Due Monday, 21 March 2011
Math 913, Spring 2011 HW 1
1. Prove that a subgroup of the symmetric group Sym(), for nite, that is
generated by a set S of transpositions (2-cycles) is a direct sum k=1 Sym(i )
i
for pairwise disjoint subsets i of .
Hint: On den

Math 810, Spring 2011 HW 1
Due Friday, 4 February 2011
You may talk with each other about problems but make sure you write up
your solutions separately.
Make sure you explain things carefully.
Some remarks:
1. Most problems that you will get this semester

Math 523 Total: 30 points
Homework 6
Due 5:30 pm 7/31/2008
From Chapter 4 of the text: 4.1. Construct a generator polynomial for a 3-error correcting Reed-Solomon code of length 10 over Z11 . 4.2. Construct a [4, 2] Reed-Solomon code C over Z5 . That is,

Math 523 Total: 20 points
Homework 5
Due 7/28/2008
From Section 4 of the lecture notes: 1. Give a multiplication table for GF (32 ). Find all generators for the cyclic group GF (32 ) , and nd the minimal polynomial of each generator over Z3 . 2. Find all

Math 523 Total: 40 points
Homework 4
Due 7/21/2008
From Section 2 of the lecture notes: In these exercises, let F be a eld, let p(x) be a nonconstant polynomial in F [x], and let R = F [x]/ p(x) . 1. (5 pts) If f (x) is any polynomial, show that [f (x)] =

48
Chapter 4
Hamming Codes
In the late 1940's Claude Shannon was developing information theory and coding as a mathematical model for communication. At the same time, Richard Hamming, a colleague of Shannon's at Bell Laboratories, found a need for error c

A GRS decoding example
Consider the code GRS10,4 (, v) over F11 with
= v = (10, 9, 8, 7, 6, 5, 4, 3, 2, 1) ,
hence (by direct calculation or Problems 5.1.3/5.1.5) we may take
u = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1) .
Note that here r = 10 4 = 6, so we can cor

62
Chapter 5
Generalized Reed-Solomon Codes
In 1960, I.S. Reed and G. Solomon introduced a family of error-correcting codes that are doubly blessed. The codes and their generalizations are useful in practice, and the mathematics that lies behind them is i

A Primer on Finite Fields
In (1)-(7) F will denote a nite eld.
(1) F contains a copy of Zp = Fp , for some prime p. (This prime is called
the characteristic of F .)
(2) There is a positive integer d with |F | = pd .
Proof. From the denitions, F is a vecto

A Euclidean Algorithm example
We now calculate gcd(z 6 , 9z 5 + 8z 4 + 2z 3 + 7z 2 + 6) = 1 over F11 using the
Euclidean Algorithm.
At Step i we dene qi (z ), ri (z ), si (z ), and ti (z ) using
ri2 (z )
=
qi (z )ri1 (z ) + ri (z )
si (z )
=
si2 (z ) qi (

100
Chapter 8
Cyclic Codes
Among the first 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 first indication that a

ECE 598 Fall 2006
Lecture 18: CDMA
What is Multiple Access?
Multiple users want to communicate in a common geographic area Cellular Example: Many people want to talk on their cell phones. Each phone must communicate with a base station. Imagine if only on

A-126
A.1
A.1.1
Basic Algebra
Fields
In doing coding theory it is advantageous for our alphabet to have a certain amount of mathematical structure. We are familiar at the bit level with boolean addition (EXCLUSIVE OR) and multiplication (AND) within the s

-124
Appendix A
Some Algebra
This appendix is broken into three sections. The first section discusses doing basic arithmetic and vector algebra over arbitrary fields of coefficients, rather than restricting to the usual rational, real, or complex fields.

Math 523
Notes on Coding Theory
July, 2008
The denition of BCH and RS codes Denition. A primitive nth root of unity is a root of xn 1 that has multiplicative order n.
1 Example. Over the real numbers the 6th roots of unity are 1, 2 23 i. A cyclic group of

Probabilities and Random Variables
This is an elementary overview of the basic concepts of probability theory. I. The Probability Space The purpose of probability theory is to model random experiments so that we can draw inferences about them. The fundame

Conditional densities, mass functions, and expectations
Jason Swanson April 22, 2007
1
Discrete random variables
Suppose that X is a discrete random variable with range cfw_x1 , x2 , x3 , . . ., and that Y is also a discrete random variable with range cfw

632 Introduction to Stochastic Processes Fall 2008 Part of Homework 2
1. You are trying to cross a busy highway. You need time c to cross safely. Interarrival times between cars are i.i.d. with common distribution pk = pq k1 for k N. Find the expected tim

Math 810
Error-Correcting Codes
Spring 2011
MWF 11:3012:20
C107 Wells Hall
J. Hall
D219 Wells Hall
x34653
[email protected]
Oce hours:
MWF 10:20
or by arrangement
There is no text for the course, but class notes (which include much of the
material) are a

Math 810, Spring 2011 Final Exam
Corrections
Due by noon, Thursday, 5 May 2011
hand in at D219 Wells Hall
Problem (B-4). Line 2 of this problem should begin:
GRSn1,k1 ( , w)
Problem (B-5). Line 3 of this problem should read:
= (0, 1, 1 , 2 , 3 , 4 , 5 ,

Math 810, Spring 2011 HW 6
Due Wednesday, 27 April 2011
1. Let C and D be linear codes over the eld F with C = D . Let P C be
the code C punctured at its last coordinate position, and let SD be the code D
shortened at its last coordinate position. Prove t

Math 810, Spring 2011 HW 5
Due Friday, 15 April 2011
Make sure you justify your answers appropriately. For these problems, that
means you should show your calculations in enough detail that I can follow
them (and can try to locate any mistakes). Please do

Due Wednesday, 2 March 2011
Math 810, Spring 2011 HW 3
1. (a) Give a syndrome table (syndrome
code E with the following check matrix:
11111
0 0 1 1 1
0 0 0 1 1
01010
dictionary) for the [9, 5] binary linear
1
0
1
1
1
0
1
0
1
1
0
1
0
1
.
0
0
(b) Use your t

Math 810, Spring 2011 HW 2
Due Monday, 21 February 2011
As before, you may talk with each other about problems but make sure you
write up your solutions separately. Also always explain things carefully
and thoroughly.
1. Problem 3.1.3 from the Notes:
http

Math 810, Spring 2011 HW 1
Due Friday, 4 February 2011
You may talk with each other about problems but make sure you write up
your solutions separately.
Make sure you explain things carefully.
Some remarks:
1. Most problems that you will get this semester

A GRS decoding example
Consider the code GRS10,4 (, v) over F11 with
= v = (10, 9, 8, 7, 6, 5, 4, 3, 2, 1) ,
hence (by direct calculation or Problems 5.1.3/5.1.5) we may take
u = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1) .
Note that here r = 10 4 = 6, so we can cor

A Primer on Finite Fields
In (1)-(7) F will denote a nite eld.
(1) F contains a copy of Zp = Fp , for some prime p. (This prime is called
the characteristic of F .)
(2) There is a positive integer d with |F | = pd .
Proof. From the denitions, F is a vecto