# Example: Systematic Encoding (1) Systematic Cyclic Codes. Systematic Encoding. Example: Systematic Encoding (2)

Save this PDF as:

Size: px
Start display at page:

Download "Example: Systematic Encoding (1) Systematic Cyclic Codes. Systematic Encoding. Example: Systematic Encoding (2)"

## Transcription

1 S Cyclic Codes 1 S Cyclic Codes 3 Example: Systematic Encoding (1) Systematic Cyclic Codes Polynomial multiplication encoding for cyclic linear codes is easy. Unfortunately, the codes obtained are in most cases not systematic. Systematic cyclic codes can be obtained through a procedure that is only slightly more complicated than the polynomial multiplication procedure. We consider the (7, 3) binary cyclic code with generator polynomial g(x) = x 4 + x 3 + x discussed in a previous example, and encode 101 = 1 + x 2 = m(x). Step 1. x n k m(x) = x 4 (x 2 + 1) = x 6 + x 4. Step 2. x 6 + x 4 = (x 4 + x 3 + x 2 + 1)(x 2 + x + 1) + (x + 1), so d(x) = x + 1 (Carry out the necessary division in the same way as you learnt in elementary school!). Step 3. c(x) = x 6 + x 4 (x + 1) = 1 + x + x 4 + x 6, and the transmitted codeword is S Cyclic Codes 2 Systematic Encoding Consider an (n, k) cyclic code C with generator polynomial g(x). The k-symbol message block is given by the message polynomial m(x). Step 1. Multiply the message polynomial m(x) by x n k. Step 2. Divide the result of Step 1 by the generator polynomial g(x). Let d(x) be the remainder. Step 3. Set c(x) = x n k m(x) d(x). This encoding works, as (1) c(x) is a multiple of g(x) and therefore a codeword, (2) the first n k coefficients of x n k m(x) are zero, and (3) only the first n k coefficients of d(x) are nonzero (the degree of g(x) is n k). S Cyclic Codes 4 Example: Systematic Encoding (2) The systematic generator matrix is obtained by selecting as rows the codewords associated with the messages 100, 010, and 001. The parity check matrix is obtained using the basic result (presented earlier) that H = [I n k P T ] with G = [P I k ]. In the current example, we get G = , H =

2 S Cyclic Codes 5 S Cyclic Codes 7 Implementations of Cyclic Codes Data rates are very high in many applications only very fast decoders and encoders can be used. Fast circuits are, for example, simple exclusive OR (XOR) gates, switches, and shift registers. For nonbinary encoders and decoders, finite-field adder and multiplier circuits are needed. We now focus on shift-register (SR) encoders and decoders for cyclic codes. Addition and SRs in Extension Fields Elements α, β GF(2 m ) are represented as binary m-tuples (a 0, a 1,...,a m 1 ) and (b 0, b 1,...,b m 1 ), respectively. Then the addition of α and β gives (a 0 + b 0, a 1 + b 1,...,a m 1 + b m 1 ), where + is binary addition. The nonbinary addition circuit is shown in [Wic, Fig. 5-2]. The non-binary shift-register cells are implemented with one flip-flop for each coordinate in the m-tuple; see [Wic, Fig. 5.4]. S Cyclic Codes 6 S Cyclic Codes 8 Operational Elements in Shift Registers Multiplication in Extension Fields The symbology used is depicted in [Wic, Fig. 5-1]. half-adder Adds the input values without carry. In the binary case, XOR. SR cell Flip-flops. In the binary case, one. fixed multiplier Multiplies the input value with a given value. In the binary case, existence or absence of connection. In the nonbinary case, we assume that the field is a binary extension field: GF(p m ) with p = 2. The circuits are substantially more complicated when p 2. As an example, we consider multiplication in GF(2 4 ) of an arbitrary value β = b 0 + b 1 α + b 2 α 2 + b 3 α 3 by a fixed value g = 1 + α, where α is a root of the primitive polynomial x 4 + x + 1. Then β g = (b 0 + b 1 α + b 2 α 2 + b 3 α 3 )(1 + α) = b 0 + (b 0 + b 1 )α + (b 1 + b 2 )α 2 + (b 2 + b 3 )α 3 + b 3 α 4 = b 0 + (b 0 + b 1 )α + (b 1 + b 2 )α 2 + (b 2 + b 3 )α 3 + b 3 (α + 1) = (b 0 + b 3 ) + (b 0 + b 1 + b 3 )α + (b 1 + b 2 )α 2 + (b 2 + b 3 )α 3. The corresponding multiplier circuit is illustrated in [Wic, Fig. 5-3].

3 S Cyclic Codes 9 S Cyclic Codes 11 Error Detection for Systematic Codes Nonsystematic Encoders With message polynomial m(x) = m 0 + m 1 x + + m k 1 x k 1 and generator polynomial g(x), the codeword polynomial is c(x) = m(x)g(x) = m 0 g(x) + m 1 xg(x) + + m k 1 x k 1 g(x). The corresponding SR circuit is shown in [Wic, Fig. 5-5]. The transmitted codeword of a systematic cyclic code has the form c = (c 0, c 1,...,c n ) = ( d 0, d 1,..., d } {{ n k 1, m } 0, m 1,...,m k 1 ). } {{ } remainder block message block Error detection is performed on a received word r as follows. 1. Denote the values in the message and parity positions of the received word r by m and d, respectively. 2. Encode m using an encoder identical to that used by the transmitter, and denote the remainder block obtained in this way by d. 3. Compare d with d. If they are different, then the received word contains errors. S Cyclic Codes 10 Systematic Encoders S Cyclic Codes 12 Syndrome Computation for Systematic Codes Step 1. (Multiply m(x) by x n k.) Easy, shown in [Wic, Fig. 5-8]. Step 2. (Divide the result of Step 1 by g(x), and let d(x) be the remainder.) Polynomial division is carried out through the use of a linear feedback shift register (LFSR) as shown in [Wic, Fig. 5-9], where a(x) is divided by g(x), and q(x) and d(x) are the quotient and remainder, respectively. Step 3. (Set c(x) = x n k m(x) d(x).) Achieved by combining the two SR circuits for the previous steps, as shown in [Wic, Fig. 5-12]. An alternative encoder for cyclic codes, not considered here, is presented in [Wic, Fig. 5-13]. Denote the received word by r with m and d in the message and parity positions, respectively. Let d be a valid parity block of message m (cf. previous slide), and denote this valid word by r. s = rh T = (r r )H T (as r H T = 0) = (d 0 d 0, d 1 d 1,...,d n k 1 d T n k 1, 0, 0,..., 0)H } {{ } d d = d d, since the parity check matrix has the form H = [I n k P T ]. Syndromes for nonsystematic codes can also be computed through the use of shift registers.

4 S Cyclic Codes 13 S Cyclic Codes 15 Error-Correction Approaches Error correction has earlier been discussed for general linear codes. A standard array has q n entries. A syndrome table has q n k entries. We shall see that the number of entries of a syndrome table for cyclic linear codes can be reduced to approximatively q n k /n. With more (algebraic) structure of the codes, even more powerful decoding is possible (to be discussed in forthcoming lectures). Decoding Algorithm for Cyclic Codes 1. Let i := 0. Compute the syndrome s for a received vector r. 2. If s is in the syndrome look-up table, goto Step Let i := i + 1. Enter a 0 into the SR input, computing s i. 4. If s i is not in the syndrome look-up table, goto Step Let e i be the error pattern corresponding to the syndrome s i. Determine e by cyclically shifting e i i times to the left. 6. Let c := r e. Output c. S Cyclic Codes 14 S Cyclic Codes 16 Syndrome Decoding for Cyclic Codes Theorem 5-3. Let s(x) be the syndrome polynomial corresponding to a received polynomial r(x). Let r i (x) be the polynomial obtained by cyclically shifting the coefficients of r(x) i steps to the right. Then the remainder obtained when dividing xs(x) by g(x) is the syndrome s 1 (x) corresponding to r 1 (x). Having computed the syndrome s with an SR division circuit, we get s i (x) after the input of i 0s into the circuit! We then need only store one syndrome s for an error pattern e and all cyclic shifts of e. Example: Error Correction of (7,4) Cyclic Code Consider the (7,4) binary cyclic code generated by g(x) = x 3 + x + 1, with parity check polynomial h(x) = (x 7 + 1)/g(x) = x 4 + x 2 + x + 1, and with parity check matrix H = This is a one-error-correcting Hamming code, so all correctable error patterns are cyclic shifts of An SR error-correction circuit for this code is displayed in [Wic, Fig. 5-14].

5 S Cyclic Codes 17 S Cyclic Codes 19 Some Generator Polynomials Error Detection in Practice The most frequently used error control techniques in the history of computers and communication networks are: one-bit parity check Very simple, but yet important. CRC codes Shortened cyclic codes that have extremely simple and fast encoder and decoder implementations. CRC-4 g 4 (x) = x 4 + x 3 + x 2 + x + 1 CRC-12 g 12 (x) = (x 11 + x 2 + 1)(x + 1) CRC-ANSI g A = (x 15 + x + 1)(x + 1) CRC-CCITT g C = (x 15 + x 14 + x 13 + x 12 + x 4 + x 3 + x 2 + x + 1) (x + 1) Example. The polynomial g 12 (x) divides x but no polynomial x m 1 with smaller degree, so it defines a cyclic code of length 2047 and dimension = So, CRC-12 encodes up to 2035 message bits, generating 12 bits of redundancy. S Cyclic Codes 18 S Cyclic Codes 20 Properties of CRC Codes Cyclic redundancy check (CRC) codes are shortened cyclic codes obtained by deleting the j rightmost coordinates in the codewords. CRC codes are generally not cyclic. CRC codes can have the same SR encoders and decoders as the original cyclic code. CRC codes have error detection and correction capabilities that are at least as good as those of the original cyclic code. CRC codes have good burst-error detection capabilities. Error Detection Performance Analysis The error detection performance of codes depends of the type of errors. In performance analysis, the following three situations are most often considered. 1. Total corruption of words. 2. Burst errors. These are errors that occur over several consecutive transmitted symbols. 3. The binary symmetric channel.

6 S Cyclic Codes 21 S Cyclic Codes 23 Total Corruption of Words When an (n, k) code is used, total corruption leads to a decoder error with probability q k q n = qk n. Note that this probability is solely a function of the number of redundant symbols in the transmitted codewords. The Binary Symmetric Channel An exact determination of the performance of a CRC code over the binary symmetric channel requires knowledge of the weight distribution of the code. Example. With CRC-12, an error is detected with probability in case of total corruption. S Cyclic Codes 22 Burst-Error Detection A burst-error pattern of length b starts and ends with nonzero symbols; the intervening symbols may be take on any value, including zero. Theorems 5-4, 5-5, and 5-6. A q-ary cyclic or shortened cyclic codes with generator polynomial g(x) of degree r can detect all burst error patterns of length r or less; the fraction 1 q 1 r /(q 1) of burst error patterns of length r + 1; and the fraction 1 q r of burst error patterns of length greater than r + 1. Example. With CRC-12, all bursts of length at most 12, 99.95% of bursts of length 13, and % of longer bursts are detected.

### Cyclic Codes Introduction Binary cyclic codes form a subclass of linear block codes. Easier to encode and decode

Cyclic Codes Introduction Binary cyclic codes form a subclass of linear block codes. Easier to encode and decode Definition A n, k linear block code C is called a cyclic code if. The sum of any two codewords

### Introduction to Computer Networks. Codes - Notations. Error Detecting & Correcting Codes. Parity. Two-Dimensional Bit Parity

Introduction to Computer Networks Error Detecting & Correcting Codes Codes - Notations K bits of data encoded into n bits of information. n-k bits of redundancy The information data is of length K The

### Reed-Solomon Codes. by Bernard Sklar

Reed-Solomon Codes by Bernard Sklar Introduction In 1960, Irving Reed and Gus Solomon published a paper in the Journal of the Society for Industrial and Applied Mathematics [1]. This paper described a

### AN INTRODUCTION TO ERROR CORRECTING CODES Part 1

AN INTRODUCTION TO ERROR CORRECTING CODES Part 1 Jack Keil Wolf ECE 154C Spring 2008 Noisy Communications Noise in a communications channel can cause errors in the transmission of binary digits. Transmit:

### Introduction to Algebraic Coding Theory

Introduction to Algebraic Coding Theory Supplementary material for Math 336 Cornell University Sarah A. Spence Contents 1 Introduction 1 2 Basics 2 2.1 Important code parameters..................... 4

### Sheet 7 (Chapter 10)

King Saud University College of Computer and Information Sciences Department of Information Technology CAP240 First semester 1430/1431 Multiple-choice Questions Sheet 7 (Chapter 10) 1. Which error detection

### The finite field with 2 elements The simplest finite field is

The finite field with 2 elements The simplest finite field is GF (2) = F 2 = {0, 1} = Z/2 It has addition and multiplication + and defined to be 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 0 0 = 0 0 1 = 0

### Module 3. Data Link control. Version 2 CSE IIT, Kharagpur

Module 3 Data Link control Lesson 2 Error Detection and Correction Special Instructional Objectives: On completion of this lesson, the student will be able to: Explain the need for error detection and

### An Introduction to Galois Fields and Reed-Solomon Coding

An Introduction to Galois Fields and Reed-Solomon Coding James Westall James Martin School of Computing Clemson University Clemson, SC 29634-1906 October 4, 2010 1 Fields A field is a set of elements on

### Introduction to polynomials

Worksheet 4.5 Polynomials Section 1 Introduction to polynomials A polynomial is an expression of the form p(x) = p 0 + p 1 x + p 2 x 2 + + p n x n, (n N) where p 0, p 1,..., p n are constants and x os

### INTRODUCTION TO CODING THEORY: BASIC CODES AND SHANNON S THEOREM

INTRODUCTION TO CODING THEORY: BASIC CODES AND SHANNON S THEOREM SIDDHARTHA BISWAS Abstract. Coding theory originated in the late 1940 s and took its roots in engineering. However, it has developed and

### Linear Codes. Chapter 3. 3.1 Basics

Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

### Nonlinear Multi-Error Correction Codes for Reliable NAND Flash Memories

1 Nonlinear Multi-Error Correction Codes for Reliable NAND Flash Memories Zhen Wang, Student Member, IEEE, Mark Karpovsky, Fellow, IEEE, and Ajay Joshi, Member, IEEE Abstract Multi-level cell (MLC NAND

### Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2)

Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Kevin Broughan University of Waikato, Hamilton, New Zealand May 13, 2010 Remainder and Factor Theorem 15 Definition of factor If f (x)

### Notes 11: List Decoding Folded Reed-Solomon Codes

Introduction to Coding Theory CMU: Spring 2010 Notes 11: List Decoding Folded Reed-Solomon Codes April 2010 Lecturer: Venkatesan Guruswami Scribe: Venkatesan Guruswami At the end of the previous notes,

### ECEN 5682 Theory and Practice of Error Control Codes

ECEN 5682 Theory and Practice of Error Control Codes Convolutional Codes University of Colorado Spring 2007 Linear (n, k) block codes take k data symbols at a time and encode them into n code symbols.

### Worksheet 4.7. Polynomials. Section 1. Introduction to Polynomials. A polynomial is an expression of the form

Worksheet 4.7 Polynomials Section 1 Introduction to Polynomials A polynomial is an expression of the form p(x) = p 0 + p 1 x + p 2 x 2 + + p n x n (n N) where p 0, p 1,..., p n are constants and x is a

### The Essentials of Computer Organization and Architecture. Linda Null and Julia Lobur Jones and Bartlett Publishers, 2003

The Essentials of Computer Organization and Architecture Linda Null and Julia Lobur Jones and Bartlett Publishers, 2003 Chapter 2 Instructor's Manual Chapter Objectives Chapter 2, Data Representation,

### The Division Algorithm for Polynomials Handout Monday March 5, 2012

The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,

### Applications of Linear Algebra to Coding Theory. Presented by:- Prof.Surjeet kaur Dept of Mathematics SIES College.Sion(W)

Applications of Linear Algebra to Coding Theory Presented by:- Prof.Surjeet kaur Dept of Mathematics SIES College.Sion(W) Outline Introduction AIM Coding Theory Vs Cryptography Coding Theory Binary Symmetric

Data Link Layer Overview Date link layer deals with two basic issues: Part I How data frames can be reliably transmitted, and Part II How a shared communication medium can be accessed In many networks,

### 1 Lecture: Integration of rational functions by decomposition

Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

### minimal polyonomial Example

Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We

### Background. Linear Convolutional Encoders (2) Output as Function of Input (1) Linear Convolutional Encoders (1)

S-723410 Convolutional Codes (1) 1 S-723410 Convolutional Codes (1) 3 Background Convolutional codes do not segment the data stream, but convert the entire data stream into another stream (codeword) Some

### R&D White Paper WHP 031. Reed-Solomon error correction. Research & Development BRITISH BROADCASTING CORPORATION. July 2002. C.K.P.

R&D White Paper WHP 03 July 00 Reed-olomon error correction C.K.P. Clarke Research & Deelopment BRITIH BROADCATING CORPORATION BBC Research & Deelopment White Paper WHP 03 Reed-olomon Error Correction

### Data Link Layer(1) Principal service: Transferring data from the network layer of the source machine to the one of the destination machine

Data Link Layer(1) Principal service: Transferring data from the network layer of the source machine to the one of the destination machine Virtual communication versus actual communication: Specific functions

### Sect 3.2 Synthetic Division

94 Objective 1: Sect 3.2 Synthetic Division Division Algorithm Recall that when dividing two numbers, we can check our answer by the whole number (quotient) times the divisor plus the remainder. This should

### STAR : An Efficient Coding Scheme for Correcting Triple Storage Node Failures

STAR : An Efficient Coding Scheme for Correcting Triple Storage Node Failures Cheng Huang, Member, IEEE, and Lihao Xu, Senior Member, IEEE Abstract Proper data placement schemes based on erasure correcting

### Error detection using CRC

Error detection using CRC The goal of this task is to become familiar with functional principles of CRC (Cyclic Redundancy Check): - math background - error detection features - practical implementation

### Coding and decoding with convolutional codes. The Viterbi Algor

Coding and decoding with convolutional codes. The Viterbi Algorithm. 8 Block codes: main ideas Principles st point of view: infinite length block code nd point of view: convolutions Some examples Repetition

### Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if

1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c

### Situation: Dividing Linear Expressions

Situation: Dividing Linear Expressions Date last revised: June 4, 203 Michael Ferra, Nicolina Scarpelli, Mary Ellen Graves, and Sydney Roberts Prompt: An Algebra II class has been examining the product

### Galois Fields and Hardware Design

Galois Fields and Hardware Design Construction of Galois Fields, Basic Properties, Uniqueness, Containment, Closure, Polynomial Functions over Galois Fields Priyank Kalla Associate Professor Electrical

### CHANNEL. 1 Fast encoding of information. 2 Easy transmission of encoded messages. 3 Fast decoding of received messages.

CHAPTER : Basics of coding theory ABSTRACT Part I Basics of coding theory Coding theory - theory of error correcting codes - is one of the most interesting and applied part of mathematics and informatics.

### Introduction to Binary Linear Block Codes

Introduction to Binary Linear Block Codes Yunghsiang S. Han Graduate Institute of Communication Engineering, National Taipei University Taiwan E-mail: yshan@mail.ntpu.edu.tw Y. S. Han Introduction to Binary

### N O T E S. A Reed Solomon Code Magic Trick. The magic trick T O D D D. M A T E E R

N O T E S A Reed Solomon Code Magic Trick T O D D D. M A T E E R Howard Community College Columbia, Maryland 21044 tmateer@howardcc.edu Richard Ehrenborg [1] has provided a nice magic trick that can be

### 6.02 Fall 2012 Lecture #5

6.2 Fall 22 Lecture #5 Error correction for linear block codes - Syndrome decoding Burst errors and interleaving 6.2 Fall 22 Lecture 5, Slide # Matrix Notation for Linear Block Codes Task: given k-bit

### COEN 6521 VLSI Testing: Built-In Self Test

COEN 6521 VLSI Testing: Built-In Self Test Zeljko Zilic (In Ottawa) zeljko@ece.mcgill.ca Some slides based on the material provided by Bushnell and Agrawal Overview Introduction to BIST Test Pattern Generators

### The processor can usually address a memory space that is much larger than the memory space covered by an individual memory chip.

Memory ddress Decoding The processor can usually address a memory space that is much larger than the memory space covered by an individual memory chip. In order to splice a memory device into the address

### Lecture 8 Binary Numbers & Logic Operations The focus of the last lecture was on the microprocessor

Lecture 8 Binary Numbers & Logic Operations The focus of the last lecture was on the microprocessor During that lecture we learnt about the function of the central component of a computer, the microprocessor

### Digital Fundamentals

Digital Fundamentals with PLD Programming Floyd Chapter 2 29 Pearson Education Decimal Numbers The position of each digit in a weighted number system is assigned a weight based on the base or radix of

### a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

### CMPE 150 Winter 2009

CMPE 150 Winter 2009 Lecture 6 January 22, 2009 P.E. Mantey CMPE 150 -- Introduction to Computer Networks Instructor: Patrick Mantey mantey@soe.ucsc.edu http://www.soe.ucsc.edu/~mantey/ / t / Office: Engr.

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### Dividing Polynomials

4.3 Dividing Polynomials Essential Question Essential Question How can you use the factors of a cubic polynomial to solve a division problem involving the polynomial? Dividing Polynomials Work with a partner.

### Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

### Digital Logic: Boolean Algebra and Gates

Digital Logic: Boolean Algebra and Gates Textbook Chapter 3 CMPE2 Summer 28 Basic Logic Gates CMPE2 Summer 28 Slides by ADB 2 Truth Table The most basic representation of a logic function Lists the output

### Factoring Polynomials

Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent

### Basics of Digital Logic Design

CSE 675.2: Introduction to Computer Architecture Basics of Digital Logic Design Presentation D Study: B., B2, B.3 Slides by Gojko Babi From transistors to chips Chips from the bottom up: Basic building

### Lecture 8: Stream ciphers - LFSR sequences

Lecture 8: Stream ciphers - LFSR sequences Thomas Johansson T. Johansson (Lund University) 1 / 42 Introduction Symmetric encryption algorithms are divided into two main categories, block ciphers and stream

### 1. A Sequential Parity Checker

Chapter 13: Analysis of Clocked Sequential Circuits 1. A Sequential Parity Checker When binary data is transmitted or stored, an extra bit (called a parity bit) is frequently added for purposes of error

### Rational Polynomial Functions

Rational Polynomial Functions Rational Polynomial Functions and Their Domains Today we discuss rational polynomial functions. A function f(x) is a rational polynomial function if it is the quotient of

### Systems of Linear Equations

A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Systems of Linear Equations Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### Coding Theory. Decimal Codes

Coding Theory Massoud Malek Decimal Codes To make error correcting codes easier to use and analyze, it is necessary to impose some algebraic structure on them. It is especially useful to have an alphabet

### Real Roots of Univariate Polynomials with Real Coefficients

Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### Chapter 3: Sample Questions, Problems and Solutions Bölüm 3: Örnek Sorular, Problemler ve Çözümleri

Chapter 3: Sample Questions, Problems and Solutions Bölüm 3: Örnek Sorular, Problemler ve Çözümleri Örnek Sorular (Sample Questions): What is an unacknowledged connectionless service? What is an acknowledged

### 7. Some irreducible polynomials

7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of

### Unit 2: Number Systems, Codes and Logic Functions

Unit 2: Number Systems, Codes and Logic Functions Introduction A digital computer manipulates discrete elements of data and that these elements are represented in the binary forms. Operands used for calculations

### 9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.

9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Introduction to Finite Fields (cont.)

Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number

### 8 Polynomials Worksheet

8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions - Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph

### Finite Fields and Error-Correcting Codes

Lecture Notes in Mathematics Finite Fields and Error-Correcting Codes Karl-Gustav Andersson (Lund University) (version 1.013-16 September 2015) Translated from Swedish by Sigmundur Gudmundsson Contents

### CHAPTER 2 Data Representation in Computer Systems

CHAPTER 2 Data Representation in Computer Systems 2.1 Introduction 47 2.2 Positional Numbering Systems 48 2.3 Converting Between Bases 48 2.3.1 Converting Unsigned Whole Numbers 49 2.3.2 Converting Fractions

### 1.10 (a) Effects of logic gates AND, OR, NOT on binary signals in a processor

Chapter 1.10 Logic Gates 1.10 (a) Effects of logic gates AND, OR, NOT on binary signals in a processor Microprocessors are the central hardware that runs computers. There are several components that make

### MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

### SOLVING POLYNOMIAL EQUATIONS

C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

This page intentionally left blank Coding Theory A First Course Coding theory is concerned with successfully transmitting data through a noisy channel and correcting errors in corrupted messages. It is

### 3.2 The Factor Theorem and The Remainder Theorem

3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial

### Exercises with solutions (1)

Exercises with solutions (). Investigate the relationship between independence and correlation. (a) Two random variables X and Y are said to be correlated if and only if their covariance C XY is not equal

### CHAPTER 3 Boolean Algebra and Digital Logic

CHAPTER 3 Boolean Algebra and Digital Logic 3.1 Introduction 121 3.2 Boolean Algebra 122 3.2.1 Boolean Expressions 123 3.2.2 Boolean Identities 124 3.2.3 Simplification of Boolean Expressions 126 3.2.4

### An Insight into Division Algorithm, Remainder and Factor Theorem

An Insight into Division Algorithm, Remainder and Factor Theorem Division Algorithm Recall division of a positive integer by another positive integer For eample, 78 7, we get and remainder We confine the

### 1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).

.7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational

### 6.4 The Remainder Theorem

6.4. THE REMAINDER THEOREM 6.3.2 Check the two divisions we performed in Problem 6.12 by multiplying the quotient by the divisor, then adding the remainder. 6.3.3 Find the quotient and remainder when x

### Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.

Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:

### 10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES

58 CHAPTER NUMERICAL METHODS. POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 you saw that the eigenvalues of an n n matrix A are obtained by solving its characteristic equation n c nn c nn...

### PROBLEM SET 6: POLYNOMIALS

PROBLEM SET 6: POLYNOMIALS 1. introduction In this problem set we will consider polynomials with coefficients in K, where K is the real numbers R, the complex numbers C, the rational numbers Q or any other

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### (Or Gate) (And Gate) (Not Gate)

6 March 2015 Today we will be abstracting away the physical representation of logical gates so we can more easily represent more complex circuits. Specifically, we use the shapes themselves to represent

### Coding Theory Lecture Notes

Coding Theory Lecture Notes Nathan Kaplan and members of the tutorial September 7, 2011 These are the notes for the 2011 Summer Tutorial on Coding Theory. I have not gone through and given citations or

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### Linear Equations in Linear Algebra

1 Linear Equations in Linear Algebra 1.2 Row Reduction and Echelon Forms ECHELON FORM A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties: 1. All nonzero

### Discrete Mathematics: Homework 7 solution. Due: 2011.6.03

EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### Lecture 6: Finite Fields (PART 3) PART 3: Polynomial Arithmetic. Theoretical Underpinnings of Modern Cryptography

Lecture 6: Finite Fields (PART 3) PART 3: Polynomial Arithmetic Theoretical Underpinnings of Modern Cryptography Lecture Notes on Computer and Network Security by Avi Kak (kak@purdue.edu) January 29, 2015

### = 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

### ALPHABET SOUP ROBERT GREEN

ALPHABET SOUP ROBERT GREEN Abstract. This paper is an expository piece on algebraic methods in the theory of error-correcting codes. We will assume familiarity with Galois theory and linear algebra as

### Solving Systems of Linear Equations Using Matrices

Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Polynomials and Cryptography

.... Polynomials and Cryptography Michele Elia Dipartimento di Elettronica Politecnico di Torino Bunny 1 Trento, 10 marzo 2011 Preamble Polynomials have always occupied a prominent position in mathematics.

### 8-bit RISC Microcontroller. Application Note. AVR236: CRC Check of Program Memory

AVR236: CRC Check of Program Memory Features CRC Generation and Checking of Program Memory Supports all AVR Controllers with LPM Instruction Compact Code Size, 44 Words (CRC Generation and CRC Checking)

### Definition: Group A group is a set G together with a binary operation on G, satisfying the following axioms: a (b c) = (a b) c.

Algebraic Structures Abstract algebra is the study of algebraic structures. Such a structure consists of a set together with one or more binary operations, which are required to satisfy certain axioms.

### Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.

Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method