1 / 16

Fields: Defns

Fields: Defns. “Closed”: a,b in F  a+b , a.b in F Properties: Commutative: a+b = b+a , a.b = b.a Associative: a+(b+c )=( a+b)+c , a.(b.c ) = ( a.b).c Distributive: a.(b+c )= a.b+a.c a+0=0+a=a, a.1=1.a=a a+(-a)=0, a.a -1 =1. Facts about fields.

meghan
Download Presentation

Fields: Defns

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Fields: Defns • “Closed”: a,b in F a+b, a.b in F • Properties: • Commutative: a+b=b+a, a.b=b.a • Associative: a+(b+c)=(a+b)+c, a.(b.c) = (a.b).c • Distributive: a.(b+c)=a.b+a.c • a+0=0+a=a, a.1=1.a=a • a+(-a)=0, a.a-1=1

  2. Facts about fields • Examples: Q, R, C, P(x)/Q(x) if P(x),Q(x) in F(x),… • Non-examples: Z, P(x) in F(x), … • Algebraically closed: C • roots of P(x) in C(x) must be in C (Fundamental theorem of algebra) • Not algebraically closed: C • roots of P(x) in R(x) may not be in C

  3. Q1. “Useful facts” about finite F • Characteristic: • Finite (else infinite field) • Prime (else exist non-zero a,bs.t. a.b = 0) • Closed set under + and scalar ., other props “Must be” n copies of set of characteristic p. • Let the set (“group”) generated by powers of a be H. Then all sets of the form aH have the same size and are disjoint (bijection). Hence |H| divides |F|. Hence… • Eg: 3 in F7, but not 2.

  4. Q2. Prime-order fields • (a+b)mod(p), (a.b)mod(p) • … • -a = p-a, a-1 = a|F|-1 (why?) • Hint: Binomial theorem, mod p,… • Keep dividing P(x) by (x-ri). Not closed eg: x2+x+1 over F2

  5. Q2. Prime-order fields (contd.) • a±b (a±b)mod(p), cost O(log(p)) • a.b (a.b)mod(p), cost O(log2(p)) (why?) • ab (ab)mod(p), cost O(log3(p)) (generate a, a2 ,a4,… in time O(log3(p)), then multiply subset also in time O(log3(p)) ) • logabHARD (brute force, O(p.poly(log(p)) • a/ba. b-1 • mb+np=1 (Euclid’s algorithm, find m) O(…?) • b|F|-1 ,cost O(log3(p))

  6. Q3.Prime-power-order fields • Analogue • a≅a(x) (with coeffs from Fp) • p≅p(x) (prime≅“irreducible” (no factors)) • … • If p(x) irreducible, consider F(x)(modp(x))… • Eg: x2+1 no solutions over R, but over C=R(x)/(x2+1)… • Bits…

  7. Q4. Linear algebra over finite fields • Yes • Yes • Yes • Yes • No. Example: (1 1) over F2. • No. • Yes • Yes • Yes

  8. S-Z Lemma (easy case) • If P(x) has degree d, then at most n roots. • Pra in F(P(a) = 0) ≤d/q • If P(x1,x2,…,xk) has degree d, then • Pra1,a2,…,ak in F(P(a1,a2,…,ak) = 0) ≤d/q • (Proof by Induction) • degree(x2y5+x4y4) = 8 by definition

  9. Q5. Rank of random matrices • m/q • mxm matrix M=(xij). • Det(M) polynomial of degree m • (1-q-n) (1-q-n+1)…(1-q-n+m+1)≥(1-q-n+m+1)m ≥1-mq-n+m+1 If n>(1+ε)m, ≈1-mq-mε

  10. Q6. BEC(p) • Prev question, q=2, R=…? • Approx pn bits erased • Complexity • Encoding time = O(n2) (Why?) • Decoding time = O(n3) (Why?) • Storage O(n2) • Design time O(n2)

  11. Q7. Prop. of Linear codes • x=Gm, 0=Hx • No. GT and T’H, for any invertible T, T’ • [G -I].[HT IT]T =[0] • x,y in C means (x-y) in C (why?) • Complexity: • Encoding: O(n2) • BSC(p) decoding: O(exp(n)) (naïve)

  12. Q8. Linear GV codes • Let xi be codeword with “low” weight d= dmin. • # codewords of weight at most d ~2nH(d) • PrG(Gx≠0 for all x of low wt) < (2nH(d). 2-n). 2-nR • Probabilistic method…

  13. Q9. Singleton Bound n qn-d+1≤qnR d-1 n-d+1

  14. Q10. Reed-Solomon encoding • Determinant(Vandermonde matrix) = ri distinct, q≥n. nR (x-x’) (m-m’) 0 m=m’ nR=n-dmin = n-nR=dmin

  15. 11. q-BSC(p) • Say q=2m, • Append (say) m’ = m1/2 zeroes to each packet. • Detect errors (w.p. ~ 2m’). • Use erasure code to decode. • Random vs. worst-case noise • Naïve: O(n2), O(n3), O(n), O(n) • (Can “cleverly” do O(n.log(n)), O(n.log(n)), O(1), O(1) – how?)

  16. 12. Reed-Solomon decoding • Note • xi = M(ri). • Define “error-locator polynomial” E(ri)= • Define q(r,y) = E(r)(y-M(r)) • q(ri,yi)=0 (why?) • E(ri)yi=E(ri)M(ri)=T(ri) (definition) • T(.) of degree k+t-1 in r, and E(ri) of degree t, hence # unknown coefficients k+2t+1 ≤ n, linear transform • Not unique (null-space), but only interested in T(r)/E(r). • This unique since T(ri)E’(ri)yi=T’(ri)E(ri)yi. • If yi= 0, then T(ri)=T’(ri) • If yi≠ 0, then T(ri)/E(ri)=T’(ri)/E’(ri) • Degree of M(r) = T(t)/E(r) at most k-1, hence must be equal.

More Related