Correcting Errors Beyond the GuruswamiSudan Radius Farzad Parvaresh

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Correcting Errors Beyond the Guruswami-Sudan
RadiusFarzad Parvaresh Alexander Vardy

• Presented by Efrat Bank

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General plan of this lecture

• Some bounds for list-decoding algorithms.
• PV algorithm for two variables.
• PV algorithm for more than two variables.

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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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A Reminder

• List-decoding
• Let C be a code,
• C is called (t, 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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Previous bounds

• Johnson bound
• Let be the maximal number of codewords in
  the Hamming ball of radius e for a code of
  minimum distance d.
• If then
• Guruswami-sudan bound
• Explicitly finds a (1-e,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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Previous bounds cont.

• Non-explicit bound
• Let .
• There exists a code (1-e,L)-List decodable with
  rate
• Parvaresh -Vardy bound
• Explicitly finds a (1-e,L)-List decodable with

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Comparing the bounds
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PV Encoding algorithm

• (for two variables)

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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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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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The Encoder mapping

• .
• Given a message construct the polynomial
• Compute
• The codeword is given by

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• RS

• PV

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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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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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• Hasse derivative
• Let be a polynomial with three variables
  over .
• For the corresponding Hasse derivative of
  is

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• The polynomial is said to have zero of
  multiplicity m at a point if
• for all such 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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• 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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• Lemma 3

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• 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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GS list-decoding for RS codes

• Given a vector consider the interpolation
  set and find the interpolation polynomial
  .
• Think of as .
• Find all the roots of in .
• Return the roots that satisfy the following
• .
• agrees with the interpolation set on a
  sufficient number of points.

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• RS encoding

• RS decoding

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

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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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Decoding Algorithm

• Given a vector consider the
  interpolation set and compute the
  interpolation polynomial
• Compute
• interperted as an element of
• Compute
• Output the roots of

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(No Transcript)
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• PV decoding

PV encoding
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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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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.
• and are not the zero polynomials. For now,
  lets assume that this is the case.

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• Lemma 8 Suppose that
• and denote and .
• Then .
• Given a codeword we know from the encoding
  algorithm that
• and that .
• Put in other words, If and
• then . Thus, if then is a root of
  the polynomial

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• Definition Let and
• define,
• Lemma 4 Let be the interpolation
  polynomial with respect to the set .
• If satisfy
• .
• .
• Then,

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• 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.
• Then satisfies

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• 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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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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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 twoIn fact, PV can be extended to any
  numer of polynomials. This requires a minor
  change of the code parameters.

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Parameters for multivariate interpolation

• q, k, n, m,
• Basis- A fixed basis for over
• Degrees

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• PV encoding

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• PV decoding

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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 ?