2.Encodings

7/27/2019 2.Encodings

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2

Overview

Review of some basic math

Error correcting codes

Low degree polynomials

H.W



3

Review - FieldsDef (field): A set F with two binary operations +

(addition) and · (multiplication) is called a field if

6 a,bF, a·bF

7 a,b,cF, (a·b)·c=a·(b·c) 8 a,bF, a·b=b·a

9 1F, aF, a·1=a

10 a0F, a-1F, a·a-1=1

1 a,bF, a+bF

2 a,

b,cF, (a+b)+c=a+(b+c) 3 a,bF, a+b=b+a

4 0F, aF, a+0=a

5 aF, -aF, a+(-a)=0

11 a,b,cF, a·(b+c)=a·b+a·c

+,·,0, 1,

-a and a-1

are only notations!



4

Finite FieldsDef (finite field): A finite set F with two binary

operations + (addition) and · (multiplication) iscalled a finite field if it is a field.

Example: Zp denotes {0,1,...,p-1}. We define + and · asthe addition and multiplication modulo p respectively.

One can prove that (Zp,+,·) is a field iff p is prime.Throughout the presentations we’ll usually referto Zp when we’ll mention finite fields.



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 f be a function f : D R. Then f can bedescribed as a string in R|D|, spelling f’s value

on each point of D.



6

Strings & Functions - ExampleFor example, let f be a function f : Z5 Z5, and let = Z5.

14

43

4211

00

)( x f

"01141"

2)( x x f





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 ofthe 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 thefraction of entries on which x and y differ.

Note that :mmR+ is indeed a distance

function , because it satisfies:

(1) x,ym (x,y)0 and (x,y)=0 iff x=y

(2) x,ym (x,y)=(y,x)

(3) x,y,zm (x,z)(x,y)+(y,z)



9

Example – a simple error

correcting code

011

:3

n

Consider the following code: for every n , letE(,k)=^k (the same word repeated k times, hencem=kn).

E(,4)

110110110110)4,( E



10

Example – a simple error

correcting codeBecause every two words n were different on atleast one coordinate to begin with, the distance of thecode (1-alpha) is:

nkn

nk k y E k x E

y x n

1)1(11)),(),,((

:





12

Reed-Solomon codes

We shall now use polynomials over finite fields to build abetter generic code (larger distance between words)

1

0

)(r

j

j

j xa x P

Note: A polynomialwhose degree-bound isr is of degree at most

r-1 !

Def (univariate polynomial): a polynomial in x over a fieldF is a function P:FF, which can be written as

for some series of coefficients a0,...,ar-1F.

The natural number r is called the degree-bound of thepolynomial.







15

Reed-Solomon codes

Def (the Reed-Solomon code):

Set F to be the finite field Zp for some prime p, andassume for simplicity that = F and m = p.

Givenn

, let E(

) be the string of the function f : F F that satisfies:f is the unique polynomial of degree-bound n suchthat f(i) = i for all 0 i n-1.



16

Reed-Solomon codes

E() can be interpolated from any n points.

Hence, for any

, E(

) and E() may agree on atmost n – 1 points.

Therefore, E is an (n – 1) / m – code, that is a code

with distance of:

m

n 11



17

Reed-Solomon codes

p = m = 5, n = 2

= 1, 2

= 3, 1f(x) = x + 1 f(x) = 3x + 3

E() = 1, 2, 3, 4, 0 E() = 3, 1, 4, 2, 0



18

Strings & Functions (2)

We can describe any string as a function f:Hd

H (H is a finite field, d is a positive

integer). Given a n we’ll achieve that by choosing

H=Zq, where q is the smallest prime greaterthan ||, and d=logqn.





20

Error correcting Codes

Home Assignment We’ve seen that Reed-Solomon codes using

polynomials with degree-bound r have distanceof:

Next

What is the distance of error correcting codes thatuse multivariate polynomials (over a finite field F,with degree-bound h in each variable and dimensiond)?

m

n 11



21

Low Degree Extension (LDE)Def: (low degree extension): Let : Hd H

be a string (where H is some finite field).

Given a finite field F, which is a superset ofH, we define a low degree extension of toF as a polynomial LDE : Fd F whichsatisfies:

LDE agrees with on Hd (extension). The degree-bound of LDE is |H| in each

variable (low degree).



22

Low Degree Extension (LDE)

Goal: To be able to find the value of anLDE in any point (set of points) of Fd.

LDEx

LDE(x)







25

Consistent Readers

In the upcoming lectures we’ll see how tobuild readers which:

access only a small number of thevariables each time.

detect inconsistency with high

probability.

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