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

2. 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

3. 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

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

5. 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

6. 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

7. 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

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

9. 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

10. 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

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

12. 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

13. 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

14. 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

15. 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

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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

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

27. 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

28. 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

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