + 1 g But recall that k + d n + 1 So for genus 0 curves, k + d = n + 1 Unfortunately, for genus zero curves, n q + 1, so over a fixed alphabet, we can t send very large messages. Goal of coding theory: To construct asymptotically good curves

48 Asymptotically Good Curves Linear Definition: For an [n, k, d] code: R = k/n is called the information rate, and δ = d/n is called the relative distance

49 Asymptotically Good Curves Linear Definition: For an [n, k, d] code: R = k/n is called the information rate, and δ = d/n is called the relative distance Theorem (Gilbert-Varshamov bound): For any fixed q and δ 1 1/q, and an arbitrarily small ɛ > 0 there exists an infinite family of codes with R 1 h q (δ) ɛ where h q (x) is the entropy function: h q (x) := xlog q (q 1) xlog q (x) (1 x)log q (1 x)

50 Residues of Differentials Let P be a place of F with local parameter t. Linear

51 Residues of Differentials Linear Let P be a place of F with local parameter t. Claim: Any x F can be written uniquely as i=m a it i with a i F q for some integer m.

52 Residues of Differentials Linear Let P be a place of F with local parameter t. Claim: Any x F can be written uniquely as i=m a it i with a i F q for some integer m. Definition: For any differential ω of F, write ω = f dt f = a i t i i=m Define res P (ω) := a 1

53 Residues of Differentials Linear Let P be a place of F with local parameter t. Claim: Any x F can be written uniquely as i=m a it i with a i F q for some integer m. Definition: For any differential ω of F, write ω = f dt f = a i t i i=m Define res P (ω) := a 1 Claim: Residue is well-defined.

54 Another Code Linear Given: P 1,..., P n distinct places of F of degree one D = P P n a divisor of X G a divisor of X such that Supp G Supp D = as before, define C Ω (D, G) := {(res P1 (ω),..., res Pn (ω)) ω Ω F (G D)}

55 Duals Linear Proposition: C Ω (D, G) = C L (D, G)

56 Duals Linear Proposition: C Ω (D, G) = C L (D, G) Proposition: There exists a Weil Differential η which can be explicitly computed such that C Ω (D, G) = C L (D, G) = C Ω (D, D G + (η))

57 Linear Let F := F q 2(x, y) with y q + y = x q+1

58 Linear Let F := F q 2(x, y) with y q + y = x q+1 P α,β be the unique place such that x(p α,β ) = α and y(p α,β ) = β and P be the common pole of x and y.

59 Linear Let F := F q 2(x, y) with y q + y = x q+1 P α,β be the unique place such that x(p α,β ) = α and y(p α,β ) = β and P be the common pole of x and y. D = β q +β=α q+1 P α,β For each 0 < r < q 3 + q 2 q 2 define C r := C L (D, rp )

60 Generating Linear Proposition: For each r 0, the elements of the form x i y j with 0 i 0 j q 1 iq + j(q + 1) r form a basis for L (rp )

61 Generating Linear Proposition: For each r 0, the elements of the form x i y j with 0 i 0 j q 1 iq + j(q + 1) r form a basis for L (rp ) Corollary: The generating matrix for C r is the matrix whose rows are (α i β j ) β q +β=α q+1 where i and j satisfy the above conditions.

62 Bounds Linear Let N(r) be the number of pairs i, j satisfying the properties previously mentioned. Proposition: n = q 3 k = dim C r = d n r { N(r) 0 r < q 3 n N(r) q 3 r q 3 + q 2 q 2

63 Numerical q = 2 Linear Let F 4 = {0, 1, a, a + 1}, C defined by y 2 z + yz 2 + x 3, i.e. F = Frac(F q [x, y, z]/(y 2 z + yz 2 + x 3 ))

64 Numerical q = 2 Linear Let F 4 = {0, 1, a, a + 1}, C defined by y 2 z + yz 2 + x 3, i.e. F = Frac(F q [x, y, z]/(y 2 z + yz 2 + x 3 )) Rational points: [(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1), (1 : a : 1), (1 : a + 1 : 1), (a : a : 1), (a : a + 1 : 1), (a + 1 : a : 1), (a + 1 : a + 1 : 1)]

65 Numerical q = 2 Linear Let F 4 = {0, 1, a, a + 1}, C defined by y 2 z + yz 2 + x 3, i.e. F = Frac(F q [x, y, z]/(y 2 z + yz 2 + x 3 )) Rational points: [(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1), (1 : a : 1), (1 : a + 1 : 1), (a : a : 1), (a : a + 1 : 1), (a + 1 : a : 1), (a + 1 : a + 1 : 1)] Let r = 5. Then the basis for L (r(0 : 1 : 0)) is {1, x, y, x 2, xy}

66 Numerical generating matrix Linear Generating matrix for C: a a + 1 a a + 1 a a a a a a + 1 a a + 1 a + 1 a 0

67 Numerical generating matrix Linear Generating matrix for C: a a + 1 a a + 1 a a a a a a + 1 a a + 1 a + 1 a 0 Sum of third and fifth rows is ( ) So d = 3.

68 Overview Fix t 0 and let a = c + e with c C Ω and wt(e) t Linear

69 Overview Linear Fix t 0 and let a = c + e with c C Ω and wt(e) t We want an algorithm to compute c given a

70 Overview Linear Fix t 0 and let a = c + e with c C Ω and wt(e) t We want an algorithm to compute c given a Write e = (e 1,..., e n )

71 Overview Linear Fix t 0 and let a = c + e with c C Ω and wt(e) t We want an algorithm to compute c given a Write e = (e 1,..., e n ) Algorithm Steps: 1 Construct f F such that f (P i ) = 0 when e i 0.

72 Overview Linear Fix t 0 and let a = c + e with c C Ω and wt(e) t We want an algorithm to compute c given a Write e = (e 1,..., e n ) Algorithm Steps: 1 Construct f F such that f (P i ) = 0 when e i 0. 2 Compute e i for all i such that f (P i ) = 0.

73 Overview Linear Fix t 0 and let a = c + e with c C Ω and wt(e) t We want an algorithm to compute c given a Write e = (e 1,..., e n ) Algorithm Steps: 1 Construct f F such that f (P i ) = 0 when e i 0. 2 Compute e i for all i such that f (P i ) = 0. 3 Since e i = 0 for all i such that f (P i ) 0, we have computed e. Then c = a e.

74 Algorithm Conditions Linear Let G 1 be a divisor such that Supp G 1 Supp D = deg G 1 < deg G (2g 2) t l(g 1 ) > t

75 Algorithm Conditions Linear Let G 1 be a divisor such that Supp G 1 Supp D = deg G 1 < deg G (2g 2) t l(g 1 ) > t and let n [a, f ] := a i f (P i ) i=1

76 Algorithm Conditions Linear Let G 1 be a divisor such that Supp G 1 Supp D = deg G 1 < deg G (2g 2) t l(g 1 ) > t and let [a, f ] := Fix bases n a i f (P i ) i=1 {f 1,..., f l } of L (G 1 ) {g 1,..., g k } of L (G G 1 ) {h 1,..., h m } of L (G)

77 Algorithm Specifics Linear Proposition: For each 1 ρ k, the linear system l [a, f λ g ρ ] x λ = 0 λ=1 has a nontrivial solution (α 1,..., α l ).

78 Algorithm Specifics Linear Proposition: For each 1 ρ k, the linear system l [a, f λ g ρ ] x λ = 0 λ=1 has a nontrivial solution (α 1,..., α l ). Let l f := α λ f λ λ=1 Then f (P i ) = 0 whenever e i 0

79 Algorithm Specifics Linear Proposition: For each 1 µ m, the linear system h µ (P i ) z i = [a, h µ ] {i f (P i )=0} has a unique solution agreeing with e on all i such that f (P i ) = 0.

80 Algorithm Specifics Linear Proposition: For each 1 µ m, the linear system h µ (P i ) z i = [a, h µ ] {i f (P i )=0} has a unique solution agreeing with e on all i such that f (P i ) = 0. Conclusion: Solving the two linear systems completely computes e and hence we can recover c = a e.

81 Linear Proposition: The dual of C r is C q 3 +q 2 q 2 r

82 Linear Proposition: The dual of C r is C q 3 +q 2 q 2 r Corollary: C L (D, rp ) = C Ω (D, (q 3 + q 2 q 2 r)p )

83 Linear Proposition: The dual of C r is C q 3 +q 2 q 2 r Corollary: C L (D, rp ) = C Ω (D, (q 3 + q 2 q 2 r)p ) We can decode C Ω (D, (q 3 + q 2 q 2 r)p ) with the algorithm given.