Algebraic Codes and Invariance Madhu Sudan Harvard April 30, 2016 AAD3: Algebraic Codes and Invariance

AAD3: Algebraic Codes and Invariance Disclaimer Very little new work in this talk! Mainly: Coding theorist’s perspective on Algebraic and Algebraic-Geometric Codes What additional properties it would be nice to have in algebraic-geometry codes. Ex- April 30, 2016 AAD3: Algebraic Codes and Invariance

AAD3: Algebraic Codes and Invariance 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

Part 1: Basic Definitions April 30, 2016 AAD3: Algebraic Codes and Invariance

Error-correcting codes Notation: 𝔽 𝑞 - finite field of cardinality 𝑞 Encoding function: messages ↦ codewords 𝐸: 𝔽 𝑞 𝑘 → 𝔽 𝑞 𝑛 ; associated Code 𝐶≜ 𝐸 𝑚 𝑚∈ 𝔽 𝑞 𝑘 } Key parameters: Rate 𝑅 𝐶 = 𝑘/𝑛 ; Distance 𝛿 𝐶 = min 𝑥≠𝑦∈𝐶 {𝛿 𝑥,𝑦 } . 𝛿 𝑥,𝑦 = 𝑖 𝑥 𝑖 ≠ 𝑦 𝑖 | 𝑛 Pigeonhole Principle ⇒𝑅 𝐶 +𝛿 𝐶 ≤1+ 1 𝑛 Algebraic codes: Get very close to this limit! April 30, 2016 AAD3: Algebraic Codes and Invariance

AAD3: Algebraic Codes and Invariance Algorithmic tasks Encoding: Compute 𝐸 𝑚 given 𝑚. Testing: Given 𝑟∈ 𝔽 𝑞 𝑛 , decide if ∃ 𝑚 s.t. 𝑟=𝐸(𝑚)? Decoding: Given 𝑟∈ 𝔽 𝑞 𝑛 , compute 𝑚 minimizing 𝛿(𝐸 𝑚 ,𝑟) [All linear codes nicely encodable, but algebraic ones efficiently (list) decodable.] Perform tasks (testing/decoding) in 𝑜 𝑛 time. Assume random access to 𝑟 Suffices to decode 𝑚 𝑖 , for given 𝑖∈[𝑘] [Many algebraic codes locally decodable and testable!] Locality April 30, 2016 AAD3: Algebraic Codes and Invariance

AAD3: Algebraic Codes and Invariance Reed-Solomon Codes: Domain = (subset of) 𝔽 𝑞 Messages = 𝔽 𝑞 ≤𝑘 [𝑥] (univ. poly. of deg. 𝑘) General paradigm: Message space = (Vector) space of functions Coordinates = Subset of domain of functions Encoding = evaluations of message on domain Examples: Reed-Solomon Codes Reed-Muller Codes Algebraic-Geometric Codes Others (BCH codes, dual BCH codes …) Reed-Muller Codes: Domain = 𝔽 𝑞 𝑚 Messages = 𝔽 𝑞 ≤𝑟 [ 𝑥 1 ,…, 𝑥 𝑚 ] (𝑚-var poly. of deg. 𝑟) Algebraic-Geometry Codes: Domain = Rational points of irred. curve in 𝔽 𝑞 𝑚 Messages = Functions of bounded “order” April 30, 2016 AAD3: Algebraic Codes and Invariance

Part 2: Combinatorics ⇐ Fund. Thm April 30, 2016 AAD3: Algebraic Codes and Invariance

Essence of combinatorics Rate of code ⇐ Dimension of vector space Distance of code ⇐ Scarcity of roots Univ poly of deg ≤𝑘 has ≤𝑘 roots. Multiv poly of deg ≤𝑘 has ≤ 𝑘 𝑞 fraction roots. Functions of order ≤𝑘 have fewer than ≤𝑘 roots [Bezout, Riemann-Roch, Ihara, Drinfeld-Vladuts] [Goppa, Tsfasman-Vladuts-Zink, Garcia-Stichtenoth] April 30, 2016 AAD3: Algebraic Codes and Invariance

AAD3: Algebraic Codes and Invariance Consequences 𝑞≥𝑛⇒∃ codes 𝐶 satisfying 𝑅 𝐶 +𝛿 𝐶 =1+ 1 𝑛 For infinitely many 𝑞, there exist infinitely many 𝑛, and codes 𝐶 𝑞,𝑛 over 𝔽 𝑞 satisfying 𝑅 𝐶 𝑞,𝑛 +𝛿 𝐶 𝑞,𝑛 ≥1− 1 𝑞 −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

Part 3: Algorithmics ⇐ Product property April 30, 2016 AAD3: Algebraic Codes and Invariance

Remarkable algorithmics Combinatorial implications: Code of distance 𝛿 Corrects 𝛿 2 fraction errors uniquely. Corrects 1− 1−𝛿 fraction errors with small lists. Algorithmically? For all known algebraic codes, above can be matched! Why? April 30, 2016 AAD3: Algebraic Codes and Invariance

AAD3: Algebraic Codes and Invariance Product property Reed-Solomon Codes: 𝐶= deg 𝑘 poly 𝐸= deg 𝑛−𝑘 2 poly 𝐸∗𝐶= deg 𝑛+𝑘 2 poly For vectors 𝑢,𝑣∈ 𝔽 𝑞 𝑛 , let 𝑢∗𝑣∈ 𝔽 𝑞 𝑛 denote their coordinate-wise product. For linear codes 𝐴,𝐵≤ 𝔽 𝑞 𝑛 , let 𝐴∗𝐵=span 𝑎∗𝑏 𝑎∈𝐴, 𝑏∈𝐵} Obvious, but remarkable, feature: For every known algebraic code 𝐶 of distance 𝛿 ∃ code 𝐸 of dimension ≈ 𝛿 2 𝑛 s.t. 𝐸∗𝐶 is a code of distance 𝛿 2 . Terminology: 𝐸,𝐸∗𝐶 ≜ error-locating pair for 𝐶 April 30, 2016 AAD3: Algebraic Codes and Invariance

Unique decoding by error-locating pairs Given 𝑟∈ 𝔽 𝑞 𝑛 : Find 𝑥∈𝐶 s.t. 𝛿 𝑥,𝑟 ≤ 𝛿 2 Algorithm Step 1: Find 𝑒∈𝐸, 𝑓∈𝐸∗𝐶 s.t. 𝑒∗𝑟=𝑓 [Linear system] Step 2: Find 𝑥 ∈𝐶 s.t. 𝑒∗ 𝑥 =𝑓 [Linear system again!] Analysis: Solution to Step 1 exists? Yes – provided dim 𝐸 > #errors Solution to Step 2 𝑥 =𝑥? Yes – Provided 𝛿 𝐸∗𝐶 large enough. [Pellikaan], [Koetter], [Duursma] – 90s April 30, 2016 AAD3: Algebraic Codes and Invariance

List decoding abstraction Increasing basis sequence 𝑏 1 , 𝑏 2 , … 𝐶 𝑖 ≜span 𝑏 1 ,…, 𝑏 𝑖 𝛿 𝐶 𝑖 ≈𝑛−𝑖+𝑜 𝑛 𝑏 𝑖 ∗ 𝑏 𝑗 ∈ 𝐶 𝑖+𝑗 (⇔ 𝐶 𝑖 ∗ 𝐶 𝑗 ⊆ 𝐶 𝑖+𝑗 ) [Guruswami, Sahai, S. ‘2000s] List-decoding algorithms correcting 1− 1−𝛿 fraction errors exist for codes with increasing basis sequences. (Increasing basis ⇒ Error-locating pairs) Reed-Solomon Codes: 𝑏 𝑖 = 𝑥 𝑖 (or its evaluations etc.) April 30, 2016 AAD3: Algebraic Codes and Invariance

Part 4: Locality ⇐ Invariances April 30, 2016 AAD3: Algebraic Codes and Invariance

AAD3: Algebraic Codes and Invariance 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

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 = low-degree polynomials. April 30, 2016 AAD3: Algebraic Codes and Invariance

Main Example: Reed-Muller Codes Message = multivariate polynomial; Encoding = evaluations everywhere. RM 𝑚,𝑟,𝑞 ={ 𝑓 𝛼 𝛼∈ 𝔽 𝑞 𝑚 | 𝑓∈ 𝔽 q 𝑥 1 ,…, 𝑥 𝑚 , deg 𝑓 ≤𝑟} Locality? (when 𝑟<𝑞) Restrictions of low-degree polynomials to lines yield low-degree (univ.) polys. Random lines sample 𝔽 𝑞 𝑚 uniformly (pairwise ind’ly) Locality ~ 𝑞 April 30, 2016 AAD3: Algebraic Codes and Invariance

AAD3: Algebraic Codes and Invariance Locality ⇐ ? Necessary condition: Small (local) constraints. Examples deg 𝑓 ≤1⇔𝑓 𝑥 +𝑓 𝑥+2𝑦 =2𝑓 𝑥+𝑦 𝑓 ​ 𝑙𝑖𝑛𝑒 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

AAD3: Algebraic Codes and Invariance Symmetry in codes 𝐴𝑢𝑡 𝐶 ≜ 𝜋∈ 𝑆 𝑛 ∀ 𝑥∈ 𝐶, 𝑥 𝜋 ∈ 𝐶} Well-studied concept. “Cyclic codes” Basic algebraic codes (Reed-Solomon, Reed-Muller, BCH) symmetric under affine group Domain is vector space 𝔽 𝑄 𝑡 (where 𝑄 𝑡 =𝑛) Code invariant under non-singular affine transforms from 𝔽 𝑄 𝑡 → 𝔽 𝑄 𝑡 April 30, 2016 AAD3: Algebraic Codes and Invariance

AAD3: Algebraic Codes and Invariance Symmetry and Locality Code has ℓ-local constraint + 2-transitive ⇒ Code is ℓ-locally decodable from 𝑂 1 ℓ -fraction errors. 2-transitive? – ∀𝑖≠𝑗, 𝑘≠𝑙 ∃𝜋∈Aut 𝐶 s.t. 𝜋 𝑖 =𝑘, 𝜋 𝑗 =𝑙 Why? Suppose constraint 𝑓 𝑎 =𝑓 𝑏 +𝑓 𝑐 +𝑓 𝑑 Wish to determine 𝑓 𝑥 Find random 𝜋∈Aut 𝐶 s.t. 𝜋 𝑎 =𝑥; 𝑓 𝑥 =𝑓 𝜋 𝑏 +𝑓 𝜋 𝑐 +𝑓(𝜋 𝑑 ); 𝜋 𝑏 , 𝜋 𝑐 , 𝜋 (𝑑) random, ind. of 𝑥 April 30, 2016 AAD3: Algebraic Codes and Invariance

Symmetry + Locality - II Local constraint + affine-invariance ⇒ Local testing … specifically Theorem [Kaufman-S.’08]: 𝐶 ℓ-local constraint & is 𝔽 𝑄 𝑡 -affine-invariant ⇒ 𝐶 is ℓ′(ℓ,𝑄)-locally testable. Theorem [Ben-Sasson,Kaplan,Kopparty,Meir] 𝐶 has product property & 1-transitive ⇒ ∃ 𝐶 ′ 1-transitive and locally testable and product property Theorem [B-S,K,K,M,Stichtenoth] Such 𝐶 exists. (𝐶 = AG code) April 30, 2016 AAD3: Algebraic Codes and Invariance

Aside: Recent Progress in Locality - 1 [Yekhanin,Efremenko ‘06]: 3-Locally decodable codes of subexponential length. [Kopparty-Meir-RonZewi-Saraf ’15]: 𝑛 𝑜(1) -locally decodable codes w. 𝑅+𝛿→1 log 𝑛 log 𝑛 -locally testable codes w. 𝑅+𝛿→1 Codes not symmetric, but based on symmetric codes. April 30, 2016 AAD3: Algebraic Codes and Invariance

Aside – 2: Symmetric Ingredients … Lifted Codes [Guo-Kopparty-S.’13] 𝐶 𝑚,𝑑,𝑞 = 𝑓: 𝔽 𝑞 𝑚 → 𝔽 𝑞 | deg 𝑓 ​ 𝑙𝑖𝑛𝑒 ≤𝑑 ∀ 𝑙𝑖𝑛𝑒 Multiplicity Codes [Kopparty-Saraf-Yekhanin’10] Message = biv. polynomial Encode 𝑓 via evaluations of (𝑓, 𝑓 𝑥 , 𝑓 𝑦 ) April 30, 2016 AAD3: Algebraic Codes and Invariance

AAD3: Algebraic Codes and Invariance Part 5: Conclusions April 30, 2016 AAD3: Algebraic Codes and Invariance

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

AAD3: Algebraic Codes and Invariance 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

AAD3: Algebraic Codes and Invariance Thank You! April 30, 2016 AAD3: Algebraic Codes and Invariance