Barak Pinhas

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List Decoding Of RS Codes

• Barak Pinhas
• ECC Seminar
• Tel-Aviv University

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Unique Decoding

• Let be an code
• We saw that if there are less than errors,
  we can find a unique decoding for any given input
• Let us be more flexible

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List Decoding

• What if there are more than errors ?
• We need that
• - There would be a few possible decodings
• - We should be able to find them efficiently

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Combinatorial List Decoding

• We would say that a code is
  -code if it satisfies the following condition
• For every
• Where ,

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Algorithmic List Decoding

• Given a code
• If , we will have possible decodings
  for every string
• We would like to find the list of possible
  decodings in polynomial time, so we need

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RS Codes Revisited

• Given a field we define the space of
  univariate polynomials over and denote
  it by
• Encoder
• Given an input
  , create
• Choose n distinct values
• Transmit
• Decoder
• Unique Decoding Algorithm (BelekampWelch)
• List Decoding Algorithm (Sudan)

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RS Codes Properties

• RS Codes are linear
• RS Codes meet the singleton bound (without
  proof), i.e. they are
  codes.

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Problem Description

• Input
• Values
• The degree, ,of the message polynomial
• The minimum number of agreements

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The Goal

• Find a list of polynomials
  of degree at most , s.t
• These are the polynomials which represent
  messages that are candidate decodings of .
• Lets see an example

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Sudans Algorithm - Simplified

• Find all functions such that
• is of small degree
• is not identically zero
• Factor into irreducible factors
• Output a list of all small degree factors that
  have a large agreement with the input.
• Lets go back to our example

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Definition Weighted Degree

• For a polynomial we define its
  weighted degree as follows
• Note that
• has weight 1
• has weight

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Sudans Algorithm Detailed

• Init and , two integers - we will see later
  what should be their values
• Find all functions such that
• is of weighted degree at most
  
• is not identically zero
• Factor into irreducible factors
• Output a list of all polynomials such that
• and

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Step 1 Finding Q

• If a function satisfies B, it can be found
  in polynomial time.
• Proof
• Consider the following system of linear
  equations.
• For each given point we have a constraint
• Linear system with variables
• And with Constraints
• We need more variables than constraints
• Linear algebra guarantees a solution, and we can
  efficiently find it.

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Step 2 Factoring Q

• Well known problem
• Studied a lot
• There are efficient deterministic polynomial time
  algorithms to solve this problem
• We wont get into details

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Step 3 Why Every Solution Divides Q?

• Claim 2
• If is a solution for some low degree
  polynomial
• and , then
• Proof
• Any monomial in is of the form
• Now let
• We set
• We have and now
• Hence
• But , so

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Step 3 Cont.

• How does implies ?
• We start by showing the theorem over by
  simple linear algebra theorem
• Applying the theorem over will give us
  the required result

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Whats Left?

• What are the optimal values for ?
• Sudan
• What is the range of values of the agreement
  parameter that the algorithm will work for?
• Sudan
• Previous algorithms
• Consider the ratio

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Improved List Decoding RS

• Algorithm By Guruswami Sudan

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The Goal

• We want to correct more errors
• Hence, we want to deal with smaller agreement
  values
• We still need the algorithm to run in polynomial
  time

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Shifted Polynomials

• Let be a polynomial with variables
• Define the shifted polynomial by
• Denote the shift of to by

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The Coefficients Of

• Note that if are the coefficients of
  , then

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The AlgorithmStep 1

• Compute parameters such that
• 1.
• 2.
• We will see what are the values for later on.

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Step 2

• Find a polynomial that satisfies
• 1. is not identically 0
• 2. All coefficients of with total degree less
  than
• are 0. That is

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Step 2 Cont.

• 3. is of weighted degree
• Reminder

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Steps 3 and 4

• Factor into irreducible factors
• Output all polynomials such that
• 1.
• 2.
• 3.

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Algorithm AnalysisStep 2 How Do We Find Q

• Again, a system of linear equations
• Number of variables equal to the number of
  monomials in , which is

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Step 2 Cont.

• The number of constraints is all the integer
  solutions for the inequality . It applies
  for each of the input points
• Using some basic combinatorics, it can be shown
  that
• the number of constraints is

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Step 2 Conclusion

• If then we can find in
• polynomial time.

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Step 3 Factoring Q

• We have already seen that this problem can be
  solved in polynomial time
• Again, we wont get into details

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Step 4 Why Every Solution Is A Factor Of Q?

• Claim 1
• If for an input point
• Then
• Where
• Proof On Board

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Step 4 Cont.

• Claim 2
• If is a polynomial of degree at most
• and has agreement number with the
  input points,
• Then
• Proof On Board

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Optimal Values For
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The agreement number

• The algorithm works for
• So we can tolerate up to
  errors.
• Great improvement over Sudan for

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Summary

• We presented Sudans efficient and well-known
  algorithm for list decoding RS codes
• Guruswami Sudan provided an improved version of
  the algorithm for the same problem
• Lets go back to our example for the last time