List Decoding Of RS Codes Barak Pinhas ECC

List Decoding Of RS Codes Barak Pinhas ECC Seminar Tel-Aviv University 1 9/9/2020 List Decoding Of RS Codes

Unique Decoding l l l 2 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 9/9/2020 List Decoding Of RS Codes

List Decoding l l 3 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 9/9/2020 List Decoding Of RS Codes

Combinatorial List Decoding 4 l We would say that a code satisfies the following condition: For every : l Where: is -code if it , 9/9/2020 List Decoding Of RS Codes

Algorithmic List Decoding l l l 5 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: 9/9/2020 List Decoding Of RS Codes

RS Codes: Revisited l 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 9/9/2020 List Decoding Of RS Codes

RS Codes: Properties 7 l RS Codes are linear: l RS Codes meet the singleton bound (without proof), i. e. they are codes. 9/9/2020 List Decoding Of RS Codes

Problem Description Input: l Values 8 l The degree, , of the message polynomial l The minimum number of agreements: 9/9/2020 List Decoding Of RS Codes

The Goal Find a list of polynomials 9 of degree at most , s. t: l These are the polynomials which represent messages that are candidate decodings of. l Let’s see an example 9/9/2020 List Decoding Of RS Codes

Sudan’s Algorithm - Simplified A. Find all functions l l is of small degree is not identically zero C. Factor into irreducible factors Output a list of all small degree factors that have a large agreement with the input. l Let’s go back to our example B. 10 such that: 9/9/2020 List Decoding Of RS Codes

Sudan’s Algorithm: Detailed A. B. Init and , two integers - we will see later what should be their values Find all functions such that: l is of weighted degree at most (Will be explained later) l l C. D. 11 is not identically zero Factor into irreducible factors Output a list of all polynomials such that and 9/9/2020 List Decoding Of RS Codes

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: l l l 12 l Linear system with variables And with Constraints We need more variables than constraints Linear algebra guarantees a solution, and we can efficiently find it. 9/9/2020 List Decoding Of RS Codes

Step 2: Factoring Q l l 13 Well known problem Studied a lot There are efficient deterministic polynomial time algorithms to solve this problem We won’t get into details 9/9/2020 List Decoding Of RS Codes

Claim 2: If is a solution and , then Step 3: Why Every Solution Divides Q? for some low degree polynomial Proof: l Any monomial in l Now let l We set l We have l Hence l But , so 14 is of the form and now 9/9/2020 List Decoding Of RS Codes

Step 3: Cont. 15 l How does implies l We start by showing theorem over l Applying theorem over required result 9/9/2020 ? will give us the List Decoding Of RS Codes

What’s Left? l What are the optimal values for Sudan: l What is the range of values of the agreement parameter that the algorithm will work for? Sudan: Previous algorithms: Consider the ratio l 16 9/9/2020 ? List Decoding Of RS Codes

Summary l l l 17 We presented Sudan’s efficient and wellknown 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 9/9/2020 List Decoding Of RS Codes