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Correcting Errors Beyond the Guruswami-Sudan Radius Farzad Parvaresh & Alexander Vardy Presented by Efrat Bank

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2 General plan of this lecture 1.Some bounds for list-decoding algorithms. 2.PV algorithm for two variables. 3.PV algorithm for more than two variables.

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3 Main goal The goal is to present a family of error-correcting codes that have polynomial time encoder and polynomial time list-decoder, correcting a fraction of adversarial errors up to: where R is the rate of the code and is an arbitrary integer.

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4 A Reminder List-decoding: Let C be a code, C is called L)-list decodable if for any vector the decoder outputs a list of L vectors such that for every codeword with there exists i such that

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Known bounds for list decoding

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6 Previous bounds Johnson bound –Let be the maximal number of codewords in the Hamming ball of radius e for a codeof minimum distance d. –If then Guruswami-sudan bound –Explicitly finds a (1- ,L)-List decodable with –Remember that RS codes are –Thus, if we take RS code and use the GS algorithm we see that we meet the Johnson bound

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7 Previous bounds cont. Non-explicit bound Let. There exists a code (1- ,L)-List decodable with rate Parvaresh -Vardy bound Explicitly finds a (1- ,L)-List decodable with

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8 Comparing the bounds LRelation between Rate and Johnson GS PV LNon-explicit

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PV Encoding algorithm (for two variables)

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10 Reminder: RS codes RS assigned for any message a polynomial with one variable. As I see it, there are two “natural” ways to generalize this construction: –For any message assign a multivariate polynomial. – RM. –For any message assign more than one polynomial - PV

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11 Code parameters q, k, n – as usual. Evaluation set- Basis- A fixed basis for over m- A positive integer which determines the multiplicity. a – A large enough positive integer which determines the list size. An irreducible polynomial of degree k.

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12 The Encoder mapping. Given a messageconstruct the polynomial Compute The codeword is given by

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13 RS PV

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14 Analysis Rate: Minimum distance: Computation time is polynomial. Note: In most cases the code is NOT linear.

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PV Decoding algorithm (for two variables)

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16 Some definitions before we start Interpolation set: Given a vector and some basis for over we may write where The interpolation set for and is the set

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17 Hasse derivative: Letbe a polynomial with three variables over. For the corresponding Hasse derivative of is:

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18 The polynomial is said to have zero of multiplicity m at a point if: for allsuch that The (1,k-1,k-1)- weighted degree of a monomial is: For a polynomial the w.d. is defined as the maximal w.d. of its monomials.

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19 The Interpolation polynomial with respect to an interpolation set is the least w.d. nonzero polynomial that has a zero of multiplicity m at each point of. We denote this polynomial by. can be computed in polynomial time in n,m.

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20 Lemma 3:

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21 Now, we are almost ready to see the decoding algorithm… Before we do that, lets recall the Guruswami- Sudan algorithm for list-decoding RS codes.

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22 GS list-decoding for RS codes 1.Given a vector consider the interpolation set and find the interpolation polynomial. 2.Think of as. 3.Find all the roots of in. 4.Return the roots that satisfy the following:. agrees with the interpolation set on a sufficient number of points.

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23 RS encoding RS decoding

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24 Can anyone guess what will be the decoding algorithm for PV?

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25 PV Encoding

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26 Reminder: Code parameters q, k, n – as usual. Evaluation set- Basis- A fixed basis for over m- A positive integer which determines the multiplicity. a – A large enough positive integer which determines the list size. An irreducible polynomial of degree k.

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27 Decoding Algorithm 1.Given a vector consider the interpolation set and compute the interpolation polynomial 2.Compute interperted as an element of 3.Compute 4.Output the roots of

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29 PV decodingPV encoding

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30 Analysis The algorithm runs in polynomial time in n,m: –One can compute in polynomial time. –Finding the roots of also can be done in polynomial time by a result of Shoup.

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31 Correctnass of the algorithm We need to show that when given any vector and any codeword that is “close enough” to v, the decoding algorithm outputs u. However, before we do that we need to verify that the algorithm is well defined, i.e. andare not the zero polynomials. For now, lets assume that this is the case.

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32 Lemma 8: Suppose that and denote and. Then. Given a codewordwe know from the encoding algorithm that and that. Put in other words, If and then. Thus, ifthen is a root of the polynomial

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33 Definition: Let and define, Lemma 4: Let be the interpolation polynomial with respect to the set. If satisfy: 1.. 2.. Then,

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34 Lemma 6: (corollary of Lemma 4): If a codeword differs from a given vector in at most positions, and let denote the message polynomials that produce the codeword. Thensatisfies

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35 Lemma 7: is not the zero polymonial. –Proof on the board. Lemma 9: is not the zero polynomial. –Proof on the board.

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36 Putting it all together Theorem 10: Given a vector the decoding algorithm outputs in polynomial time the list of all codewords that differ from v in at most positions, where is an arbitrary multiplicity parameter. The size of the list is at most

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PV for many variables

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38 Extension to multivariate interpolation RS code assigns one polynomial for each message. We have presented the PV code which assigns two polynomial for each message. Of course, there is nothing magical about the number two…In fact, PV can be extended to any numer of polynomials. This requires a minor change of the code parameters.

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39 Parameters for multivariate interpolation q, k, n, m, Basis- A fixed basis for over Degrees:

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40 PV encoding

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41 PV decoding

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42 PV Main result Theorem 1: Let q be a power of a prime. Then for all positive integers there is a code C of length n and rate over, Such that C is List-decodable where,

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Thanks

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