Algebraic Codes and Invariance
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1 Algebraic Codes and Invariance Madhu Sudan Harvard April 30, 2016 AAD3: Algebraic Codes and Invariance 1 of 29
2 Disclaimer Very little new work in this talk! Mainly: Ex- Coding theorist s perspective on Algebraic and Algebraic-Geometric Codes What additional properties it would be nice to have in algebraic-geometry codes. April 30, 2016 AAD3: Algebraic Codes and Invariance 2 of 29
3 Outline of the talk Part 1: Codes and Algebraic Codes Part 2: Combinatorics of Algebraic Codes Fundamental theorem(s) of algebra Part 3: Algorithmics of Algebraic Codes Product property Part 4: Locality of (some) Algebraic Codes Invariances Part 5: Conclusions April 30, 2016 AAD3: Algebraic Codes and Invariance 3 of 29
4 Part 1: Basic Definitions April 30, 2016 AAD3: Algebraic Codes and Invariance 4 of 29
5 Error-correcting codes Notation: FF qq - finite field of cardinality qq Encoding function: messages codewords EE: FF qq kk FF qq nn ; associated Code CC EE mm mm FF qq kk } Key parameters: Rate RR CC = kk/nn ; Distance δδ CC = min {δδ xx, yy }. xx yy CC δδ xx, yy = ii xx ii yy ii nn Pigeonhole Principle RR CC + δδ CC nn Algebraic codes: Get very close to this limit! April 30, 2016 AAD3: Algebraic Codes and Invariance 5 of 29
6 Algorithmic tasks Encoding: Compute EE mm given mm. Testing: Given rr FF qq nn, decide if mm s. t. rr = EE(mm)? Decoding: Given rr FF qq nn, compute mm minimizing δδ(ee mm, rr) [All linear codes nicely encodable, but algebraic ones efficiently (list) decodable.] Locality Perform tasks (testing/decoding) in oo nn time. Assume random access to rr Suffices to decode mm ii, for given ii [kk] [Many algebraic codes locally decodable and testable!] April 30, 2016 AAD3: Algebraic Codes and Invariance 6 of 29
7 Algebraic Codes? General paradigm: Message space = (Vector) space of functions Coordinates = Subset of domain of functions Encoding Messages = evaluations = FF rr qq [xx 1, of, xxmessage mm ] on domain Examples: Reed-Solomon Codes: Domain Reed-Muller = (subset Codes: of) FF qq Messages = mm Domain = FF FF kk qq qq [xx] (univ. poly. of deg. kk) Algebraic-Geometry Codes: mm Domain = Rational points of irred. curve in FF qq (mm-var poly. of deg. rr) Messages = Functions of bounded order Reed-Solomon Codes Reed-Muller Codes Algebraic-Geometric Codes Others (BCH codes, dual BCH codes ) April 30, 2016 AAD3: Algebraic Codes and Invariance 7 of 29
8 Part 2: Combinatorics Fund. Thm April 30, 2016 AAD3: Algebraic Codes and Invariance 8 of 29
9 Essence of combinatorics Rate of code Dimension of vector space Distance of code Scarcity of roots Univ poly of deg kk has kk roots. Multiv poly of deg kk has kk qq fraction roots. Functions of order kk have fewer than kk roots [Bezout, Riemann-Roch, Ihara, Drinfeld- Vladuts] [Goppa, Tsfasman-Vladuts-Zink, Garcia- Stichtenoth] April 30, 2016 AAD3: Algebraic Codes and Invariance 9 of 29
10 Consequences qq nn codes CC satisfying RR CC + δδ CC = nn For infinitely many qq, there exist infinitely many nn, and codes CC qq,nn over FF qq satisfying RR CC qq,nn + δδ CC qq,nn 1 1 qq 1 Many codes that are better than random codes Reed-Solomon, Reed-Muller of order 1, AG, BCH, dual BCH Moral: Distance property Algebra! April 30, 2016 AAD3: Algebraic Codes and Invariance 10 of 29
11 Part 3: Algorithmics Product property April 30, 2016 AAD3: Algebraic Codes and Invariance 11 of 29
12 Remarkable algorithmics Combinatorial implications: Code of distance δδ Corrects δδ 2 fraction errors uniquely. Corrects 1 lists. Algorithmically? 1 δδ fraction errors with small For all known algebraic codes, above can be matched! Why? April 30, 2016 AAD3: Algebraic Codes and Invariance 12 of 29
13 Product property Reed-Solomon Codes: For vectors uu, vv FF nn nn qq, let uu vv FF qq denote their coordinate-wise CC = deg kk poly product. EE = deg nn kk poly 2 For linear codes AA, BB FF nn qq, let AA BB = span aa bb aa AA, bb BB} EE CC = deg nn+kk 2 poly Obvious, but remarkable, feature: For every known algebraic code CC of distance δδ code EE of dimension δδ 2 nn s.t. EE CC is a code of distance δδ 2. Terminology: EE, EE CC error-locating pair for CC April 30, 2016 AAD3: Algebraic Codes and Invariance 13 of 29
14 Unique decoding by error-locating pairs nn Given rr FF qq : Find xx CC s.t. δδ xx, rr δδ 2 Algorithm Step 1: Find ee EE, ff EE CC s.t. ee rr = ff [Linear system] [Pellikaan], [Koetter], [Duursma] 90s Step 2: Find xx CC s.t. ee xx = ff [Linear system again!] Analysis: Solution to Step 1 exists? Yes provided dim EE > #errors Solution to Step 2 xx = xx? Yes Provided δδ EE CC large enough. April 30, 2016 AAD3: Algebraic Codes and Invariance 14 of 29
15 List decoding abstraction Increasing basis sequence bb 1, bb 2, CC ii span bb 1,, bb ii δδ CC ii nn ii + oo nn bb ii bb jj CC ii+jj ( CC ii CC jj CC ii+jj ) Reed-Solomon Codes: [Guruswami, Sahai, S. 2000s] List-decoding bb algorithms ii = xx ii (or its evaluations etc.) correcting 1 1 δδ fraction errors exist for codes with increasing basis sequences. (Increasing basis Error-locating pairs) April 30, 2016 AAD3: Algebraic Codes and Invariance 15 of 29
16 Part 4: Locality Invariances April 30, 2016 AAD3: Algebraic Codes and Invariance 16 of 29
17 Locality in Codes General motivation: Does correcting linear fraction of errors require scanning the whole code? Does testing? Deterministically: Yes! Probabilistically? Not necessarily!! If possible, potentially a very useful concept Definitely in other mathematical settings PCPs, Small-set expanders, Hardness amplification, Private information retrieval Maybe even in practice April 30, 2016 AAD3: Algebraic Codes and Invariance 17 of 29
18 Locality of some algebraic codes Locality is a rare phenomenon. Reed-Solomon codes are not local. Random codes are not local. AG codes are (usually) not local. Basic examples are algebraic and a few composition operators preserve it. Canonical example: Reed-Muller Codes = lowdegree polynomials. April 30, 2016 AAD3: Algebraic Codes and Invariance 18 of 29
19 Main Example: Reed-Muller Codes Message = multivariate polynomial; Encoding = evaluations everywhere. RM mm, rr, qq = { ff αα αα FFqq mm ff FF q xx 1,, xx mm, deg ff rr} Locality? (when rr < qq) Restrictions of low-degree polynomials to lines yield low-degree (univ.) polys. Random lines sample FF qq mm uniformly (pairwise ind ly) Locality ~ qq April 30, 2016 AAD3: Algebraic Codes and Invariance 19 of 29
20 Locality? Necessary condition: Small (local) constraints. Examples deg ff 1 ff xx + ff xx + 2yy = 2ff xx + yy ff llllllll has low-degree (so values on line are not arbitrary) Local Constraints local decoding/testing? No! Transitivity + locality? April 30, 2016 AAD3: Algebraic Codes and Invariance 20 of 29
21 Symmetry in codes AAAAAA CC ππ SS nn xx CC, xx ππ CC} Well-studied concept. Cyclic codes Basic algebraic codes (Reed-Solomon, Reed- Muller, BCH) symmetric under affine group Domain is vector space FF QQ tt (where QQ tt = nn) Code invariant under non-singular affine transforms from FF QQ tt FF QQ tt April 30, 2016 AAD3: Algebraic Codes and Invariance 21 of 29
22 Symmetry and Locality Code has l-local constraint + 2-transitive Code is l-locally decodable from OO 1 -fraction errors. l 2-transitive? ii jj, kk ll ππ Aut CC s. t. ππ ii = kk, ππ jj = ll Why? Suppose constraint ff aa = ff bb + ff cc + ff dd Wish to determine ff xx Find random ππ Aut CC s.t. ππ aa = xx; ff xx = ff ππ bb + ff ππ cc + ff(ππ dd ); ππ bb, ππ cc, ππ (dd) random, ind. of xx April 30, 2016 AAD3: Algebraic Codes and Invariance 22 of 29
23 Symmetry + Locality - II Local constraint + affine-invariance Local testing specifically Theorem [Kaufman-S. 08]: CC l-local constraint & is FF QQ tt -affine-invariant CC is l (l, QQ)-locally testable. Theorem [Ben-Sasson,Kaplan,Kopparty,Meir] CC has product property & 1-transitive CC 1- transitive and locally testable and product property Theorem [B-S,K,K,M,Stichtenoth] Such CC exists. (CC = AG code) April 30, 2016 AAD3: Algebraic Codes and Invariance 23 of 29
24 Aside: Recent Progress in Locality - 1 [Yekhanin,Efremenko 06]: 3-Locally decodable codes of subexponential length. [Kopparty-Meir-RonZewi-Saraf 15]: nn oo(1) -locally decodable codes w. RR + δδ 1 log nn log nn -locally testable codes w. RR + δδ 1 Codes not symmetric, but based on symmetric codes. April 30, 2016 AAD3: Algebraic Codes and Invariance 24 of 29
25 Aside 2: Symmetric Ingredients Lifted Codes [Guo-Kopparty-S. 13] CC mm,dd,qq = ff: FF mm qq FF qq deg ff llllllll dd llllllll Multiplicity Codes [Kopparty-Saraf-Yekhanin 10] Message = biv. polynomial Encode ff via evaluations of (ff, ff xx, ff yy ) April 30, 2016 AAD3: Algebraic Codes and Invariance 25 of 29
26 Part 5: Conclusions April 30, 2016 AAD3: Algebraic Codes and Invariance 26 of 29
27 Remarkable properties of Algebraic Codes Strikingly strong combinatorially: Often only proof that extreme choices of parameters are feasible. Algorithmically tractable! The product property! Surprisingly versatile Broad search space (domain, space of functions) April 30, 2016 AAD3: Algebraic Codes and Invariance 27 of 29
28 Quest for future Construct algebraic geometric codes with rich symmetries. In general points on curve have few(er) symmetries. Can we construct curve carefully? Symmetry inherently? Symmetry by design? Still work to be done for specific applications. April 30, 2016 AAD3: Algebraic Codes and Invariance 28 of 29
29 Thank You! April 30, 2016 AAD3: Algebraic Codes and Invariance 29 of 29
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