ECE 635 Error Detecting and Correcting Systems Design and Hardware Implementation
By
Nagi M. El Naga
Department of Electrical and Computer Engineering California State University
September, 2009
C Nagi El Naga
1
Course Objectives
- To get you familiar wit
TCOM 370
NOTES 99-9
CYCLIC CODES, AND THE CRC (CYCLIC REDUNDANCY CHECK) CODE
1. CYCLIC CODES Cyclic codes are a special type of linear block code that are popular because they are very effective for error detection and correction and their coders and deco
An overview on error correcting codes
Linear codes
Cyclic codes
Encoding and decoding with cyclic codes
An introduction to cyclic codes
Emanuele Betti1 , Emmanuela Orsini2
1 bettie@posso.dm.unipi.it Department of Mathematics, University of Florence, Italy
EE 229B Solutions for Homework 2
ERROR CONTROL CODING
Spring 2005
1. (Weights of codewords in a cyclic code) Let g (X ) be the generator polynomial of a binary cyclic code of length n. (a) Show that if g (X ) has X + 1 as a factor then the code contains n
1.Introduction:
Cyclic codes form an important subclass of linear codes. These codes are attractive for two
reasons: first, encoding and decoding can be implemented easily by employing shift-registers
with feedback connections (as will be seen later); and
YASAMAN DIANATPEY
ECE 635
Dr. N. El Naga
Homework #1
1.
Solve the following simultaneous equations of X,Y,Z, and W with modulo-2 arithmetic:
1
2
3
4
X + Y + W = 1,
X + Z + W = 0,
X + Y + Z + W = 1,
Y + Z + W = 0.
Solution no1:
4 -2 = X+Y=0
1= 0+W=1 W=1
3=
By
Nagi M. El Naga
Department of Electrical and Computer Engineering
California State University
September,2003
Nagi El Naga
ECE 635
Error Detecting and Correcting Systems Design
and Hardware Implementation
Part Two
2
ECE 635
Error Detecting and Correcti
By
Nagi M. El Naga
Department of Electrical and Computer Engineering
California State University
Nagi El Naga
ECE 635
Error Detecting and Correcting Systems Design
and Hardware Implementation
Course Objectives
To get you familiar with various error detec
ECE 635
Dr. N. El Naga
Homework # 2
2.1 Let be a primitive element in GF(24). Use the table given in class to solve the following
simultaneous equations for X, Y, and Z:
X + 5Y + Z = 7,
X + Y + 7Z = 9,
2X + Y+ 6Z =
4
Using the primitive polynomial H(x)=
Assignment on Cyclic Codes
EE512: Error Control Coding Questions marked (Q) or (F) are questions from previous quizzes or nal exams, respectively.
1. What is the ideal describing the cyclic code cfw_0000, 0101, 1010, 1111? 2. Describe the smallest cyclic
EE 387, John Gill, Stanford University
Notes #5, October 31, Handout #21
Cyclic codes: overview
A linear block code is called cyclic if every cyclic shift of a codeword is a codeword. Cyclic codes have many advantages. Elegant algebraic descriptions: c(x)
IV054 CHAPTER 3: Cyclic and convolution codes
Cyclic codes are of interest and importance because
They posses rich algebraic structure that can be utilized in a variety
of ways. They have extremely concise specifications.
They can be efficiently impleme
ECE 635 Homework #1 1.
Dr. N. El Naga
Solve the following simultaneous equations of X,Y,Z, and W with modulo-2 arithmetic: X+ Y + X +Z+ X+ Y+ Z + Y+ Z + W= W= W= W= 1, 0, 1, 0.
2. 3. 4. (a) (b) (c) (d)
5 3 Show that X + X + 1 is irreducible over GF(2). Fi
ECE 635 Homework # 2
2.1
Dr. N. El Naga
Let be a primitive element in GF(24). Use the table given in class to solve the following simultaneous equations for X, Y, and Z: X X 2X + 5Y + Y +Y +Z + 7Z + 6Z = 7, = 9, =
2.2
Construct the vector space V5 of all
ECE 635 Homework # 3
3.1 Consider a (7,4) code whose generator matrix is: 1 0 G= 0 0 (a) (b) 3.2 0 1 0 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 0
Dr. N. El Naga
Find all the code vectors of this code. Find the parity check matrix H of this code.
Consider a
ECE 635 Homework # 4
4.1
Dr. N. El Naga
Let g(X) = 1 + X + X2 + X4 + X5 + X8 + X10 be the generator polynomial of a (15,5) cyclic code. (a) (b) Find the parity polynomial h(X) of this code. Find the code polynomial for the message m(X) = 1 + X + X4 (in sy
ECE 635 Homework # 5
5.1 5.2
Dr. N. El Naga
List the code words of the (7,3) code with g(X) = X4 + X3 + X2 +1. Find dmin. X15 + 1 = (X + 1)(X4 + X + 1)(X4 + X3 +1)(X4 +X3 + X2 + X + 1).(x2 + X +1) Determine g(X) for a (15,9) code. Determine g(X) for a (15
ECE 635 Homework # 6
6.1 For the code generated by:
Dr. N. El Naga
g(X) = X4 + X + 1, encode m(X) = X5 + 1. Add error pattern e(X) = X3 + X2 +X + 1. Calculate syndrome. After doing problem 1, show that if the degree of e(X) is less than the degree of g(X)
Chapter 4 Linear Cyclic Codes
4.1 Definition of Cyclic Code
An (n, k) linear code C is called a cyclic code if any cyclic shift of a codeword is another codeword. That is, if c = ( c 0 , c 1 ,L , c n- 1 ) C
( 1) then c = ( c n- 1 , c0 , c 1 ,L , c n- 2 )
Chapter 8
Cyclic Codes
Among the rst codes used practically were the cyclic codes which were generated using shift registers. It was quickly noticed by Prange that the class of cyclic codes has a rich algebraic structure, the rst indication that algebra w
ECE 635
Dr. N. El Naga
Homework # 3
3.1
Consider a (7,4) code whose generator matrix is:
1
0
G =
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
1
1
0
1
0
1
1
1
1
0
1
(a)
(b)
3.2
Find all the code vectors of this code.
Find the parity check matrix H of this code.
Consider