Download presentation

Presentation is loading. Please wait.

Published byCali Binkley Modified over 2 years ago

1
1

2
2 Overview Review of some basic math Review of some basic math Error correcting codes Error correcting codes Low degree polynomials Low degree polynomials H.W

3
3 Review - Fields Def (field): A set F with two binary operations + (addition) and · (multiplication) is called a field if 6 a,b F, a·b F 7 a,b,c F, (a·b)·c=a·(b·c) 8 a,b F, a·b=b·a 9 1 F, a F, a·1=a 10 a 0 F, a -1 F, a·a -1 =1 1 a,b F, a+b F 2 a,b,c F, (a+b)+c=a+(b+c) 3 a,b F, a+b=b+a 4 0 F, a F, a+0=a 5 a F, -a F, a+(-a)=0 11 a,b,c F, a·(b+c)=a·b+a·c +,·,0, 1, -a and a -1 are only notations!

4
4 Finite Fields Def (finite field): A finite set F with two binary operations + (addition) and · (multiplication) is called a finite field if it is a field. Example: Z p denotes {0,1,...,p-1}. We define + and · as the addition and multiplication modulo p respectively. One can prove that (Z p,+,·) is a field iff p is prime. Throughout the presentations we’ll usually refer to Z p when we’ll mention finite fields.

5
5 Strings & Functions (1) Let = 0 2... n-1, where i . We can describe the string as a function : {0…n-1} , such that i (i) = i. Let = 0 2... n-1, where i . We can describe the string as a function : {0…n-1} , such that i (i) = i. Let f be a function f : D R. Then f can be described as a string in R |D|, spelling f’s value on each point of D. Let f be a function f : D R. Then f can be described as a string in R |D|, spelling f’s value on each point of D.

6
6 Strings & Functions - Example For example, let f be a function f : Z 5 Z 5, and let = Z 5.

7
7 1001110 Introduction to Error Correcting Codes Motivation: communication line original message 1001110 received message 1101110 1 “noise” We’d like to still be able to reconstruct the original message

8
8 Error Correcting Codes Def (encoding): An encoding E is a function E : n m, where m >> n. Def (code word): A code word w is a member of the image of the encoding E : n m. Def ( -code): An encoding E is an -code if n (E( ),E( )) 1 - , where (x,y) (the Hamming distance), denotes the fraction of entries on which x and y differ. Note that : m m R + is indeed a distance function, because it satisfies: (1) x,y m (x,y) 0 and (x,y)=0 iff x=y (2) x,y m (x,y)= (y,x) (3) x,y,z m (x,z) (x,y)+ (y,z)

9
9 Example – a simple error correcting code Consider the following code: for every n, let E( ,k)= ^k (the same word repeated k times, hence m=kn). ,4) E( ,4)

10
10 Example – a simple error correcting code Because every two words n were different on at least one coordinate to begin with, the distance of the code (1-alpha) is:

11
11 Example – a simple error correcting code E 1- =1/n D R

12
12 Reed-Solomon codes We shall now use polynomials over finite fields to build a better generic code (larger distance between words) Note: A polynomial whose degree-bound is r is of degree at most r-1 ! Def (univariate polynomial): a polynomial in x over a field F is a function P:F F, which can be written as for some series of coefficients a 0,...,a r-1 F. The natural number r is called the degree-bound of the polynomial.

13
13 Reed-Solomon codes Thm : Given x 0,y 0,...,x r-1,y r-1 F there is a single univariate polynomial P and degree-bound r, which satisfies 0 k r-1 P(x k )=y k Existence: We shall build such a polynomial using Lagrange’s formula: Proof : Uniqueness: If there are two such polynomials: p1 & p2, then p1-p2 is a polynomial with degree-bound r, which has r roots. This contradicts the fundamental theorem of Algebra!

14
14 Reed-Solomon codes Let’s check the value of this polynomial in x = x t for some 0 t r-1: Since the degree-bound of this polynomial is r, we in fact proved the correctness of the formula a-b denotes a+(-b) a/b denoted a(b -1 ) 0 ytyt

15
15 Reed-Solomon codes Def (the Reed-Solomon code): Set F to be the finite field Z p for some prime p, and assume for simplicity that = F and m = p. Given n, let E( ) be the string of the function f : F F that satisfies: f is the unique polynomial of degree-bound n such that f (i) = i for all 0 i n-1.

16
16 Reed-Solomon codes E( ) can be interpolated from any n points. E( ) can be interpolated from any n points. Hence, for any , E( ) and E( ) may agree on at most n – 1 points. Hence, for any , E( ) and E( ) may agree on at most n – 1 points. Therefore, E is an (n – 1) / m – code, that is a code with distance of: Therefore, E is an (n – 1) / m – code, that is a code with distance of:

17
17 Reed-Solomon codes p = m = 5, n = 2 = 1, 2 = 3, 1 f (x) = x + 1 f (x) = 3x + 3 E( ) = 1, 2, 3, 4, 0 E( ) = 3, 1, 4, 2, 0

18
18 Strings & Functions (2) We can describe any string as a function f:H d H (H is a finite field, d is a positive integer). We can describe any string as a function f:H d H (H is a finite field, d is a positive integer). Given a n we’ll achieve that by choosing H=Z q, where q is the smallest prime greater than | |, and d= log q n . Given a n we’ll achieve that by choosing H=Z q, where q is the smallest prime greater than | |, and d= log q n .

19
19 Reed-Muller Codes Def (multivariate polynomial): Let F be a field and let d be some positive integer number. A function p:F d F is a multivariate polynomial if it can be written as for some series of coefficients in the field. h is the degree-bound on each one of the variables. The total-degree of the polynomial is max{ i 0 +…+i d-1 : a i 0 … i d-1 0 }.

20
20 Error correcting Codes Home Assignment We’ve seen that Reed-Solomon codes using polynomials with degree-bound r have distance of: We’ve seen that Reed-Solomon codes using polynomials with degree-bound r have distance of: Next What is the distance of error correcting codes that use multivariate polynomials (over a finite field F, with degree-bound h in each variable and dimension d)? What is the distance of error correcting codes that use multivariate polynomials (over a finite field F, with degree-bound h in each variable and dimension d)?

21
21 Low Degree Extension (LDE) Def: (low degree extension): Let : H d H be a string (where H is some finite field). Given a finite field F, which is a superset of H, we define a low degree extension of to F as a polynomial LDE : F d F which satisfies: LDE agrees with on H d (extension). The degree-bound of LDE is |H| in each variable (low degree).

22
22 Low Degree Extension (LDE) Goal: To be able to find the value of an LDE in any point (set of points) of F d. LDE x LDE(x)

23
23 Low Degree Extension (LDE) x LDE(x) Straightforward approach: Represent the LDE by its coefficients. Alas, this will require access to |H| d variables, log|F| bits each, each time! the coefficients of the dimension- d, degree-bound- |H| LDE

24
24 Low Degree Extension (LDE) x LDE(x) the value of the LDE in every point in F d Second approach: Represent the LDE by its values in the points of F d. Now we only need access to one variable (log|F| bits) each time. But now we encounter a new problem: we cannot be sure the values we are given are consistent, i.e. correspond to a single dimension-d, degree- bound-|H| polynomial.

25
25 Consistent Readers In the upcoming lectures we’ll see how to build readers which: access only a small number of the variables each time. access only a small number of the variables each time. detect inconsistency with high probability. detect inconsistency with high probability.

Similar presentations

OK

Complexity 19-1 Complexity Andrei Bulatov More Probabilistic Algorithms.

Complexity 19-1 Complexity Andrei Bulatov More Probabilistic Algorithms.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on polynomials and coordinate geometry practice Ppt on sound navigation and ranging system of equations Ppt on statistics in maths what is the factor Free download ppt on modern periodic table Ppt on causes of 1857 revolt in india Ppt on cross site scripting virus Ppt on job rotation spreadsheet Ppt on natural and artificial satellites list Ppt on describing words for class 1 Ppt on trial and error examples