Algebraic Codes and Invariance - Madhu Sudanmadhu.seas.harvard.edu/talks/2016/AAD3-Alg-Inv.pdf · efficiently (list) decodable.] Perform tasks (testing/decoding) in 𝑜𝑜𝑛𝑛

of 29 April 30, 2016 AAD3: Algebraic Codes and Invariance 1

Algebraic Codes and Invariance

Madhu Sudan Harvard

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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.

April 30, 2016 AAD3: Algebraic Codes and Invariance 2

Ex-

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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


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Part 1: Basic Definitions


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Error-correcting codes

Notation: 𝔽𝔽𝑞𝑞- finite field of cardinality 𝑞𝑞 Encoding function: messages ↦ codewords

𝐸𝐸:𝔽𝔽𝑞𝑞𝑘𝑘 → 𝔽𝔽𝑞𝑞𝑛𝑛; associated Code 𝐶𝐶 ≜ 𝐸𝐸 𝑚𝑚 𝑚𝑚 ∈ 𝔽𝔽𝑞𝑞𝑘𝑘} Key parameters:

Rate 𝑅𝑅 𝐶𝐶 = 𝑘𝑘/𝑛𝑛 ; Distance 𝛿𝛿 𝐶𝐶 = min

𝑥𝑥≠𝑦𝑦∈𝐶𝐶{𝛿𝛿 𝑥𝑥,𝑦𝑦 }.

𝛿𝛿 𝑥𝑥,𝑦𝑦 = 𝑖𝑖 𝑥𝑥𝑖𝑖≠𝑦𝑦𝑖𝑖 |𝑛𝑛

Pigeonhole Principle ⇒ 𝑅𝑅 𝐶𝐶 + 𝛿𝛿 𝐶𝐶 ≤ 1 + 1𝑛𝑛

Algebraic codes: Get very close to this limit! April 30, 2016 AAD3: Algebraic Codes and Invariance 5

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Algorithmic tasks

Encoding: Compute 𝐸𝐸 𝑚𝑚 given 𝑚𝑚. Testing: Given 𝑟𝑟 ∈ 𝔽𝔽𝑞𝑞𝑛𝑛, decide if ∃ 𝑚𝑚 s. t. 𝑟𝑟 = 𝐸𝐸(𝑚𝑚)? Decoding: Given 𝑟𝑟 ∈ 𝔽𝔽𝑞𝑞𝑛𝑛, compute 𝑚𝑚 minimizing 𝛿𝛿(𝐸𝐸 𝑚𝑚 , 𝑟𝑟) [All linear codes nicely encodable, but algebraic ones

efficiently (list) decodable.]

Perform tasks (testing/decoding) in 𝑜𝑜 𝑛𝑛 time.

Assume random access to 𝑟𝑟 Suffices to decode 𝑚𝑚𝑖𝑖, for given 𝑖𝑖 ∈ [𝑘𝑘]

[Many algebraic codes locally decodable and testable!]


Locality

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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 …)

Algebraic Codes?


Reed-Solomon Codes: Domain = (subset of) 𝔽𝔽𝑞𝑞 Messages = 𝔽𝔽𝑞𝑞≤𝑘𝑘[𝑥𝑥] (univ. poly. of deg. 𝑘𝑘) Reed-Muller Codes: Domain = 𝔽𝔽𝑞𝑞𝑚𝑚 Messages = 𝔽𝔽𝑞𝑞≤𝑟𝑟[𝑥𝑥1, … , 𝑥𝑥𝑚𝑚] (𝑚𝑚-var poly. of deg. 𝑟𝑟)

Algebraic-Geometry Codes: Domain = Rational points of irred. curve in 𝔽𝔽𝑞𝑞𝑚𝑚 Messages = Functions of bounded “order”

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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 ≤ 𝑘𝑘 has ≤ 𝑘𝑘 roots.

Multiv poly of deg ≤ 𝑘𝑘 has ≤ 𝑘𝑘𝑞𝑞 fraction roots.

Functions of order ≤ 𝑘𝑘 have fewer than ≤ 𝑘𝑘 roots [Bezout, Riemann-Roch, Ihara, Drinfeld-

Vladuts] [Goppa, Tsfasman-Vladuts-Zink, Garcia-

Stichtenoth]


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Consequences

𝑞𝑞 ≥ 𝑛𝑛 ⇒ ∃ codes 𝐶𝐶 satisfying 𝑅𝑅 𝐶𝐶 + 𝛿𝛿 𝐶𝐶 = 1 + 1𝑛𝑛

For infinitely many 𝑞𝑞, there exist infinitely many 𝑛𝑛, and codes 𝐶𝐶𝑞𝑞,𝑛𝑛 over 𝔽𝔽𝑞𝑞 satisfying

𝑅𝑅 𝐶𝐶𝑞𝑞,𝑛𝑛 + 𝛿𝛿 𝐶𝐶𝑞𝑞,𝑛𝑛 ≥ 1 −1𝑞𝑞 − 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 𝛿𝛿

Corrects 𝛿𝛿2 fraction errors uniquely.

Corrects 1 − 1 − 𝛿𝛿 fraction errors with small lists.

Algorithmically? For all known algebraic codes, above can be

matched! Why?


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For vectors 𝑢𝑢,𝑣𝑣 ∈ 𝔽𝔽𝑞𝑞𝑛𝑛, let 𝑢𝑢 ∗ 𝑣𝑣 ∈ 𝔽𝔽𝑞𝑞𝑛𝑛 denote their coordinate-wise product.

For linear codes 𝐴𝐴,𝐵𝐵 ≤ 𝔽𝔽𝑞𝑞𝑛𝑛, let 𝐴𝐴 ∗ 𝐵𝐵 = span 𝑎𝑎 ∗ 𝑏𝑏 𝑎𝑎 ∈ 𝐴𝐴, 𝑏𝑏 ∈ 𝐵𝐵}

Obvious, but remarkable, feature:

For every known algebraic code 𝐶𝐶 of distance 𝛿𝛿

∃ code 𝐸𝐸 of dimension ≈ 𝛿𝛿2𝑛𝑛 s.t.

𝐸𝐸 ∗ 𝐶𝐶 is a code of distance 𝛿𝛿2.

Terminology: 𝐸𝐸,𝐸𝐸 ∗ 𝐶𝐶 ≜ error-locating pair for 𝐶𝐶

Product property


Reed-Solomon Codes: 𝐶𝐶 = deg 𝑘𝑘 poly

𝐸𝐸 = deg 𝑛𝑛−𝑘𝑘2

poly

𝐸𝐸 ∗ 𝐶𝐶 = deg 𝑛𝑛+𝑘𝑘2

poly

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Unique decoding by error-locating pairs

Given 𝑟𝑟 ∈ 𝔽𝔽𝑞𝑞𝑛𝑛 : Find 𝑥𝑥 ∈ 𝐶𝐶 s.t. 𝛿𝛿 𝑥𝑥, 𝑟𝑟 ≤ 𝛿𝛿2

Algorithm Step 1: Find 𝑒𝑒 ∈ 𝐸𝐸,𝑓𝑓 ∈ 𝐸𝐸 ∗ 𝐶𝐶 s.t. 𝑒𝑒 ∗ 𝑟𝑟 = 𝑓𝑓

[Linear system] 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.


[Pellikaan], [Koetter], [Duursma] – 90s

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List decoding abstraction

Increasing basis sequence 𝑏𝑏1, 𝑏𝑏2, … 𝐶𝐶𝑖𝑖 ≜ span 𝑏𝑏1, … , 𝑏𝑏𝑖𝑖 𝛿𝛿 𝐶𝐶𝑖𝑖 ≈ 𝑛𝑛 − 𝑖𝑖 + 𝑜𝑜 𝑛𝑛 𝑏𝑏𝑖𝑖 ∗ 𝑏𝑏𝑗𝑗 ∈ 𝐶𝐶𝑖𝑖+𝑗𝑗 (⇔ 𝐶𝐶𝑖𝑖 ∗ 𝐶𝐶𝑗𝑗 ⊆ 𝐶𝐶𝑖𝑖+𝑗𝑗)

[Guruswami, Sahai, S. ‘2000s] List-decoding

algorithms correcting 1 − 1 − 𝛿𝛿 fraction errors exist for codes with increasing basis sequences.

(Increasing basis ⇒ Error-locating pairs)


Reed-Solomon Codes: 𝑏𝑏𝑖𝑖 = 𝑥𝑥𝑖𝑖 (or its evaluations etc.)

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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


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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.


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Message = multivariate polynomial; Encoding = evaluations everywhere.

RM 𝑚𝑚, 𝑟𝑟, 𝑞𝑞 = { 𝑓𝑓 𝛼𝛼 𝛼𝛼∈𝔽𝔽𝑞𝑞𝑚𝑚| 𝑓𝑓 ∈ 𝔽𝔽q 𝑥𝑥1, … , 𝑥𝑥𝑚𝑚 , deg 𝑓𝑓 ≤ 𝑟𝑟}

Locality? (when 𝑟𝑟 < 𝑞𝑞)

Restrictions of low-degree polynomials to lines yield low-degree (univ.) polys. Random lines sample 𝔽𝔽𝑞𝑞𝑚𝑚 uniformly (pairwise ind’ly) Locality ~ 𝑞𝑞

Main Example: Reed-Muller Codes


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Locality ⇐ ?

Necessary condition: Small (local) constraints. Examples

deg 𝑓𝑓 ≤ 1 ⇔ 𝑓𝑓 𝑥𝑥 + 𝑓𝑓 𝑥𝑥 + 2𝑦𝑦 = 2𝑓𝑓 𝑥𝑥 + 𝑦𝑦 𝑓𝑓|𝑙𝑙𝑖𝑖𝑛𝑛𝑙𝑙 has low-degree (so values on line are

not arbitrary)

Local Constraints ⇒ local decoding/testing? No!

Transitivity + locality?


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Symmetry in codes

𝐴𝐴𝑢𝑢𝐴𝐴 𝐶𝐶 ≜ 𝜋𝜋 ∈ 𝑆𝑆𝑛𝑛 ∀ 𝑥𝑥 ∈ 𝐶𝐶, 𝑥𝑥𝜋𝜋 ∈ 𝐶𝐶}

Well-studied concept. “Cyclic codes”

Basic algebraic codes (Reed-Solomon, Reed-

Muller, BCH) symmetric under affine group Domain is vector space 𝔽𝔽𝑄𝑄𝑡𝑡 (where 𝑄𝑄𝑡𝑡 = 𝑛𝑛) Code invariant under non-singular affine

transforms from 𝔽𝔽𝑄𝑄𝑡𝑡 → 𝔽𝔽𝑄𝑄𝑡𝑡


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Symmetry and Locality

Code has ℓ-local constraint + 2-transitive ⇒ Code is ℓ-locally decodable from 𝑂𝑂 1

ℓ-fraction errors.

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


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Symmetry + Locality - II

Local constraint + affine-invariance ⇒ Local testing … specifically

Theorem [Kaufman-S.’08]: 𝐶𝐶 ℓ-local constraint & is 𝔽𝔽𝑄𝑄𝑡𝑡 -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)


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Aside: Recent Progress in Locality - 1

[Yekhanin,Efremenko ‘06]: 3-Locally decodable codes of subexponential length.

[Kopparty-Meir-RonZewi-Saraf ’15]: 𝑛𝑛𝑜𝑜(1)-locally decodable codes w. 𝑅𝑅 + 𝛿𝛿 → 1 log𝑛𝑛log 𝑛𝑛-locally testable codes w. 𝑅𝑅 + 𝛿𝛿 → 1 Codes not symmetric, but based on symmetric

codes.


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Aside – 2: Symmetric Ingredients …

Lifted Codes [Guo-Kopparty-S.’13] 𝐶𝐶𝑚𝑚,𝑑𝑑,𝑞𝑞 = 𝑓𝑓:𝔽𝔽𝑞𝑞𝑚𝑚 → 𝔽𝔽𝑞𝑞| deg 𝑓𝑓|𝑙𝑙𝑖𝑖𝑛𝑛𝑙𝑙 ≤ 𝑑𝑑 ∀ 𝑙𝑙𝑖𝑖𝑛𝑛𝑒𝑒

Multiplicity Codes [Kopparty-Saraf-Yekhanin’10] Message = biv. polynomial Encode 𝑓𝑓 via evaluations of (𝑓𝑓,𝑓𝑓𝑥𝑥 ,𝑓𝑓𝑦𝑦)


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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)


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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!


Documents

Algebraic Codes and Invariance - Madhu Sudanmadhu.seas.harvard.edu/talks/2016/AAD3-Alg-Inv.pdf · efficiently (list) decodable.] Perform tasks (testing/decoding) in 𝑜𝑜𝑛𝑛