admin
	ps5 due
	ps6 out
today
	local unique decoding
recall
	f(\vx) avg case hard iff \vx\mapsto \vx,f(\vx) is a PRG
		[[ie, cannot predict last bit]]
		[[gets 1 bit stretch]]
	NisanWigderson: f (2^{\Omega(\ell)},1/2-1/2^{\Omega(\ell)}) avg case hard => PRG w/ logarithmic seed-length
				[[asking for maximal circuit size hardness]]
				[[evaluate f on *nearly* disjoint inputs]]
				[[used hybrid argument]]
			f in E => (mildly) explicit PRG => P=BPP
	goal: f worst case hard => \hat{f} avg case hard
	last time:
		f size-t worst case hard
		Enc a "good" locally decodable code
			\delta-uniquely decodable
			[[LDC picture]]
			[[LCC picture]]
		=> \hat{f}=Enc(f) is (~t,\delta) average-case hard
	constructions
		Hadamard code
			min dist = 1/2 => 1/4-unique global decoding
			prop: (1/4-\eps)-unique local decoding, poly(\log L,1/\eps) decoding time
				\-> great [[for unique dec]]		\-> great
			bad: \Enc:\bits^n\to\bits^{2^n} 
				poor rate
		Reed-Solomon codes
			rate: great!
			locality: poor (provably)
		Reed-Muller:
			interpolates between Hadamard and Reed-Solomon
			distance: ok
			locality: ok
			rate: ok
			alphabet size: fix with concatenation
	today: local decoding of reed-muller codes
reed-muller
	def(Reed-Muller):
		\Enc:\F_q^{(n+d choose d)}\to\F_q^{q^m}
		f m-variate total degree polynomial d
		f\mapsto (f(\vaa))_{\vaa\in\F_q^m}
	lem[Schwartz Zippel]
		f(\vx) degree \le d poly, non-zero
		Pr_{\vaa\in \F_q}[f(\vaa)=0]\le d/q
		=> f\ne g, Pr[f(\vaa)=g(\vaa)]\le d/q
		=> RM(d,m) has distance 1-d/q
	thm: RM(d,m) has 1/12-unique local correcting algorithm in time poly(m,q), for d\le q/9, q\ge 36
	param:
		[[can do better, but these are reasonable parameters]]
		distance: \Omega(1) [[pretty good]]
		rate: will do later
		locality: poly(m,q) vs q^m
			[[local vs global]
			[[can do better, but suffices for our parameters]]
		d\le q/9: so that 1-d/q\ge \Omega(1)
		q\ge 36: so that 1-d/q\ge \Omega(1) [[and technical point later]]
	pf1:
		idea: random self-reducability w/ lines
		hadamard: 
			random r, <a,x>=<r,x>+<a-r,x>
			ie, random r, a\in\span\{r,a-r\} [[using two points from one-dimensional line]]
		idea
			[[draw picture]]
		defn: 
			a parameterized line is \ell_{x,y}:\F\to\F^m
			with \ell_{x,y}(t)=x+ty
			[[parameterized by t]]
		lem: g|_{\ell_{x,y}}=g(\ell_{x,y}(t)):\F\to\F is degree \le d polynomial
					[[deg d ( deg 1)]]
		algo: 
			input: g:\F^m\to\F, x\in\F^m, degree d bound [[want to correct g to low degree po
			output: unique degree \le d polynomial p, with \delta(p,g)\le ..., output p(\vx)
			1) choose y\from\F^m randomly. construct \ell_{x,y}
			2) query g|_{\ell_{x,y}} on \alpha_0,\alpha_1,\ldots,\alpha_d\in\F\setminus\{0\} [[\alpha_i fixed constants]]
			3) find *unique* degree \tilde{p} \le d poly agreeing with (2)
			4) output \tilde{p}(0)
		runtime: poly(m,q) [[d\le q]]
		lem: \delta(g,p)\le 1/(3(d+1)) some p => algo outputs p(\vx) w/p \ge 2/3
		pf
			\alpha_i\in\F\setminus\{0\} fixed
			\ell_{x,y}(\alpha_i)=x+\alpha_i y, random over y
			Pr_y[g|_\ell(\alpha_i)\ne p|_\ell(\alpha_i)]\le 1/3(d+1)
			Pr_y[\exists i, g|_\ell(\alpha_i)\ne p|_\ell(\alpha_i)]\le 1/3
			Pr_y[recover p|_\ell] \ge 1-1/3=2/3 [[then evaluate p|_\ell at 0]]
		rmk: not good for large d [[we need d large for good rate]]
	pf2
		Q. how to do better?
		A. use reed-solomon decoding
			[[draw picture]]
		algo: 
			input: g:\F^m\to\F, x\in\F^m, degree d bound [[want to correct g to low degree po
			output: for unique degree \le d polynomial p, with \delta(p,g)\le ..., output p(\vx)
			1) choose y\from\F^m randomly. construct \ell_{x,y}
			2) query g|_{\ell_{x,y}} on all \alpha\in\F
			3) find *unique* degree \tilde{p} \le d poly agreeing with (2), using 1/3-unique decoding of Reed-Solomon codes
				if it exists!
			4) output \tilde{p}(0)
		runtime: poly(m,q)
		prop: deterministic algorihm for 1/3-unique decoding of Reed-Solomon codes, if if d\le q/9 [[here we use degree bound]]
		pf:
			use 1-2\sqrt{d/q}\ge 1-2*1/3=1/3 list decoding
			unique decoding radius is 1/2*(1-d/q)\ge 1/3
		lem: \delta(g,p)\le 1/12, some \deg p\le d, then such \tilde{p} is uniquely found w/p \ge 2/3
			[[need to prove is unique]]
			[[need to prove is found]]
		pf
			[[draw picture]]
			\E_y[\delta(g|_\ell,p|_\ell)]\le 1/q+1/12
							\le 1/36+1/12=1/9	[[use field size]]
			Markov => Pr_\ell[\delta(g|_\ell,p|_\ell)\ge 1/3]\le 1/3
				=> \Pr_\ell[unique \tilde{p}]\ge 1-1/3=2/3
	got: \Omega(1)-locally correctible code with reasonable parameters
	need: 			*decodable*
		[[give systematic encoding]]
	defn: Enc:\bits^L\to\F^{\hat{L}} is *systematic* if first L coordinates are identity function
		[[systematic LCC are LDC]]
	idea:
						f \deg \le d \mapsto \{f(\vaa)\}_{\vaa\in\F_q^m}
		 \{f(\vaa)\}_{\vaa\in S}	\mapsto
		 				\=> statement about polynomial interpolation
	lem: f:\bits^\ell\to\bits. exist unique polynomial \hat{f}\in\F[x_1,\ldots,x_\ell] with
		\deg_{x_i}\hat{f} \le 1
		\hat{f}|_{\bits^\ell}=f
		[[\hat{f} is called the low-degree extension of f]]
	pf:
		existence:
			\hat{f}(\vx)=\sum_{\vaa\in\bits^\ell} f(\vaa) \ind{\vx=\vaa}
									=\prod_{\alpha_i=1} x_i \prod_{\alpha_i=0} (1-x_i)
									\->	1 	for \vx=\vaa, for \vx\in\bits^\ell
										0	else
			\hat{f}(\vaa)=f(\vaa)
		uniqueness:
			V={polynomials with \deg_{x_i}\hat{f} \le 1}
				linear space
				dimension 2^n
			W={functions f:\bits^\ell\to\F}
				linear space
				dimension 2^n
			eval:V\mapsto W[[sends polynomial to its evaluation]]
			lem: \eval is linear: eval(f+g)=eval(f)+eval(g)
			existence => eval is surjective
			Q. injectivity?
			dimension count + surjective => bijectivity
	cor: S\subseteq\F. f:S^\ell\to \F. exists unique polynomial \hat{f} with
		\deg_{x_i}\hat{f} < |S|
		\hat{f}|_{S^\ell}=f
	prop: explicit Enc:\bits^L\to\Sigma^{\hat{L}}
		\hat{L}=\poly(L)
		|\Sigma|=\polylog(L)
		1/12 uniquely locally decodable in time polylog(L)
	pf.	
		f:\bits^\ell\to\bits	\mapsto g:S^{\ell'}\to bits, if |S|^{\ell'}\ge 2^\ell
		take |S|=\ceil{\sqrt{q}}, m=\ceil{\ell/\log |S|}
		\hat{\ell}=m\cdot \log q=O(\ell)
		encode g as \hat{g}
			individual degree <|S|
			total degree \le\ell*(|S|-1)\le \ell\sqrt{q}
		local correction if q\ge 81\ell^2
		encoding in time poly(L)=poly(2^\ell)
	cor: explicit Enc:\bits^L\to\bits^{\hat{L}}
		\hat{L}=\poly(L)
		\Sigma=\bits
		1/48 uniquely locally decodable in time polylog(L)
	pf
		concatenate with Hadamard
		1/4-uniquely globally decodable
				\-> inner code is small [[so loss is acceptable]]
	cor: f:\bits^\ell\to\bits in E=TIME(2^{O(\ell)}) t(\ell)-worst-case hard
		=> \hat{f}:\bits^{O(\ell)}\to\bits in E that is (t(\ell)/\poly(\ell),1/48)-avg case hard
next time:
	local list decoding