1. Zeev Dvir (Princeton) Shachar Lovett (IAS)
Subspace evasive sets
Zeev Dvir (Princeton)
Shachar Lovett (IAS)
STOC 2012

2. Subspace evasive sets
is (k,c) subspace evasive if for any k-dimensional linear subspace V,

3. Motivation
is (k,c) subspace evasive (SE) if for any k-dimension subspace V,
[Pudlák-Rödl’94]: (n/2,c) SE over
 bipartite Ramsey graphs
[Guruswami’11]: (k,c) SE over large fields
 Reduce list-size for list decodable codes

4. Talk overview Construction of subspace evasive sets
Application to list decodable codes
Proof idea
Further research

5. Talk overview Construction of subspace evasive sets
Application to list decodable codes
Proof idea
Further research

6. Example: line-evasive sets (k=1)
Claim : for any line V,
[Gabizon-Raz’08]
Proof:
Line for
corresponds to
Nonzero polynomial of degree n has ≤n solutions

7. Main result set of common solutions to
Thm: If in A=(ai,j) any k columns are independent then S is (k,c=nk) subspace evasive,

8. Main result
Main theorem: explicit (k,c=nk) subspace evasive set of size
Application to list decoding requires k,c=O(1) but can handle
Solution: bucket variables, apply main theorem on each bucket

9. Main result Let m=k/ Bucket variables: (x1,…,xm),(xm+1,…,x2m),…
Apply main construction to each bucket
S is (k,(k/)k) subspace evasive,
List size independent of number of vars

10. Talk overview Construction of subspace evasive sets
Application to list decodable codes
Proof idea
Further research

11. List decodable codes Codes: Unique decoding: returns best message
Encode messages to codewords
Decode received word (with errors)
Error model: worst case error
Unique decoding: returns best message
List decoding: returns list of potential messages
List decoding can potentially correct twice as many errors as unique decoding

12. Capacity achieving codes
A code is capacity achieving if it has optimal tradeoff of rate vs distance
Unique decoding: Reed-Solomon (60s)
List decoding: folded Reed-Solomon codes [Guruswami-Rudra’06]

13. Making list size constant
Folded Reed-Solomon code
Messages
Codewords
Subspace evasive set
List decoding
Linear subspace
[Guruswami’11]
Corrupted codeword

14. Making list size constant
Compose folded RS codes with subspace-evasive sets to obtain capacity achieving codes with constant list size
Messages
Codewords
Folded RS code
Embedding of subspace evasive set

15. Code parameters Block length n Rate r
Can list decode from (1-r-) errors
Folded RS: list size n1/
Concatenation with subspace evasive sets reduces list size to (1/)1/
Random codes: list size 1/

16. Talk overview Construction of subspace evasive sets
Application to list decodable codes
Proof idea
Further research

17. Main theorem set of common solutions to
Thm: If in A=(ai,j) any k columns are independent then S is (k,c=nk) subspace evasive,

18. Proof idea Main tool: algebraic geometry Co-dimension k variety:
In general, intersection of co-dim k variety and a generic dim k subspace is zero dimensional
Need to prove: for our construction, this holds for all dimension k subspace
Bézout theorem  Bound on list size

19. Proof idea V – dim k subspace (in algebraic closure) co-dim k variety
Need to prove is finite
Two proofs:
Combinatorial proof
Algebraic-geometric proof (w. János Kollár), allows to extend subspaces to low-degree dimension k varieties

20. Talk overview Construction of subspace evasive sets
Application to list decodable codes
Proof idea
Further research

21. Main theorem set of common solutions to
Thm: If in A=(ai,j) any k columns are independent then S is (k,c=nk) subspace evasive,

22. THANK YOU! Further research Problem 1: coding theory
reduce list size from (1/)1/ (random construction - 1/)
Problem 2: algebraic constructions in graph theory
Extend to small fields (Ramsey graphs)
Reduce degree of polynomials (Zarankiewicz problem)
THANK YOU!