coding3 - 1 Coding for Communications Lecture 3...

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Unformatted text preview: 1 Coding for Communications Lecture 3 Introduction to linear block codes Linear block codes 2 Channel coding Theorem (r ,,r n-1 ) (u ,,u k-1 ) (c ,,c n-1 ) (m ,,m k-1 ) What are linear block codes? Theorem 3 Linear block codes Let us review some basic definitions first that are useful in understanding Linear block codes. The binary field GF(2) Binary field : The set {0,1}, under modulo 2 binary addition and multiplication forms a field. Binary field is also called Galois field, GF(2), or Z 2 . 1 1 1 1 1 1 = = = = 1 1 1 1 1 = = = = Addition Multiplication 4 Fields Let F be a set of objects on which two operations + and . are defined. F is said to be a field if and only if 1. F forms a commutative group under + operation. The additive identity element is labeled 0. 1. F-{0} forms a commutative group under . Operation. The multiplicative identity element is labeled 1. 1. The operations + and . are distributive: F a b b a F b a + = + , F a b b a F b a = , ) ( ) ( ) ( c a b a c b a + = + Vector space Let V be a set of vectors and F a fields of elements called scalars . V forms a vector space over F if: 1. Commutative: 2. 3. Distributive: 4. Associative: 5. V u v V v = a F a , v u v u v v v + = + + = + a a a b a b a ) ( and ) ( F V + = + u v v u v u, ) ( ) ( , , v v v = b a b a V F b a v v V v = 1 , 5 Vector spaces and subspaces Examples of vector spaces over GF(2) The set of binary n-tuples, denoted by Vector subspace: A subset S of the vector space is called a subspace if: The all-zero vector is in S . The sum of any two vectors in S is also in S . Example: . of subspace a is )} 1111 ( ), 1010 ( ), 0101 ( ), 0000 {( 4 H n H n H )} 1111 ( ), 1101 ( ), 1100 ( ), 1011 ( ), 1010 ( ), 1001 ( ), 1000 ( ), 0111 ( ), 0101 ( ), 0100 ( ), 0011 ( ), 0010 ( ), 0001 ( ), 0000 {( 4 = H Bases Spanning set: A collection of vectors , is said to be a spanning set for H or to span H if linear combinations of the vectors in V include all vectors in the vector space H, Example: Bases: The spanning set of V that has minimal cardinality is called the basis for H....
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This note was uploaded on 04/24/2011 for the course EE 4421 taught by Professor Mondin during the Spring '11 term at California State University Los Angeles .

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coding3 - 1 Coding for Communications Lecture 3...

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