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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