
/ :
1. A group can be constructed by using the rotations and reections of a pentagon into itself.
(a) How many elements are in this group?
(b) Is it an abelian group?
(c) Construct the group?
(d) Find a subgroup with ve elements and a subgroup
Introduction to Algebraic Coding Theory
Supplementary material for Math 336 Cornell University
Sarah A. Spence
Contents
1 Introduction
1
2 Basics
2.1 Important code parameters . .
2.2 Correcting and detecting errors
2.3 Spherepacking bound . . . . .
2.4
Introduction to Algebraic Coding Theory
With Gap
Fall 2006
Sarah Spence Adams
January 11, 2008
The rst versions of this book were written in Fall 2001 and June 2002 at Cornell University, respectively supported
by an NSF VIGRE Grant and a Department of M
D. Keffer, ChE 505 ,University of Tennessee, August, 1999
Advanced Analytical Techniques for the Solution of
Single and MultiDimensional Integral Equations
David Keffer
Department of Chemical Engineering
University of Tennessee, Knoxville
August 1999
Ta
Kevin Buckley  2010
ECE8771, Summer 2010, Info. Theory & Coding .  Lect.
1
compressed
information bits
information
xj
ai
information
source
source
encoder
channel
decoder
^
xj
Ck
received
signal
information
output
^
ai
estimated
compressed
information b
Chapter 3
Linear Codes
In order to dene codes that we can encode and decode eciently, we add more
structure to the codespace. We shall be mainly interested in linear codes. A
linear code of length n over the eld F is a subspace of F n . Thus the words of
Math 350
Spring, 2009
HOMEWORK #5
Do 50 points of the following problems (due 2/25/09).
20 pts.
1 List the (Slepian) array for the binary code with the generator matrix
listed below, and decode 101011, 111111, and 000100 using your scheme.
Also calculate
Introduction to Coding Theory  Solutions to Exercise 2
November 19, 2009
1. (a) Let C1 and C2 be two linear codes over Fq . Show that C = cfw_(c1 c2 )  c1 C1 , c2 C2
(where  stands for concatenations) is a linear code with d = mincfw_d1 , d2 .
(b) Le
A RankMetric Approach to Error Control in
Random Network Coding
Danilo Silva, Student Member, IEEE, Frank R. Kschischang, Fellow, IEEE,
arXiv:0711.0708v2 [cs.IT] 10 Jun 2008
and Ralf K tter, Senior Member, IEEE
o
Abstract
The problem of error control in
MATH32031: Coding Theory
Sheet 2: Linear codes
2.1. Show that A2 (3, 2) = 4 by writing down a suitable code with four
elements and then showing that it is impossible to nd one with more
elements. [The last part might be slightly easier if you use Lemma
6.
4
Kevin Buckley  2010
ECE8771
Information Theory & Coding for
Digital Communications
Villanova University
ECE Department
Prof. Kevin M. Buckley
Lecture Set 1
Review of Digital Communications,
Introduction to Information Theory
compressed
information bit
198
Kevin Buckley  2010
ECE8771
Information Theory & Coding for
Digital Communications
Villanova University
ECE Department
Prof. Kevin M. Buckley
Lecture Set 4
Turbo & LDPC Codes, SpaceTime
Coding, TCM
xj ; j=1,2, . , N
xj ; j=1,2, . , N
Convolutional
E
169
Kevin Buckley  2010
ECE8771
Information Theory & Coding for
Digital Communications
Villanova University
ECE Department
Prof. Kevin M. Buckley
Lecture Set 3
Convolutional Codes
x
c
(a)
(0,0)
000
010
101 111
000
010
101 111
110
100
011 001
(0,1)
.
.
10
Information Theory and Coding Course
Problem Set #1
1. A source emits one of four symbols s0, s1, s2, and s3 with probabilities 1/3, 1/6, 1/4, and
1/4, respectively. The successive symbols emitted by the source are statistically
independent. Calculate the
109
Kevin Buckley  2010
ECE8771
Information Theory & Coding for
Digital Communications
Villanova University
ECE Department
Prof. Kevin M. Buckley
Lecture Set 2
Block Codes
m
m
m
GF(2 ) multiplier
GF(2 ) adder
GF(2 ) element
register
g
R
1
g0
.
g1
g2T1
2
Chapter 2
2.3 Since m is not a prime, it can be factored as the product of two integers a and b, m=ab with 1 < a, b < m. It is clear that both a and b are in the set cfw_1, 2, , m 1. It follows from the denition of modulom multiplication that a b = 0.
Si
Information Theory and Coding Course
Problem Set #1
1. A source emits one of four symbols s0, s1, s2, and s3 with probabilities 1/3, 1/6, 1/4, and
1/4, respectively. The successive symbols emitted by the source are statistically
independent. Calculate the
LDPC Codes: An Introduction
Amin Shokrollahi
Digital Fountain, Inc. 39141 Civic Center Drive, Fremont, CA 94538
amin@digitalfountain.com
April 2, 2003
Abstract LDPC codes are one of the hottest topics in coding theory today. Originally invented in the ear
University of California San Diego
ECE 259A: Problem Set #1
0. Send an email to avardy@ucsd.edu, stating your name, your general research interests, your
research advisor (if you have one), and what brings you to the course on algebraic coding theory.
2.

/ :
1. The set of all 7 by 7 matrices over GF(2) is a vector space. Give a basis for this space. How
many vectors are in the space? Is the set of all such matrices with zeros on the diagonal
a subspace? If so, how many vectors are in this subs

/ :
1. Let H be a systematic parity check matrix of an (n, k, 2t + 1) code C. Assume that y was
received from the channel and the syndrome s = Hy has weight t. Show that the only
possible error pattern of weight t is e = (0k s), where 0k deno

/ :
1. An erasure is an error whose location is known. Prove that a code C over Fq with minimum
distance d can correct errors and erasures, provided d 2 + + 1.
2. Find all the binary linear MDS codes. Prove your answer.
3. Consider the binary
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