6/20/2015List Decoding Of RS Codes 1 Barak Pinhas ECC Seminar Tel-Aviv University

6/20/2015List Decoding Of RS Codes 2 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

6/20/2015List Decoding Of RS Codes 3 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

6/20/2015List Decoding Of RS Codes 4 Combinatorial List Decoding We would say that a code is -code if it satisfies the following condition: For every : Where:,

6/20/2015List Decoding Of RS Codes 5 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:

6/20/2015List Decoding Of RS Codes 6 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 (Belekamp&Welch) * List Decoding Algorithm (Sudan)

6/20/2015List Decoding Of RS Codes 7 RS Codes: Properties RS Codes are linear: RS Codes meet the singleton bound (without proof), i.e. they are codes.

6/20/2015List Decoding Of RS Codes 8 Problem Description Input: Values The degree,,of the message polynomial The minimum number of agreements:

6/20/2015List Decoding Of RS Codes 9 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. Let’s see an example

6/20/2015List Decoding Of RS Codes 10 Sudan’s Algorithm - Simplified A. Find all functionssuch that: is of small degree is not identically zero B. Factor into irreducible factors C. Output a list of all small degree factors that have a large agreement with the input. Let’s go back to our example

6/20/2015List Decoding Of RS Codes 11 Sudan’s Algorithm: Detailed A. Init and, two integers - we will see later what should be their values B. Find all functionssuch that: is of weighted degree at most (Will be explained later) is not identically zero C. Factor into irreducible factors D. Output a list of all polynomials such that and

6/20/2015List Decoding Of RS Codes 12 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.

6/20/2015List Decoding Of RS Codes 13 Step 2: Factoring Q Well known problem Studied a lot There are efficient deterministic polynomial time algorithms to solve this problem We won’t get into details

6/20/2015List Decoding Of RS Codes 14 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

6/20/2015List Decoding Of RS Codes 15 Step 3: Cont. How does implies ? We start by showing the theorem over Applying the theorem over will give us the required result

6/20/2015List Decoding Of RS Codes 16 What’s 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

6/20/2015List Decoding Of RS Codes 17 Summary We presented Sudan’s efficient and well- known algorithm for list decoding RS codes Guruswami & Sudan provided an improved algorithm for the same problem Let’s go back to our example for the last time