ECEN 5682 Theory and Practice of Error Control Codes Short - - PowerPoint PPT Presentation

Introduction

ECEN 5682 Theory and Practice of Error Control Codes

Short Introduction Peter Mathys

University of Colorado

Spring 2007

Peter Mathys ECEN 5682 Theory and Practice of Error Control Codes

Introduction Symbols, Algebraic Operations

Definition: A code C over some alphabet A is a set of codewords C = {c0, c1, . . . , cM−1} , where M may be finite or infinite, and each codeword is a vector consisting of one or more symbols from the alphabet A. Codewords may be of finite, semi-infinite, or infinite length. Some codes require that all codewords have the same length, others allow for a variable codeword length. Example: Let A = {0, 1}. Then C = {000, 001, 010, 011, 100, 101, 110, 111} is a binary code with M = 8 and fixed codewordlength 3.

Peter Mathys ECEN 5682 Theory and Practice of Error Control Codes

Introduction Symbols, Algebraic Operations

Example: Let A = {a, b}. Then C = {a, baa, bab, bb} is a binary code with M = 4 and variable codewordlength. Example: Let A = {0, 1, 2, 3, 4}. Then C = { 0000, 1430, 2310, 3240, 4120, 0143, 1023, 2403, 3333, 4213, 0231, 1111, 2041, 3421, 4301, 0324, 1204, 2134, 3014, 4444, 0412, 1342, 2222, 3102, 4032 } is a 5-ary code with M = 25 and fixed codewordlength 4.

Peter Mathys ECEN 5682 Theory and Practice of Error Control Codes

Introduction Symbols, Algebraic Operations

Definition: Two codes which are the same except for a permutation of codeword components are called equivalent. Example: The two binary codes C1 = { 00000, 01011, 10101, 11110 } C2 = { 00000, 11100, 00111, 11011 } are equivalent.

Peter Mathys ECEN 5682 Theory and Practice of Error Control Codes

Introduction Symbols, Algebraic Operations

If appropriate algebraic operations are defined, then

◮ Codewords can be scaled. ◮ Codewords can be added and subtracted. ◮ Linear combinations of codewords can be taken.

The mathematical definitions and tools for this come from the theory of groups, rings, fields, and vector spaces.

Peter Mathys ECEN 5682 Theory and Practice of Error Control Codes

Introduction Symbols, Algebraic Operations

Example: Addition and multiplication modulo 2 + 1 1 1 1 × 1 1 1 Example: Addition and multiplication modulo 5 + 1 2 3 4 1 2 3 4 1 1 2 3 4 2 2 3 4 1 3 3 4 1 2 4 4 1 2 3 × 1 2 3 4 1 1 2 3 4 2 2 4 1 3 3 3 1 4 2 4 4 3 2 1

Peter Mathys ECEN 5682 Theory and Practice of Error Control Codes

Introduction Symbols, Algebraic Operations

Example: Addition and multiplication modulo 6 + 1 2 3 4 5 1 2 3 4 5 1 1 2 3 4 5 2 2 3 4 5 1 3 3 4 5 1 2 4 4 5 1 2 3 5 5 1 2 3 4 × 1 2 3 4 5 1 1 2 3 4 5 2 2 4 2 4 3 3 3 3 4 4 2 4 2 5 5 4 3 2 1

Peter Mathys ECEN 5682 Theory and Practice of Error Control Codes