Error control coding EEE 470 (3+0) credits

Error control coding EEE 470 (3+0) creditsSpring 2015Instructor: Dr.Hafiz M.Asif

Introduction to algebra:Groups:G is closed under additionOrder of the group: number of elementsFinite groupExample 2.2

Subgroup:Definition: A subgroup of a group G is a subset H of G that is itself a group under the operation of G:H is closed under the operation of G.Example:Example:G1 = Z16 = {0, 1, . . . , 15} with mod 16 addition.G1 has one subgroup with 8 elements,the set of even integers {0, 2, . . . , 14}.

Coset:A subgroup H can be thought of as a smaller dimensional subspace of G.H can be translated by adding a fixed element g (in G) to every element of HThese translates are called cosets.

Coset:Every element in G appears in one and only one coset of HAll the distinct cosets of H are disjointThe union of all the distinct cosets of H forms the group GFields:

Example 2.5:

Binary field:GF(2) is a binary field under modulo-2 +/.1+1=2=0, 1=-1

Polynomials:GF(2m) is an extension field of GF(2) (analogous to real vs. complex numbers)Consists of 2m elements Formed using polynomialsPolynomial over GF(2) means polynomial with coefficients from GF(2)Irreducible polynomial:Primitive polynomial:Primitive polynomial:Construction of GF(2m):Example 2.7:

Basic properties of GF(2m):Conjugate of the root:

Article 2.6:Vector space:

Vector space:Elements of V vectors (V is a vector space over field F,e.g., GF(2))Elements of F scalarsExample 2.12Subspace S:S a subset of vector space V (also a vector space over F)Example 2.13

Subspace and linear combination:

Linearly independent vectors:

Subspace formation:

Matrix representation:

Practice problems:Home Work Suggestions:

2.1-3,2.10,2.13,2.14,2.17,2.19,2.25,2.26