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Algebraic Codes and Invariance
Madhu Sudan Harvard
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Disclaimer
Very little new work in this talk!
Mainly: Coding theoristโs perspective on Algebraic and
Algebraic-Geometric Codes What additional properties it would be nice to
have in algebraic-geometry codes.
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Ex-
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Outline of the talk
Part 1: Codes and Algebraic Codes Part 2: Combinatorics of Algebraic Codes
โ Fundamental theorem(s) of algebra Part 3: Algorithmics of Algebraic Codes
โ Product property Part 4: Locality of (some) Algebraic Codes
โ Invariances Part 5: Conclusions
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Error-correcting codes
Notation: ๐ฝ๐ฝ๐๐- finite field of cardinality ๐๐ Encoding function: messages โฆ codewords
๐ธ๐ธ:๐ฝ๐ฝ๐๐๐๐ โ ๐ฝ๐ฝ๐๐๐๐; associated Code ๐ถ๐ถ โ ๐ธ๐ธ ๐๐ ๐๐ โ ๐ฝ๐ฝ๐๐๐๐} Key parameters:
Rate ๐ ๐ ๐ถ๐ถ = ๐๐/๐๐ ; Distance ๐ฟ๐ฟ ๐ถ๐ถ = min
๐ฅ๐ฅโ ๐ฆ๐ฆโ๐ถ๐ถ{๐ฟ๐ฟ ๐ฅ๐ฅ,๐ฆ๐ฆ }.
๐ฟ๐ฟ ๐ฅ๐ฅ,๐ฆ๐ฆ = ๐๐ ๐ฅ๐ฅ๐๐โ ๐ฆ๐ฆ๐๐ |๐๐
Pigeonhole Principle โ ๐ ๐ ๐ถ๐ถ + ๐ฟ๐ฟ ๐ถ๐ถ โค 1 + 1๐๐
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 ๐ธ๐ธ ๐๐ given ๐๐. Testing: Given ๐๐ โ ๐ฝ๐ฝ๐๐๐๐, decide if โ ๐๐ s. t. ๐๐ = ๐ธ๐ธ(๐๐)? Decoding: Given ๐๐ โ ๐ฝ๐ฝ๐๐๐๐, compute ๐๐ minimizing ๐ฟ๐ฟ(๐ธ๐ธ ๐๐ , ๐๐) [All linear codes nicely encodable, but algebraic ones
efficiently (list) decodable.]
Perform tasks (testing/decoding) in ๐๐ ๐๐ time.
Assume random access to ๐๐ Suffices to decode ๐๐๐๐, for given ๐๐ โ [๐๐]
[Many algebraic codes locally decodable and testable!]
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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 โฆ)
Algebraic Codes?
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Reed-Solomon Codes: Domain = (subset of) ๐ฝ๐ฝ๐๐ Messages = ๐ฝ๐ฝ๐๐โค๐๐[๐ฅ๐ฅ] (univ. poly. of deg. ๐๐) Reed-Muller Codes: Domain = ๐ฝ๐ฝ๐๐๐๐ Messages = ๐ฝ๐ฝ๐๐โค๐๐[๐ฅ๐ฅ1, โฆ , ๐ฅ๐ฅ๐๐] (๐๐-var poly. of deg. ๐๐)
Algebraic-Geometry Codes: Domain = Rational points of irred. curve in ๐ฝ๐ฝ๐๐๐๐ Messages = Functions of bounded โorderโ
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Essence of combinatorics
Rate of code โ Dimension of vector space Distance of code โ Scarcity of roots
Univ poly of deg โค ๐๐ has โค ๐๐ roots.
Multiv poly of deg โค ๐๐ has โค ๐๐๐๐ fraction roots.
Functions of order โค ๐๐ have fewer than โค ๐๐ roots [Bezout, Riemann-Roch, Ihara, Drinfeld-
Vladuts] [Goppa, Tsfasman-Vladuts-Zink, Garcia-
Stichtenoth]
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Consequences
๐๐ โฅ ๐๐ โ โ codes ๐ถ๐ถ satisfying ๐ ๐ ๐ถ๐ถ + ๐ฟ๐ฟ ๐ถ๐ถ = 1 + 1๐๐
For infinitely many ๐๐, there exist infinitely many ๐๐, and codes ๐ถ๐ถ๐๐,๐๐ over ๐ฝ๐ฝ๐๐ satisfying
๐ ๐ ๐ถ๐ถ๐๐,๐๐ + ๐ฟ๐ฟ ๐ถ๐ถ๐๐,๐๐ โฅ 1 โ1๐๐ โ 1
Many codes that are better than random codes Reed-Solomon, Reed-Muller of order 1, AG,
BCH, dual BCH โฆ
Moral: Distance property โ Algebra!
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Part 3: Algorithmics โ Product property
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Remarkable algorithmics
Combinatorial implications: Code of distance ๐ฟ๐ฟ
Corrects ๐ฟ๐ฟ2 fraction errors uniquely.
Corrects 1 โ 1 โ ๐ฟ๐ฟ fraction errors with small lists.
Algorithmically? For all known algebraic codes, above can be
matched! Why?
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For vectors ๐ข๐ข,๐ฃ๐ฃ โ ๐ฝ๐ฝ๐๐๐๐, let ๐ข๐ข โ ๐ฃ๐ฃ โ ๐ฝ๐ฝ๐๐๐๐ denote their coordinate-wise product.
For linear codes ๐ด๐ด,๐ต๐ต โค ๐ฝ๐ฝ๐๐๐๐, let ๐ด๐ด โ ๐ต๐ต = span ๐๐ โ ๐๐ ๐๐ โ ๐ด๐ด, ๐๐ โ ๐ต๐ต}
Obvious, but remarkable, feature:
For every known algebraic code ๐ถ๐ถ of distance ๐ฟ๐ฟ
โ code ๐ธ๐ธ of dimension โ ๐ฟ๐ฟ2๐๐ s.t.
๐ธ๐ธ โ ๐ถ๐ถ is a code of distance ๐ฟ๐ฟ2.
Terminology: ๐ธ๐ธ,๐ธ๐ธ โ ๐ถ๐ถ โ error-locating pair for ๐ถ๐ถ
Product property
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Reed-Solomon Codes: ๐ถ๐ถ = deg ๐๐ poly
๐ธ๐ธ = deg ๐๐โ๐๐2
poly
๐ธ๐ธ โ ๐ถ๐ถ = deg ๐๐+๐๐2
poly
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Unique decoding by error-locating pairs
Given ๐๐ โ ๐ฝ๐ฝ๐๐๐๐ : Find ๐ฅ๐ฅ โ ๐ถ๐ถ s.t. ๐ฟ๐ฟ ๐ฅ๐ฅ, ๐๐ โค ๐ฟ๐ฟ2
Algorithm Step 1: Find ๐๐ โ ๐ธ๐ธ,๐๐ โ ๐ธ๐ธ โ ๐ถ๐ถ s.t. ๐๐ โ ๐๐ = ๐๐
[Linear system] Step 2: Find ๐ฅ๐ฅ๏ฟฝ โ ๐ถ๐ถ s.t. ๐๐ โ ๐ฅ๐ฅ๏ฟฝ = ๐๐
[Linear system again!] Analysis:
Solution to Step 1 exists? Yes โ provided dim ๐ธ๐ธ > #errors
Solution to Step 2 ๐ฅ๐ฅ๏ฟฝ = ๐ฅ๐ฅ? Yes โ Provided ๐ฟ๐ฟ ๐ธ๐ธ โ ๐ถ๐ถ large enough.
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[Pellikaan], [Koetter], [Duursma] โ 90s
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List decoding abstraction
Increasing basis sequence ๐๐1, ๐๐2, โฆ ๐ถ๐ถ๐๐ โ span ๐๐1, โฆ , ๐๐๐๐ ๐ฟ๐ฟ ๐ถ๐ถ๐๐ โ ๐๐ โ ๐๐ + ๐๐ ๐๐ ๐๐๐๐ โ ๐๐๐๐ โ ๐ถ๐ถ๐๐+๐๐ (โ ๐ถ๐ถ๐๐ โ ๐ถ๐ถ๐๐ โ ๐ถ๐ถ๐๐+๐๐)
[Guruswami, Sahai, S. โ2000s] List-decoding
algorithms correcting 1 โ 1 โ ๐ฟ๐ฟ fraction errors exist for codes with increasing basis sequences.
(Increasing basis โ Error-locating pairs)
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Reed-Solomon Codes: ๐๐๐๐ = ๐ฅ๐ฅ๐๐ (or its evaluations etc.)
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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 โฆ
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 โฆ โฆ 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 ๐๐, ๐๐, ๐๐ = { ๐๐ ๐ผ๐ผ ๐ผ๐ผโ๐ฝ๐ฝ๐๐๐๐| ๐๐ โ ๐ฝ๐ฝq ๐ฅ๐ฅ1, โฆ , ๐ฅ๐ฅ๐๐ , deg ๐๐ โค ๐๐}
Locality? (when ๐๐ < ๐๐)
Restrictions of low-degree polynomials to lines yield low-degree (univ.) polys. Random lines sample ๐ฝ๐ฝ๐๐๐๐ uniformly (pairwise indโly) Locality ~ ๐๐
Main Example: Reed-Muller Codes
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Locality โ ?
Necessary condition: Small (local) constraints. Examples
deg ๐๐ โค 1 โ ๐๐ ๐ฅ๐ฅ + ๐๐ ๐ฅ๐ฅ + 2๐ฆ๐ฆ = 2๐๐ ๐ฅ๐ฅ + ๐ฆ๐ฆ ๐๐|๐๐๐๐๐๐๐๐ has low-degree (so values on line are
not arbitrary)
Local Constraints โ local decoding/testing? No!
Transitivity + locality?
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Symmetry in codes
๐ด๐ด๐ข๐ข๐ด๐ด ๐ถ๐ถ โ ๐๐ โ ๐๐๐๐ โ ๐ฅ๐ฅ โ ๐ถ๐ถ, ๐ฅ๐ฅ๐๐ โ ๐ถ๐ถ}
Well-studied concept. โCyclic codesโ
Basic algebraic codes (Reed-Solomon, Reed-
Muller, BCH) symmetric under affine group Domain is vector space ๐ฝ๐ฝ๐๐๐ก๐ก (where ๐๐๐ก๐ก = ๐๐) Code invariant under non-singular affine
transforms from ๐ฝ๐ฝ๐๐๐ก๐ก โ ๐ฝ๐ฝ๐๐๐ก๐ก
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Symmetry and Locality
Code has โ-local constraint + 2-transitive โ Code is โ-locally decodable from ๐๐ 1
โ-fraction errors.
2-transitive? โ โ๐๐ โ ๐๐,๐๐ โ ๐๐ โ๐๐ โ Aut ๐ถ๐ถ s. t. ๐๐ ๐๐ = ๐๐,๐๐ ๐๐ = ๐๐
Why? Suppose constraint ๐๐ ๐๐ = ๐๐ ๐๐ + ๐๐ ๐๐ + ๐๐ ๐๐ Wish to determine ๐๐ ๐ฅ๐ฅ Find random ๐๐ โ Aut ๐ถ๐ถ s.t. ๐๐ ๐๐ = ๐ฅ๐ฅ; ๐๐ ๐ฅ๐ฅ = ๐๐ ๐๐ ๐๐ + ๐๐ ๐๐ ๐๐ + ๐๐(๐๐ ๐๐ ); ๐๐ ๐๐ ,๐๐ ๐๐ ,๐๐ (๐๐) random, ind. of ๐ฅ๐ฅ
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Symmetry + Locality - II
Local constraint + affine-invariance โ Local testing โฆ specifically
Theorem [Kaufman-S.โ08]: ๐ถ๐ถ โ-local constraint & is ๐ฝ๐ฝ๐๐๐ก๐ก -affine-invariant โ ๐ถ๐ถ
is โโฒ(โ,๐๐)-locally testable.
Theorem [Ben-Sasson,Kaplan,Kopparty,Meir] ๐ถ๐ถ has product property & 1-transitive โ โ๐ถ๐ถโฒ 1-
transitive and locally testable and product property
Theorem [B-S,K,K,M,Stichtenoth] Such ๐ถ๐ถ exists. (๐ถ๐ถ = AG code)
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Aside: Recent Progress in Locality - 1
[Yekhanin,Efremenko โ06]: 3-Locally decodable codes of subexponential length.
[Kopparty-Meir-RonZewi-Saraf โ15]: ๐๐๐๐(1)-locally decodable codes w. ๐ ๐ + ๐ฟ๐ฟ โ 1 log๐๐log ๐๐-locally testable codes w. ๐ ๐ + ๐ฟ๐ฟ โ 1 Codes not symmetric, but based on symmetric
codes.
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Aside โ 2: Symmetric Ingredients โฆ
Lifted Codes [Guo-Kopparty-S.โ13] ๐ถ๐ถ๐๐,๐๐,๐๐ = ๐๐:๐ฝ๐ฝ๐๐๐๐ โ ๐ฝ๐ฝ๐๐| deg ๐๐|๐๐๐๐๐๐๐๐ โค ๐๐ โ ๐๐๐๐๐๐๐๐
Multiplicity Codes [Kopparty-Saraf-Yekhaninโ10] Message = biv. polynomial Encode ๐๐ via evaluations of (๐๐,๐๐๐ฅ๐ฅ ,๐๐๐ฆ๐ฆ)
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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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