4 2 binary fields and binary vector spaces

4 -2 binary fields and binary vector spaces Special Thanks to Dr. Samir Al-Ghadhban & EE 430 Students

Binary fields n n To describe codes, we use a mathematical structure called a field. A field is defined to be a set of elements A and two arithmetic operation called addition (+) and multiplication (. ) , such that; 1 - Closure For all a, b ϵ A → a+b ϵ A and a. b ϵ A 2 - Associative properties of addition and multiplication a+b +c = (a+b)+c = a+(b+c) and a. b. c = (a. b). c = a. (b. c) 3 - Identity elements a+0 = 0+a=a a. 1 = 1. a=a ; 0 is the additive identity ; 1 is the multiplicative identity

Cont. 4 - Additive inverse For any a ϵ A, there exist b such that a+b =0 → b is the additive inverse b→ -a 5 - Multiplicative inverse For any element a ϵ A , except 0, there is a multiplicative inverse , such that a. b=1 b → a-1 6 - Addition is commutative a+b = b+a 7 - Distributive property a. (b+c) = a. b + a. c (a+b). c = a. c + b. c

Boolean arithmetic n For binary source, A={0, 1} n The field is called Galois field with two elements, GF(2). Addition and multiplication are defined as: 0+0 = 0 ; 0+1 = 1+0 = 1 ; 1+1 = 0 0. 0 = 1. 0 = 0 ; 1. 1 = 1 n n n → Addition is the XOR "Exclusive or" Multiplication is the AND function

Binary vectors n n n We can construct binary vectors from the binary field GF(2) Define An to be a set with elements a= (a 0, a 1 , …, an-1) with each ai ϵ A= {0, 1}. Define two arithmetic operators 1. Vector addition: if a, b ϵ An then vector addition (+) is defined as a+b ≡ ((a 0+b 0), (a 1+b 1), …, (an-1+bn-1)) 2. Scalar multiplication: If a ϵ An and b ϵ A is a binary scalar , then b. a = a. b = (ba 0, ba 1 , …, ban-1)

Vector space n n n A vector space is a structure made of: - vectors An - scalars A - and two arithmetic operations. The sets and the arithmetic operations satisfy the following constraints: 1. Closure: For every a, b ϵ An , the sum a+b ϵ An 2. Addition is commutative 3. Addition is associative 4. An contains a vector 0 such that a+0 = a 5. Additive inverses: for every a ϵ An , there is some vector b ϵ An such that a+b = 0.

Cont. 6. For every scalar a ϵ A and every vector b ϵ An there is a vector a b ϵ An → closure of scalar multiplication. 7. Scalar multiplication is associative 8. Scalar multiplication is distributive with respect to vector addition 9. Scalar multiplication is distributive with respect to scalar addition 10. If 1 ϵ A is the scalar multiplicative identity, then for every a ϵ An , 1 a = a.

Example: 2 D Code. n n n A 2 - D code is a simple error correcting code with additional error detection capability. The rate in this example is 9/16. Let the 9 bit massage be M=[m 0, m 1, m 2, …. , m 8] , mi ϵ {0, 1} Let m 8 be the first bit transmitted, then the codeword is C = [c 0, c 1, c 2 , m 0, c 4, m 1, c 6, m 2, c 8, m 3, m 5, c 12, m 6, m 7, m 8] Where

Cont. n n n Where c 12 = m 8 + m 7 + m 6 similarly c 1, c 2, c 4, c 6 and c 8 c 0 = m 8 + m 7 + …+ m 1 + m 0 During transmission , bit error may occur, the received word is v= c + e where e = [ e 0, e 1, …. , e 15]; if ei = 1 → error ei = 0 → no error

Cont. n At the receiver, we calculate seven error check bits, which are called syndrome bits. this code can correct single bit errors , and can detect two bit errors.

Cont. n 1. 2. 3. 4. Notice that, If no errors occur, all si are zero. If an odd number of errors occur , s 0 = 1 If a single error occurs in one of the mi message bits, one of the row and one of the column syndrome bits will equal to 1. In this case, we can correct the error bit by adding 1 to the vi bit corresponding to the intersection of the row and column. If a single error occurs in one of the parity bits (ci), then either one row or one column syndrome bit (but not both) will be equal to one and the s 0 bit will be 1. in this case, no correction is necessary because none of the message bits are in error.

Cont. 5. If two errors occur, s 0=0, and multiple syndrome bits in a row or column will be 1 → we can detect two errors. but we can't guarantee to correct these errors For example, assume the errors are in v 14 and v 13, then s 2 and s 1 = 1 but s 6 =0 So, we can't tell the error location. 6. If the number of errors are greater than two: Odd number → we can't guarantee that we will not miscorrect the error. Even number → we can't guarantee to detect the error. See example 4. 2. 3 (Retransmission Probability)