LINEAR ALGEBRA APPLICATION TO CODING THEORY

Introduction Transmitted messages, like data from a satellite, are always subject to noise. Therefore, to be able to encode a message in such a way that after noise scrambles it, it can be decoded to its original form. This is done sometimes by repeating the message two or three times, something very common in human speech.

A code that detects errors in a scrambled message is called error detecting. It can correct the error it is called error correcting.

Hamming code In the 1950’s, R.H. Hamming introduced an interesting single error-correcting code that became known as the Hamming code. Before we can examine the details of that technique, we need some background from linear algebra.

Vector spaces over Z2 Closure of addition: if x, y are in F, then x+y is in F. Closure of multiplication: if x, y are in F, then xy is in F. Associative Law of Addition: if x, y, z are in F, then (x+y)+z=x+(y+z) Associative Law of Multiplication: if x, y, z are in F, then (xy)z=x(yz) Distributive Law: if x, y, z are in F, then x(y+z)=xy+yz Existence of 0: an element of F satisfying x+0=x for all x in F

Existence of 1: an element of F satisfying x.1=x for all x in F Existence of additive inverses (negatives):If x is in F, there exists y in F such that x+y=0 Existence of multiplicative inverses (reciprocals), except for: If x is in F is not the zero element, then there exists an element y in F such that xy=1. Commutative Law of Addition: If x, y are in F, then x+y=y+x Commutative Law of Multiplication: If x, y are in F, then xy=yx.

Examples of fields are Q (rational numbers), R (real numbers), C (complex numbers), and Z/pZ if p is a prime number (integers modulo a prime p): Z/pZ = {0,1,………..,(p-1)} when p=2 the field Z/2Z is denoted by Z2. It consists of two numbers only: 0 and 1: In Z2, addition and multiplication rules are defined as follows: Z2 = Z/2Z = {0,1}

In Z2, addition and multiplication rules are defined as follows: 0+0=0, 1+0=0, 0+1=1, 1+1=0 0*0=0, 1*0=0, 0*1=0, 1*1=1 Thus, addition and subtraction are the same in Z2. Recall the vector space structure of Rn (R is the set of all real numbers) over R: 1. (x1,…, xn)+ (y1,…, yn)= (x1+ y1,…, xn+yn) 2. a(x1,…, xn)= (ax1,…,a xn) if a is a real number.