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

• 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

Q1. “Useful facts” about finite F

• Characteristic:– Finite (else infinite field)– Prime (else exist non-zero a,b s.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.

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

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/ba. b-1

– mb+np=1 (Euclid’s algorithm, find m) O(…?)– b|F|-1 , cost O(log3(p))

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)(mod p(x))…– Eg: x2+1 no solutions over R, but over

C=R(x)/(x2+1)…• Bits…

Q4. Linear algebra over finite fields

• Yes• Yes• Yes• Yes• No. Example: (1 1) over F2.• No.• Yes• Yes• Yes

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

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

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)

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)

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…

€

PrG (Gr m =

r x ) =

i=1

nR

ΠPrGi(Gi

r m =

r x ) =

−nR

2

Q9. Singleton Boundn

n-d+1 d-1

qn-d+1≤qnR

Q10. Reed-Solomon encoding

nR(m-m’)(x-x’)

=

n-nR=dmin

nR=n-dmin

0m=m’

• Determinant(Vandermonde matrix) = ri distinct, q≥n. • €

(ri − rj )1≤ j< k ≤n

∏

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?)

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.

€

(r − ri)error location i

∏