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Lower bounds for constant query affine-invariant LCCs and LTCs [article]

Arnab Bhattacharyya, Sivakanth Gopi
2015 arXiv   pre-print
The dependence on n in these bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty and Sudan (ITCS '13) construct affine-invariant codes via lifting that have the same asymptotic tradeoffs.  ...  In this work, we give lower bounds on the length of locally correctable and locally testable affine-invariant codes with constant query complexity.  ...  Acknowledgements We thank Madhu Sudan for helpful pointers to previous work. The second author would like to thank his advisor, Zeev Dvir, for his guidance and encouragement.  ... 
arXiv:1511.07558v1 fatcat:nxvmhj6lovdpjneky2e65abz2a

Lower Bounds for Constant Query Affine-Invariant LCCs and LTCs

Arnab Bhattacharyya, Sivakanth Gopi
2017 ACM Transactions on Computation Theory  
The dependence on n in these bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty and Sudan (ITCS'13) construct affine-invariant codes via lifting that have the same asymptotic tradeoffs.  ...  In this work, we give lower bounds on the length of locally correctable and locally testable affine-invariant codes with constant query complexity.  ...  We thank Madhu Sudan for helpful pointers to previous work. The second author would like to thank his advisor, Zeev Dvir, for his guidance and encouragement.  ... 
doi:10.1145/3016802 fatcat:s5tvj25yqzhsnpv2icij2rhrlu

New affine-invariant codes from lifting [article]

Alan Guo, Swastik Kopparty, Madhu Sudan
2012 arXiv   pre-print
While previous lower bounds on Nikodym sets were roughly growing as q^m/2^m, we use our lifted codes to prove a lower bound of (1 - o(1))q^m for fields of constant characteristic.  ...  Affine-invariant codes are simply linear codes whose coordinates are a vector space over a field and which are invariant under affine-transformations of the coordinate space.  ...  We would like to thank Elad Haramaty for clarifying discussions on the relationship between the definition of lifting in [6] and in this work.  ... 
arXiv:1208.5413v2 fatcat:v5cfesgqwfaznorwxm67iacnbq

Limits on the Rate of Locally Testable Affine-Invariant Codes [chapter]

Eli Ben-Sasson, Madhu Sudan
2011 Lecture Notes in Computer Science  
In this work we rule out this possibility and show that general (linear) affine-invariant properties are contained in Reed-Muller codes that are testable with a slightly larger query complexity.  ...  This obstacle was removed by [Kaufman and Sudan, STOC 2008] who considered (linear) "affine-invariant properties", i.e., properties that are closed under summation, and under affine transformations of  ...  Acknowledgements We thank the anonymous referees for helpful comments. We thank Shachar Lovett for allowing us to include the simplified proof of Theorem 3 in this paper.  ... 
doi:10.1007/978-3-642-22935-0_35 fatcat:7jmyu2axkvgx7prbay6rucnkhi

New affine-invariant codes from lifting

Alan Guo, Swastik Kopparty, Madhu Sudan
2013 Proceedings of the 4th conference on Innovations in Theoretical Computer Science - ITCS '13  
Affine-invariant codes are simply linear codes whose coordinates are a vector space over a field and which are invariant under affine-transformations of the coordinate space.  ...  In this work we explore error-correcting codes derived from the "lifting" of "affine-invariant" codes.  ...  We would like to thank Elad Haramaty for clarifying discussions on the relationship between the definition of lifting in [6] and in this work.  ... 
doi:10.1145/2422436.2422494 dblp:conf/innovations/GuoKS13 fatcat:em5dxs7smbfk5fwel5egjjnlj4

A new family of locally correctable codes based on degree-lifted algebraic geometry codes

Eli Ben-Sasson, Ariel Gabizon, Yohay Kaplan, Swastik Kopparty, Shubangi Saraf
2013 Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13  
The automorphisms of the base-code are crucially used to generate query-sets, abstracting the use of affine-lines in the local correction procedure of Reed-Muller codes.  ...  rate, message length, and distance.  ...  The best known lower bound on the length of constant query LDCs isΩ k 1+δ(r) (where r is the number of queries and δ (r) < 1).  ... 
doi:10.1145/2488608.2488714 dblp:conf/stoc/Ben-SassonGKKS13 fatcat:hwixwvv265aypdsm6qrmbfwnwq