Transcript Algebraic Codes and Invariance
Algebraic Codes and Invariance
Madhu Sudan
Microsoft Research August 4, 2015 SIAM AAG: Algebraic Codes and Invariance 1 of 29
Disclaimer
Very little Algebraic Geometry 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.
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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 August 4, 2015 SIAM AAG: Algebraic Codes and Invariance 3 of 29
Part 1: Basic Definitions
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Error-correcting codes
Notation: 𝔽 𝑞 - finite field of cardinality 𝑞 Encoding function: messages ↦ codewords 𝐸: 𝔽 𝑘 𝑞 → 𝔽 𝑛 𝑞 ; associated Code 𝐶 ≜ 𝐸 𝑚 𝑚 ∈ 𝔽 𝑘 𝑞 } 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!
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Algorithmic tasks
Encoding: Compute 𝐸 𝑚 given 𝑚 .
Testing: Given 𝑟 ∈ 𝔽 𝑛 𝑞 , decide if ∃ 𝑚 Decoding: Given 𝑟 ∈ 𝔽 𝑛 𝑞 , compute 𝑚 s. t.
𝑟 = 𝐸(𝑚) minimizing ?
𝛿(𝐸 𝑚 , 𝑟) [All linear codes nicely encodable, but algebraic ones efficiently (list) decodable.]
Locality
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!] August 4, 2015 SIAM AAG: Algebraic Codes and Invariance 6 of 29
Algebraic Codes?
Message space = (Vector) space of functions 𝑞 Messages = 𝑚 𝑞 𝔽 ≤𝑘 𝑞 Encoding = evaluations of message on domain ( 𝑚 ≤𝑟 -var poly. of deg. 𝑘 ) 𝑚 𝑞 Examples: Reed-Solomon Codes Reed-Muller Codes Algebraic-Geometric Codes Others (BCH codes, dual BCH codes …) August 4, 2015 SIAM AAG: Algebraic Codes and Invariance 7 of 29
Part 2: Combinatorics
⇐
Fund. Thm
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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 Functions of order ≤ 𝑘 roots has ≤ 𝑘 fraction roots.
𝑞 have fewer than ≤ 𝑘 [Bezout, Riemann-Roch, Ihara, Drinfeld Vladuts] [Tsfasman-Vladuts-Zink, Garcia Stichtenoth] August 4, 2015 SIAM AAG: Algebraic Codes and Invariance 9 of 29
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 1 𝑅 𝐶 𝑞,𝑛 + 𝛿 𝐶 𝑞,𝑛 ≥ 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!
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Part 3: Algorithmics
⇐
Product property
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Remarkable algorithmics
Combinatorial implications: Code of distance 𝛿 Corrects 𝛿 2 fraction errors uniquely.
Corrects 1 − 1 − 𝛿 lists.
fraction errors with small Algorithmically?
For all known algebraic codes, above can be matched!
Why?
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Product property
For vectors 𝑢, 𝑣 ∈ 𝔽 𝑞 poly , let 𝑢 ∗ 𝑣 ∈ 𝔽 𝑛 𝑞 coordinate-wise product.
𝑛−𝑘 For linear codes 𝐴, 𝐵 ≤ 𝔽 𝑛 𝑞 , let denote their 𝐴 ∗ 𝐵 = span 𝑎 ∗ 𝑏 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵} 2 poly Obvious, but remarkable, feature: For every known algebraic code 𝐶 of distance 𝛿 ∃ code 𝐸 of co-dimension ≈ 𝛿 2 𝑛 s.t.
𝐸 ∗ 𝐶 Terminology : is a code of distance 𝛿 2 .
𝐸, 𝐸 ∗ 𝐶 ≜ error-locating pair for 𝐶 August 4, 2015 SIAM AAG: Algebraic Codes and Invariance 13 of 29
Unique decoding by error-locating pairs
Given 𝑟 ∈ 𝔽 𝑛 𝑞 : Find 𝑥 ∈ 𝐶 s.t.
Algorithm Step 1: Find 𝑒 ∈ 𝐸, 𝑓 ∈ 𝐸 ∗ 𝐶 𝛿 𝑥, 𝑟 ≤ 𝛿 2 s.t.
𝑒 ∗ 𝑟 = 𝑓 [Pellikaan], [Koetter], [Duursma] – 90s Step 2: Find s.t.
[Linear system again!] Analysis: Solution to Step 1 exists? Yes – provided dim 𝐸 > #errors Solution to Step 2 ? Yes – Provided 𝛿 𝐸 large enough.
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List decoding abstraction
Increasing basis sequence 𝑏 1 , 𝑏 2 , … 𝐶 𝑖 ≜ span 𝑏 1 , … , 𝑏 𝑖 𝛿 𝐶 𝑏 𝑖 𝑖 ∗ 𝑏 𝑗 ≈ 𝑛 − 𝑖 + 𝑜 𝑛 ∈ 𝐶 𝑖+𝑗 ( ⇔ 𝐶 𝑖 ∗ 𝐶 𝑗 ⊆ 𝐶 𝑖+𝑗 ) [Guruswami, Sahai, S. ‘2000s] List-decoding 𝑖 algorithms correcting 1 − 1 − 𝛿 fraction errors exist for codes with increasing basis sequences.
(Increasing basis ⇒ Error-locating pairs) August 4, 2015 SIAM AAG: Algebraic Codes and Invariance 15 of 29
Part 4: Locality
⇐
Invariances
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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 Aside: Related to LRCs from Judy Walker’s talk.
Focus here on more errors. August 4, 2015 SIAM AAG: Algebraic Codes and Invariance 17 of 29
Locality of some algebraic codes
Locality is a rare phenomenon.
Reed-Solomon codes are not.
Random codes are not.
AG codes are (usually) not.
Basic examples are algebraic … … and a few composition operators preserve it.
Canonical example: Reed-Muller Codes = low degree polynomials.
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Main Example: Reed-Muller Codes
Message = multivariate polynomial; Encoding = evaluations everywhere.
RM 𝑚, 𝑟, 𝑞 = { 𝑓 𝛼 𝛼∈𝔽 𝑚 𝑞 | 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 𝑚 𝑞 uniformly (pairwise ind’ly) Locality ~ 𝑞 June 16, 2015 ISIT: Locality in Coding Theory 19 of 29
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 ⇒ No!
local decoding/testing? Transitivity + locality?
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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 𝑡 𝑄 (where 𝑄 𝑡 = 𝑛 ) Code invariant under non-singular affine transforms from 𝔽 𝑡 𝑄 → 𝔽 𝑡 𝑄 August 4, 2015 SIAM AAG: Algebraic Codes and Invariance 21 of 29
Symmetry and Locality
Code has ℓ -local constraint + 2 -transitive ⇒ is ℓ -locally decodable from 𝑂 1 ℓ Code -fraction errors.
2 -transitive? – ∀𝑖 ≠ 𝑗, 𝑘 ≠ 𝑙 ∃𝜋 ∈ Aut 𝐶 Why?
s. t.
𝜋 𝑖 = 𝑘, 𝜋 𝑗 = 𝑙 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 𝜋 𝑏 , 𝜋 𝑐 , 𝜋 (𝑑) + 𝑓 𝜋 𝑐 + 𝑓(𝜋 𝑑 ) ; random, ind. of 𝑥 August 4, 2015 SIAM AAG: Algebraic Codes and Invariance 22 of 29
Symmetry + Locality - II
Local constraint + affine-invariance ⇒ testing … specifically Theorem [Kaufman-S.’08]: Local 𝐶 ℓ -local constraint & is 𝔽 𝑡 𝑄 is ℓ′(ℓ, 𝑄) -locally testable.
-affine-invariant ⇒ 𝐶 Theorem [Ben-Sasson,Kaplan,Kopparty,Meir] 𝐶 has product property & 1-transitive ⇒ ∃ 𝐶 ′ transitive and locally testable and product property Theorem [B-S,K,K,M,Stichtenoth] Such 𝐶 exists. ( 𝐶 = AG code) August 4, 2015 SIAM AAG: Algebraic Codes and Invariance 1 23 of 29
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.
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Aside – 2: Symmetric Ingredients …
Lifted Codes [Guo-Kopparty-S.’13] 𝐶 𝑚,𝑑,𝑞 = 𝑓: 𝔽 𝑚 𝑞 → 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 ) August 4, 2015 SIAM AAG: Algebraic Codes and Invariance 25 of 29
Part 5: Conclusions
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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) August 4, 2015 SIAM AAG: Algebraic Codes and Invariance 27 of 29
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.
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Thank You!
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