Meetings

Click on the theme item for the meeting plan for that theme.Click on the meeting item for references, exercises, and additional reading related to it.

**Theme 1 : Student Presentations**- 14 meetings- Meeting 01 : Mon, Apr 23, 04:00 pm-05:00 pm - Rahul CS
- Meeting 02 : Tue, Apr 24, 04:00 pm-05:00 pm - Balagopal
- Meeting 03 : Tue, Apr 24, 05:00 pm-06:00 pm - Mrinal Kumar
- Meeting 04 : Wed, Apr 25, 02:00 pm-03:00 pm - Prashant Vasudevan
- Meeting 05 : Wed, Apr 25, 03:00 pm-04:00 pm - Sivaramakrishnan
- Meeting 06 : Wed, Apr 25, 04:00 pm-05:00 pm - Devanathan T.
- Meeting 07 : Sat, Apr 28, 08:00 am-05:00 pm - N and J
- Meeting 08 : Mon, Apr 30, 09:00 am-10:00 am - Dinesh K.
- Meeting 09 : Mon, Apr 30, 10:00 am-11:00 am - Princy Lunawat
- Meeting 10 : Mon, Apr 30, 11:00 am-12:00 pm - Nilkamal Adak
- Meeting 11 : Mon, Apr 30, 01:30 pm-02:30 pm - Sajin Koroth
- Meeting 12 : Mon, Apr 30, 02:30 pm-03:30 pm - Krithika R.
- Meeting 13 : Mon, May 07, 04:30 pm-05:30 pm - Pavan Kumar
- Expander Codes - Venkat Guruswami's Lecture Notes.
- Expander Codes - Dan Spielman's Lecture Notes.
- Meeting 14 : Thu, May 10, 03:30 pm-04:30 pm - Saurav Kant Jha

Martingales and Talagrand inequalitites - Chapter 10 of Molloy and Reed. References Chapter 10 of Molloy and Reed. Exercises Reading Martingales and Talagrand inequalitites - Chapter 10 of Molloy and Reed.

Scribe : Pavan Kumar Sunkara (notes being written up)Scribe : Key : References : Chapter 10 of Molloy and Reed. <a href="http://tau.ac.il/~nogaa/PDFS/aghp4.pdf">Almost k-wise independent sample space constructions and lower bounds</a> - Alon-Goldreich-Hastad-Peralta. <br> <a href="./CS6845/2012/Almost_kwise_independent_spaces.pdf">PDF of Slides</a>. References Exercises Reading Almost k-wise independent sample space constructions and lower bounds - Alon-Goldreich-Hastad-Peralta.

PDF of Slides.

Scribe : Mrinal Kumar (to be edited)Scribe : Key : References : None <a href="http://www.cs.technion.ac.il/~shpilka/publications/DvirShpilka_LDC_PIT.pdf">Identity Testing of Depth-3 circuits and LDCs</a>.<br> <a href="./CS6845/2012/LDC_and_PIT.pdf">PDF of Slides</a>. References Exercises Reading Identity Testing of Depth-3 circuits and LDCs.

PDF of Slides.

Scribe : Balagopal (notes being written up)Scribe : Key : References : None FKG inequality and application - chapter 6 of alon and spencer References Exercises Reading FKG inequality and application - chapter 6 of alon and spencer

Scribe : Devanathan T. (notes being written up)Scribe : Key : References : None AB : <a href="http://www.cs.utexas.edu/~diz/395T/01/lec23.ps">Trevisan Extractor</a>. References <a href="http://www.cs.berkeley.edu/~luca/pubs/extractor-full.pdf">Trevisan Extractor</a> Exercises Reading References : Trevisan Extractor B2 : Complexity of Computational Problems in Coding Theory : <a href="http://people.csail.mit.edu/madhu/FT01/scribe/lect20.ps">Source 1</a>, <a href="http://users.cms.caltech.edu/~umans/522/lec11.ps">Source 2</a> References Exercises Reading References : None Exam + Viva : Checkout the references section of the meeting for the exact schedule. References <center><u><b>Exam+Viva Schedule :</b></u></center> Short Exam : Apr 28 Saturday morning 9am - 10:30am <br> Viva Schedule : 10:45am - 1pm and 2pm - 4:20pm. <br> The viva will have two parts : Part A and B for 20 minutes each. <br> ---------------------------------------------<br> 10:40 - 11:00 - 11:20 - Mrinal Kumar<br> 11:00 - 11:20 - 11:40 - Pavan Kumar <br> 11:20 - 11:40 - 12:00 - Balagopal <br> 11:40 - 12:00 - 12:20 - Devanathan T.<br> 12:00 - 12:20 - 12:40 - Krithika R.<br> 12:20 - 12:40 - 13:00 - Sajin Koroth<br> ---------------------------------------------<br> 14:00 - 14:20 - 14:40 - Dinesh K.<br> 14:20 - 14:40 - 15:00 - Prashant Vasudevan<br> 14:40 - 15:00 - 15:20 - Princy Lunawat<br> 15:00 - 15:20 - 15:40 - Sivaramakrishnan<br> 15:20 - 15:40 - 16:00 - Nilkamal Adak<br> 15:40 - 16:00 - 16:20 - Saurav Kant Jha<br> --------------------------------------------- Exercises Reading Exam + Viva : Checkout the references section of the meeting for the exact schedule.

Scribe : N and J (to be edited)Scribe : Key : References : Short Exam : Apr 28 Saturday morning 9am - 10:30am**Exam+Viva Schedule :**

Viva Schedule : 10:45am - 1pm and 2pm - 4:20pm.

The viva will have two parts : Part A and B for 20 minutes each.

---------------------------------------------

10:40 - 11:00 - 11:20 - Mrinal Kumar

11:00 - 11:20 - 11:40 - Pavan Kumar

11:20 - 11:40 - 12:00 - Balagopal

11:40 - 12:00 - 12:20 - Devanathan T.

12:00 - 12:20 - 12:40 - Krithika R.

12:20 - 12:40 - 13:00 - Sajin Koroth

---------------------------------------------

14:00 - 14:20 - 14:40 - Dinesh K.

14:20 - 14:40 - 15:00 - Prashant Vasudevan

14:40 - 15:00 - 15:20 - Princy Lunawat

15:00 - 15:20 - 15:40 - Sivaramakrishnan

15:20 - 15:40 - 16:00 - Nilkamal Adak

15:40 - 16:00 - 16:20 - Saurav Kant Jha

---------------------------------------------A4 : Janson's inequaliites and poisson paradigm - chapter 8 of alon and spencer <br> <a href="./CS6845/2012/Janson_Inequalities.pdf">PDF of Slides</a>. References Exercises Reading A4 : Janson's inequaliites and poisson paradigm - chapter 8 of alon and spencer

PDF of Slides.

Scribe : Princy Lunawat (notes being written up)Scribe : Key : References : None B4 : <a href="http://people.csail.mit.edu/dmoshkov/courses/codes/lec5-venkat.pdf">Construction</a> and <a href="http://people.csail.mit.edu/madhu/FT01/scribe/lect11.ps">Decoding</a> of Concatenated Codes. References Exercises Reading B4 : Construction and Decoding of Concatenated Codes.

Scribe : Dinesh K. (notes being written up)Scribe : Key : References : None A5 : <A href="http://www.eecs.harvard.edu/~michaelm/CS222/minwise.pdf">Construction of minwise independent permutation families</a> References Exercises Reading A5 : Construction of minwise independent permutation families

Scribe : Saurav Kant Jha (notes being written up)Scribe : Key : References : None B6 : <a href="http://arxiv.org/pdf/cs/0201001 ">Matrix Product and Codes</a> References Exercises Reading References : None A6 : Applications of the LLL for graph coloring applications - total coloring chapter 7 and 9 of molloy and reed References Exercises Reading A6 : Applications of the LLL for graph coloring applications - total coloring chapter 7 and 9 of molloy and reed

Scribe : Sajin Koroth (notes being written up)Scribe : Key : References : None Expander Codes and their applications. References <ul> <li><a href="http://www.cs.cmu.edu/~venkatg/teaching/codingtheory/notes/notes8.pdf">Expander Codes</a> - Venkat Guruswami's Lecture Notes. </li> <li><a href="http://www.cs.yale.edu/homes/spielman/561/lect12-09.pdf">Expander Codes </a> - Dan Spielman's Lecture Notes.</li> </ul> Exercises Reading Expander Codes and their applications.

Scribe : Rahul CS (notes being written up)Scribe : Key : References : TOPIC AND SCHEDULE SUBJECT TO CHANGE <br> B5 : <a href="http://www.cs.utexas.edu/~diz/pubs/extractor-codes.ps">Extractor Codes</a>. References Exercises Reading TOPIC AND SCHEDULE SUBJECT TO CHANGE

B5 : Extractor Codes.

Scribe : Nilkamal Adak (notes being written up)Scribe : Key : References : None **Theme 2 : Probablistic Method**- 6 meetings- Meeting 15 : Wed, Jan 04, 02:00 pm-04:00 pm - Narayanaswamy
- Meeting 16 : Fri, Jan 13, 03:00 pm-04:00 pm - Narayanaswamy
- Meeting 17 : Fri, Jan 27, 03:00 pm-05:00 pm - Narayanaswamy
- Meeting 18 : Fri, Feb 03, 03:00 pm-05:00 pm - Narayanaswamy
- Meeting 19 : Fri, Feb 17, 03:00 pm-05:00 pm - Narayanaswamy
- Meeting 20 : Fri, Feb 24, 03:00 pm-05:00 pm - Narayanaswamy

Linearity of Expectations, Applications of Markov's Inequality, Basic ones from Alon and Spencer References Alon Spencer Exercises Reading Alon Spencer References : Alon Spencer Reading : Alon Spencer Markov's inequality and application in arguing the presennce of an independent set of size at least n/sqrt(m), and the using the method of alterations to argue that there is always an independent set of size at least n^2/4m. References Exercises Reading References : None Random walks on undirected graphs. Structural properties of random walks. Random Commutes. Gobel-Jagers Bijection. Universal Traversal Sequences. References Lecture 5,6,7 from <a href="http://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.35.1132&rep=rep1&type=pdf">Tompa's Lecture Notes</a>. Exercises Reading References : Lecture 5,6,7 from Tompa's Lecture Notes. Derandomzing RL as an application of the Chebyshev inequality, the union bound, matrix norm and the connection to variation distance between distributions. Recursive doubling of the matrix, its impact on the norm, and the connection to the structure of the pseudorandom generator. Space bounded maintenance of good hash function by enumerative search. References Exercises Reading Derandomzing RL as an application of the Chebyshev inequality, the union bound, matrix norm and the connection to variation distance between distributions. Recursive doubling of the matrix, its impact on the norm, and the connection to the structure of the pseudorandom generator. Space bounded maintenance of good hash function by enumerative search.

Scribe : Mrinal KumarScribe : Key : References : None Chernoff bounds and application to analysis of oblivious routing protocols. Deterministic algorithms have an exponential lower bound on the number of rounds, whereas randomized oblivious protocols have an expected O(n) rounds. Analysis of the algorithm using chernoff bounds. References Exercises Reading Chernoff bounds and application to analysis of oblivious routing protocols. Deterministic algorithms have an exponential lower bound on the number of rounds, whereas randomized oblivious protocols have an expected O(n) rounds. Analysis of the algorithm using chernoff bounds.

Scribe : BalagopalScribe : Key : References : None The notion of a dependency graph among the events, the statement and proof of the Lovasz Local Lemma. The symmetric version of the lovasz local lemma, and application to satisfying assignments of CNF SAT in which variable occur only a bounded number of times. Mention of SAT(2) which is solvable in L. Constructivizing the LLL, the idea of resampling by Moser and Tardos. Similarities to Schoening's search for a satisfying assignment in k-SAT. Example of optimal routing without collisions, setting up the LLL instance, and the claim. References Moser and Tardos, Aravind Srinivasan's work on applications of LLL, recursive application of LLL by Leighton, Maggs, and Rao. Exercises Reading Probabilistic Method by Alon and Spencer Graph Coloring and Probabilistic Method by Molloy and Reed The notion of a dependency graph among the events, the statement and proof of the Lovasz Local Lemma. The symmetric version of the lovasz local lemma, and application to satisfying assignments of CNF SAT in which variable occur only a bounded number of times. Mention of SAT(2) which is solvable in L. Constructivizing the LLL, the idea of resampling by Moser and Tardos. Similarities to Schoening's search for a satisfying assignment in k-SAT. Example of optimal routing without collisions, setting up the LLL instance, and the claim.

Scribe : Princy LunawatScribe : Key : References : Moser and Tardos, Aravind Srinivasan's work on applications of LLL, recursive application of LLL by Leighton, Maggs, and Rao. Reading : Probabilistic Method by Alon and Spencer Graph Coloring and Probabilistic Method by Molloy and Reed **Theme 3 : Explicit Constructions**- 6 meetings- Meeting 21 : Fri, Mar 02, 03:00 pm-05:00 pm - Narayanaswamy
- Meeting 22 : Fri, Mar 09, 03:00 pm-04:00 pm - Narayanaswamy
- Meeting 23 : Fri, Mar 16, 03:00 pm-05:00 pm - Narayanaswamy
- Meeting 24 : Fri, Mar 30, 03:00 pm-05:00 pm - Narayanaswamy
- Meeting 25 : Fri, Apr 13, 03:00 pm-05:00 pm - Narayanaswamy
- Meeting 26 : Thu, Apr 19, 06:15 pm-07:45 pm - Narayanaswamy

Construction of d-wise independent sample spaces References Exercises Reading References : None Method of conditional probabilities. References Exercises Reading References : None Extracting Randomness:Motivation and Formal Defns References Nisan and Zuckerman 93 - Randomness is Linear in Space Nisan 96 - Extracting Randomness: How and Why? Exercises Reading Extracting Randomness:Motivation and Formal Defns

Scribe : Sajin Koroth (rough draft; better one coming soon)Scribe : Key : References : Nisan and Zuckerman 93 - Randomness is Linear in Space Nisan 96 - Extracting Randomness: How and Why? Construction of extractors - even suboptimal ones References Exercises Reading References : None Extractor consrtuction contd., Expander graphs, Explicit expander graph construction. Gabber Galil Expanders References Exercises Reading Extractor consrtuction contd., Expander graphs, Explicit expander graph construction. Gabber Galil Expanders

Scribe : Nilkamal Adak (notes being written up)Scribe : Key : References : None Gabber Galil Expander Graphs - Proof of Expansion. Expanders and Eigen values. References Exercises Reading Gabber Galil Expander Graphs - Proof of Expansion. Expanders and Eigen values.

Scribe : Rahul CS (notes being written up)Scribe : Key : References : None **Theme 4 : Markov Chains and Applications**- 1 meetings- Meeting 27 : Fri, Apr 20, 03:00 pm-05:00 pm - Narayanaswamy

Introduction to Markov chains. Steady-state probability distributions, canonical paths, conductance, Coupling and mixing times, approximate counting and sampling. References Exercises Reading Introduction to Markov chains. Steady-state probability distributions, canonical paths, conductance, Coupling and mixing times, approximate counting and sampling.

Scribe : Saurav Kant Jha (notes being written up)Scribe : Key : References : None **Theme 5 : Error Correcting Codes & Applications**- 12 meetings- Meeting 28 : Fri, Jan 06, 02:00 pm-03:00 pm - Jayalal Sarma
- Shannon's original paper (very readable !). Here is a local copy.

Claude E. Shannon. A Mathematical Theory of Communication. Bell Systems Technical Journal, Vol 27, pp. 379-423, 623-656, July, October, 1948. - A write-up on the context and style of Shannon's work.
- A good source on information theory is the textbook by Cover and Thomas. In particular, Chapters 2 and 8 of the book are relevant to our discussion.
- Meeting 29 : Wed, Jan 11, 02:00 pm-04:00 pm - Jayalal Sarma
- Shannon's original paper (very readable !). Here is a local copy.

Claude E. Shannon. A Mathematical Theory of Communication. Bell Systems Technical Journal, Vol 27, pp. 379-423, 623-656, July, October, 1948. - A write-up on the context and style of Shannon's work.
- A good source on information theory is the textbook by Cover and Thomas. In particular, Chapters 2 and 8 of the book are relevant to our discussion.
- Meeting 30 : Wed, Jan 18, 02:00 pm-04:00 pm - Jayalal Sarma
- Show that C is an (n,k) linear code if and only if there is a matrix H (of dimensions (n-k)xn of full row rank such that C is precisely the null space of H.
- In the last lecture we proved existence of codes with high success probability in decoding. Are there such linear codes? Attempt to carry out the same probabilistic argument for linear codes.
- Meeting 31 : Wed, Feb 01, 02:00 pm-04:00 pm - Jayalal Sarma
- Meeting 32 : Wed, Feb 08, 02:00 pm-04:00 pm - Jayalal Sarma
- Meeting 33 : Wed, Feb 15, 02:00 pm-04:00 pm - Jayalal Sarma
- Meeting 34 : Wed, Feb 22, 02:00 pm-04:00 pm - Jayalal Sarma
- Meeting 35 : Wed, Feb 29, 02:00 pm-04:00 pm - Jayalal Sarma
- Meeting 36 : Wed, Mar 14, 02:00 pm-04:00 pm - Jayalal Sarma
- Meeting 37 : Wed, Mar 21, 02:00 pm-03:30 pm - Jayalal Sarma
- Meeting 38 : Sat, Mar 24, 10:00 am-12:00 pm - Jayalal Sarma
- Meeting 39 : Wed, Mar 28, 02:00 pm-04:00 pm - Jayalal Sarma

Administrative Announcements, Course Plan, Introduction to Coding Theory, Information theoretic background, Noiseless coding theorem. References <ul> <li><a href="http://people.csail.mit.edu/madhu/FT01/scribe/lect1.ps">Madhu Sudan's Lectures Notes</a>.</li> <li>Our <a href="./CS6845/lecture-B01-02.pdf">notes</a>.</li> </ul> Exercises Reading <ul> <li><a href="http://dl.acm.org/citation.cfm?id=584093">Shannon's original paper</a> (very readable !). Here is a <a href="./CS6845/Shannon.pdf">local copy</a>.<br>Claude E. Shannon. A Mathematical Theory of Communication. Bell Systems Technical Journal, Vol 27, pp. 379-423, 623-656, July, October, 1948. </li> <li>A <a href="http://mit.edu/6.933/www/Fall2001/Shannon1.pdf"> write-up</a> on the context and style of Shannon's work.</li> <li>A good source on information theory is the textbook by Cover and Thomas. In particular, Chapters 2 and 8 of the book are relevant to our discussion.</li> </ul> Administrative Announcements, Course Plan, Introduction to Coding Theory, Information theoretic background, Noiseless coding theorem.References : Reading : Noisy Coding Theorem - Statement and Proof. Basic Coding theory goals. References <ul> <li><a href="http://people.csail.mit.edu/madhu/FT01/scribe/lect1.ps">Madhu Sudan's Lectures Notes</a>.</li> <li>Our <a href="./CS6845/lecture-B01-02.pdf">notes</a>.</li> </ul> Exercises Reading <ul> <li><a href="http://dl.acm.org/citation.cfm?id=584093">Shannon's original paper</a> (very readable !). Here is a <a href="./CS6845/Shannon.pdf">local copy</a>.<br>Claude E. Shannon. A Mathematical Theory of Communication. Bell Systems Technical Journal, Vol 27, pp. 379-423, 623-656, July, October, 1948. </li> <li>A <a href="http://mit.edu/6.933/www/Fall2001/Shannon1.pdf"> write-up</a> on the context and style of Shannon's work.</li> <li>A good source on information theory is the textbook by Cover and Thomas. In particular, Chapters 2 and 8 of the book are relevant to our discussion.</li> </ul> References : Reading : A combinatorial and constructive view : minimum distance and rate. <br> A simple application of codes : Constructing hash families from codes. <br> Basic code constructions. Hamming code. <br>Linear codes, Generator matrix and parity check matrices. References <ul> <li><a href="http://www.cs.washington.edu/education/courses/cse533/06au/lecnotes/lecture2.pdf">Lecture 1</a> of Venkat Guruswami's course at UW.</li> <li><a href="http://users.cms.caltech.edu/~umans/522/lec1.ps">Lecture 1</a> of Chris Umans's Course at Caltech.</li> </ul> Exercises <ul> <li>Show that C is an (n,k) linear code if and only if there is a matrix H (of dimensions (n-k)xn of full row rank such that C is precisely the null space of H.</li> <li>In the last lecture we proved existence of codes with high success probability in decoding. Are there such linear codes? Attempt to carry out the same probabilistic argument for linear codes.</li> </ul> Reading <a href="http://www.siam.org/pdf/news/595.pdf">Coding Theory meets theoretical computer science</a>. References : Exercises : Reading : Coding Theory meets theoretical computer science. General Hamming Bound, Generalized Hamming codes. Duals of codes, Hadamard codes, parameters. <br> Singleton Bound, Reed-Solomon Codes. Rough idea for Reed-Muller codes. References <a href="http://www.cs.cmu.edu/~venkatg/teaching/codingtheory/notes/notes1.pdf">Venkat Guruswami's Lecture</a> at CMU<br> <a href="http://people.csail.mit.edu/dmoshkov/courses/codes/lec5-venkat.pdf">Venkat Guruswami's Lecture</a> hosted at MIT. Exercises Find out the parity check matrix formulation of Reed-Solomon Codes<br> Find an example of a self-dual code. Reading <a href="http://sdsu-dspace.calstate.edu/xmlui/handle/10211.10/1743">Applications of Reed-Solomon codes</a> on optical storage media.<br> <a href="http://www.vocal.com/data_sheets/Reed_Solomon_Implementation.pdf">An actual implementation<br> References : Venkat Guruswami's Lecture at CMU

Venkat Guruswami's Lecture hosted at MIT.Exercises : Find out the parity check matrix formulation of Reed-Solomon Codes

Find an example of a self-dual code.Reading : Applications of Reed-Solomon codes on optical storage media.

An actual implementationReed-Muller Codes, Concatenation Codes - Bounds and parameters. Unique decoding problem. Decoding problem for Reed-Solomon Codes. Error locating Polynomials References Exercises Reading References : None Berlekamp-Welsh decoding algorithm for Reed-Solomon Codes. Concatenated Codes, Decoding bounds. <br> List decoding problem. Decoding for Hadamard codes. List decoding of Reed-Solomon Codes, Statement and interpretation of Sudan's list decoding. References <a href="http://arxiv.org/abs/cs/0409044">Survey by Luca Trevisan</a><br> <a href="http://people.csail.mit.edu/madhu/FT01/scribe/lect10.ps">Lecture 10</a> in Madhu Sudan's course. Exercises Show the following for list size bound for Hadamard codes. For any Boolean function f from k bits to 1 bit, there are at most 1/(4e^2) linear functions g that agree with f in at least 0.5+e fraction of inputs (0 < e < 0.5). Reading <a href="http://arxiv.org/abs/cs/0409044">Survey by Luca Trevisan</a> References : Survey by Luca Trevisan

Lecture 10 in Madhu Sudan's course.Exercises : Show the following for list size bound for Hadamard codes. For any Boolean function f from k bits to 1 bit, there are at most 1/(4e^2) linear functions g that agree with f in at least 0.5+e fraction of inputs (0 < e < 0.5). Reading : Survey by Luca Trevisan Sudan's algorithm for list-decoding of Reed-Solomon codes. Sublinear time decoding : locally decodable codes. Applications to Private information Retrieval. Local Smooth Decoder, that suits both purposes. Local Smooth decoder for Hadamard code. References Appendix of the <a href="http://arxiv.org/abs/cs/0409044">survey by Luca Trevisan</a> Exercises Identify a slackness in the proof and tune the parameters to improve error allowed (t) from 2.sqrt{nk} to sqrt{2nk} (the smaller t is it is more difficult to do list decoding because there could be more polynomials in the list, since there are lesser constraints). Reading <a href="http://arxiv.org/abs/cs/0409044">Survey by Luca Trevisan</a><br> Sudan's algorithm for list-decoding of Reed-Solomon codes. Sublinear time decoding : locally decodable codes. Applications to Private information Retrieval. Local Smooth Decoder, that suits both purposes. Local Smooth decoder for Hadamard code.

Scribe : Sivaramakrishnan (to be edited)Scribe : Key : References : Appendix of the survey by Luca Trevisan Exercises : Identify a slackness in the proof and tune the parameters to improve error allowed (t) from 2.sqrt{nk} to sqrt{2nk} (the smaller t is it is more difficult to do list decoding because there could be more polynomials in the list, since there are lesser constraints). Reading : Survey by Luca Trevisan An application of Reed-Solomon Codes to show average-case hardness.<br> An efficient algorithm computing Permanent for 1/2+epsilon fraction of input will imply an efficient randomized algorithm for permanent that works for all inputs. <br><i>Tool : Random Self reducibility + Berlekamp-Welch unique decoding of RS codes.</i> <br> Cai-Pavan-Sivakumar - Efficiently computing permanent in atleast 1/n^k fraction of inputs will imply an efficient randomized algorithm for permanent that works for all inputs. <i>Tool : Sudan's list-decoding algorithm.</i> References <a href="www.cs.cmu.edu/~venkatg/pubs/papers/ld-avg-PR.pdf">List decoding in Average Case Complexity and Pseudorandomness</a> - Survey by Venkatesan Guruswami. (We discussed the first part of this survey which addresses the average case complexity)<br> <a href="http://pages.cs.wisc.edu/~jyc/papers/permanent.pdf">Hardness of Pemanent</a> - Cai-Pavan-Sivakumar - STACS 1999. Exercises Reading <a href="www.cs.cmu.edu/~venkatg/pubs/papers/ld-avg-PR.pdf">List decoding in Average Case Complexity and Pseudorandomness</a> - Survey by Venkatesan Guruswami. An application of Reed-Solomon Codes to show average-case hardness.

An efficient algorithm computing Permanent for 1/2+epsilon fraction of input will imply an efficient randomized algorithm for permanent that works for all inputs.*Tool : Random Self reducibility + Berlekamp-Welch unique decoding of RS codes.*

Cai-Pavan-Sivakumar - Efficiently computing permanent in atleast 1/n^k fraction of inputs will imply an efficient randomized algorithm for permanent that works for all inputs.*Tool : Sudan's list-decoding algorithm.*

Scribe : Rahul CS (notes being written up)Scribe : Key : References : List decoding in Average Case Complexity and Pseudorandomness - Survey by Venkatesan Guruswami. (We discussed the first part of this survey which addresses the average case complexity)

Hardness of Pemanent - Cai-Pavan-Sivakumar - STACS 1999.Reading : List decoding in Average Case Complexity and Pseudorandomness - Survey by Venkatesan Guruswami. Locally decodable codes for hardness amplification within E. Conversion from worst-case hard Boolean function to average-case hard Boolean function using a local decoder. Improving the average case hardness by using local-list decoders. <br> Back to local decoding : Local decoding vs Local correction. References <a href="http://people.seas.harvard.edu/~salil/pseudorandomness/pseudorandomness-Apr11.pdf">Pseudorandomness</a> - survey by Salil Vadhan. See section 7.5 and 7.6. <br> <a href="http://arxiv.org/abs/cs/0409044">Survey</a> by Luca Trevisan (section 4) Exercises Reading Locally decodable codes for hardness amplification within E. Conversion from worst-case hard Boolean function to average-case hard Boolean function using a local decoder. Improving the average case hardness by using local-list decoders.

Back to local decoding : Local decoding vs Local correction.

Scribe : Nilkamal Adak (to be edited)Scribe : Key : References : Pseudorandomness - survey by Salil Vadhan. See section 7.5 and 7.6.

Survey by Luca Trevisan (section 4)<i>Shorter lecture</i> : t-query Local corrector for Reed-Muller Codes (where t is the degree bound of the polynomials). Making the code systematic. The trade-off between Query complexity and rate of the LDC. Variants which achieves constant query complexity at the expense of rate. References Section 3.3 & 3.4 in the <a href="http://arxiv.org/abs/cs/0409044">Survey by Luca Trevisan</a>. Exercises Reading *Shorter lecture*: t-query Local corrector for Reed-Muller Codes (where t is the degree bound of the polynomials). Making the code systematic. The trade-off between Query complexity and rate of the LDC. Variants which achieves constant query complexity at the expense of rate.

Scribe : Princy LunawatScribe : Key : References : Section 3.3 & 3.4 in the Survey by Luca Trevisan. Local list decoder. Goldreich-Levin Local list decoder for Hadamard codes. References Section 4.4 in the <a href="http://arxiv.org/abs/cs/0409044">Survey by Luca Trevisan</a>. Exercises Reading Local list decoder. Goldreich-Levin Local list decoder for Hadamard codes.

Scribe : Balagopal (notes being written up)Scribe : Key : References : Section 4.4 in the Survey by Luca Trevisan. Application of GL-decoder to construct hardcore predicates of one-way permutations. <br> Local list decoder for Reed-Muller Codes. References Section 4.5 in the <a href="http://arxiv.org/abs/cs/0409044">Survey by Luca Trevisan</a>. <br> <a href="people.seas.harvard.edu/~salil/cs225/spring07/lecnotes/lec21.ps">Salil Vadhan's Lecture</a> in his Pseudorandomness course. Exercises Reading References : Section 4.5 in the Survey by Luca Trevisan.

Salil Vadhan's Lecture in his Pseudorandomness course.**Theme 6 : Fourier Transform & Applications**- 3 meetings- Meeting 40 : Wed, Apr 04, 02:00 pm-04:00 pm - Jayalal Sarma
- Meeting 41 : Wed, Apr 11, 03:15 pm-04:30 pm - Jayalal Sarma
- Meeting 42 : Wed, Apr 18, 02:00 pm-03:30 pm - Jayalal Sarma
- N. Linial, Y. Mansour, N. Nisan. Constant depth circuits, Fourier transform, and learnability. 1993.

Transforms. The example of polynomial multiplication. Coefficient vs Pointwise Evaluation representations. Tranforming one from the other. Choosing the points of evaluations. FFT formulation. Inverse transform. Multiplying polynomials using O(n log n) operations. Applications to integer multiplication. References <a href="http://www.inf.ed.ac.uk/teaching/courses/ads/Lects/FFT.pdf">Fast Fourier Transforms</a>. Exercises Reading <a href="http://www.cse.psu.edu/~furer/Papers/mult.pdf">Furer's Paper</a><br> <a href="http://arxiv.org/abs/0801.1416">Fast Integer Multiplication using Modular Arithmetic</a> - DKSS 2008 Transforms. The example of polynomial multiplication. Coefficient vs Pointwise Evaluation representations. Tranforming one from the other. Choosing the points of evaluations. FFT formulation. Inverse transform. Multiplying polynomials using O(n log n) operations. Applications to integer multiplication.

Scribe : Prashant Vasudevan (notes being written up)Scribe : Key : References : Fast Fourier Transforms. Reading : Furer's Paper

Fast Integer Multiplication using Modular Arithmetic - DKSS 2008<i>Shorter Lecture</i> : The Fourier basis for the vector space of Boolean Realvalued functions. Viewing Fourier transform as a change of basis. Parseval's equality. Fourier transforms of example Boolean functions. Basics of Learning of Boolean functions. References <a href="www.math.tau.ac.il/~mansour/papers/fourier-survey.ps.gz">Learning Boolean Functions via the Fourier transforms</a> - Yishay Mansour. Exercises Reading *Shorter Lecture*: The Fourier basis for the vector space of Boolean Realvalued functions. Viewing Fourier transform as a change of basis. Parseval's equality. Fourier transforms of example Boolean functions. Basics of Learning of Boolean functions.

Scribe : Sajin Koroth (notes being written up)Scribe : Key : References : Learning Boolean Functions via the Fourier transforms - Yishay Mansour. Learning Boolean Functions. From Fourier expansion to learning. Estimating the Fourier coefficients. Low-degree learning. Fourier spectrum for decision trees. LMN theorem.<br> Property testing model. BLR Linearity Testing. View from the Fourier spectrum. References <ul> <a href="www.math.tau.ac.il/~mansour/papers/fourier-survey.ps.gz">Learning Boolean Functions via the Fourier transforms</a> - Yishay Mansour.</li> <li>N. Linial, Y. Mansour, N. Nisan. <a href="http://doi.acm.org/10.1145/174130.174138">Constant depth circuits, Fourier transform, and learnability.</a> 1993.</li> </ul> Exercises Reading Learning Boolean Functions. From Fourier expansion to learning. Estimating the Fourier coefficients. Low-degree learning. Fourier spectrum for decision trees. LMN theorem.

Property testing model. BLR Linearity Testing. View from the Fourier spectrum.

Scribe : Saurav Kant Jha (notes being written up)Scribe : Key : References : - Learning Boolean Functions via the Fourier transforms - Yishay Mansour.