Lecture 8
A (15, 7) code that corrects 2 errors
The benefit of rephrasing things in terms of polynomials, instead of matrices, is that we can use the
additional structure of polynomials (namely division) to encode additional redundancy. Consider
now the p
Lecture 17
Orders of elements and the exponent
Last time, we considered a finite group (G, , e). We proved the following results.
Proposition 17.1. If G is a finite group, and g G is an element, then the following are true.
0) The element g has an order;
Lecture 9
A (15, 7) code that corrects 2 errors, ctd.
Continuing with our 2-error correcting code, recall that if 2 errors have occurred, we were dealing
with the matrix:
i
+ j 2i + 2j
2i + 2j 3i + 3j
and we would like to recover i and j from this matri
Lecture 16
Groups ctd.
Last time we introduced the notion of a group. If (G, , e) was a group, often abbreviated simply to
G, and G consisted of finitely many elements, we said its order was the number of elements. Given
an element g G, we considered the
Lecture 12
Constructing finite fields
Last time we established the following result by explicit construction. Recall that F [x]/(p(x) was
the set of congruence classes [f (x)]p(x) with addition defined by addition of remainders and
multiplication defined
Lecture 13
Getting back to codes
Now, since our goal in constructing finite fields as above was to construct codes, we should think
back to the properties of finite fields we used in constructing codes. Recall that when we studied
F2 [x]/(x4 + x + 1), we
Last 1
First Last
Mrs. St. Amant
LA 1, Period 4
17 December 2013
My Interesting Title
Indent your first line. Here is where you will write your essay. Be sure to double
space. In order to do that for the whole essay, highlight all of the text you want to
Spring 2013
Math 370: Applied Algebra
Prof. Assaf
Solutions to Problem Set 5
1. Since f is a homomorphism, f (0) = 0 and f (1) = 1. For n N, f (n) = f (1) + + f (1) =
1 + + 1 = n, and f (n) = f (n) = n. Therefore f (n) = n for all n Z. For n = 0, n
is a u
Lecture 4
Probabilities
Last time we introduced what we called the binary symmetric channel with error probability p.
Briefly, we made the following assumptions: signals are sent in bits, i) for each bit sent a bit is
received, ii) the probability that a
Lecture 15
Groups
Last time we spent a fair bit of time studying the non-zero elements in the field F2d . Our goal was
to find an element such that every non-zero element of F2d could be realized as a power of .
Before moving on, I want to formalize some
Lecture 6
The Hamming bound
Now, suppose we have a code that is e-error correcting. We can rephrase the condition above
slightly. Under SSe , if z is any word, x is a codeword and, z Be (x), then z is decoded to x.
Given two distinct codewords x and y, an
Lecture 5
Hamming distance
Last time we introduced the Hamming distance between two words. If x and y are two words of
length n, i.e., vectors in Fn2 , then the Hamming distance is defined by:
d(x, y) = cfw_# places where x and y differ.
We observed that,
Lecture 11
Constructing fields
When p(x) = x2 + x + 1, we saw that F2 [x]/p(x) was precisely F2 [] and, in particular was a
field. The basic question we would like to ask is: when is F [x]/(p(x) a field? To answer this
question, we need to know when congr
Lecture 7
Larger finite fields
Last time, we observed that the polynomial x2 + x + 1 has no root in F2 . Nevertheless, we could
formally introduce a symbol that plays the role of a root of this equation. By definition,
satisfies the algebraic equation 2
Lecture 2
Matrix operations
Last time we introduced the finite field with 2 elements F2 , and we discussed arithmetic in this finite
field. Today, we will introduce some aspects of the theory of matrices with entries in F2 , and study
an example of a code
Lecture 14
BCH codes
The letters BCH stand for the initials of Bose, Chaudhuri, and Hocquenghem. Suppose we want to
construct a code that corrects t errors. We begin with a finite field with 2d elements. We let be a
primitive root of F2d . We then write d
Lecture 1
Introduction
The goal of this class is to provide an concrete introduction to modern algebra. Unlike MATH 410,
which focuses on abstract algebraic structures and their basic properties, this class will focus on
a series of applications. Largely,
Lecture 10
Toward a construction of general BCH codes
So far, weve been constructing codes on a case by case basis. To begin, we start with an irreducible
polynomial with coefficients in F2 . In the several cases we have considered, we constructed from
th
Lecture 3
The (7, 4) Hamming code continued
Last time, after introducing basic operations on matrices, we considered a code that I called the
(7, 4)-Hamming code. Given 4 data bits, the code produced 3 additional parity bits; the 7 stands
for the total nu
Spring 2013
Prof. Assaf
Math 370: Applied Algebra
Solutions to Problem Set 8
1. Let p(x) = x5 and q (x) = x. Then p(x) = q (x) since they have dierent degrees, but, by
Fermats Little Theorem, for any a Z, p(a) = a5 a = q (a) (mod 5).
2. Suppose p(x) Q[x]
Spring 2013
Math 370: Applied Algebra
Prof. Assaf
Solutions to Problem Set 9
1. We rst prove by induction that f (x) can be factored into a product of irreducible polynomials.
If deg(f (x) = 1, then f (x) is irreducible over F. Now assume deg(f (x) > 1 an
M idtermL-Math370
Instructions
p
s
o
The following examconsists f several roblems, omeof which havemultiple parts' There are125
points availableon the exam. To get full credit, you must do enoughproblemsto get 100 points
c
s
iwhich is also the highestposs
Homework Week 13
On the Chinese remainder theorem
1. Find the smallest positive solution, if any, of
x9
x 17
mod 16
mod 28.
2. Find the smallest positive solution, if any, of
x 10
mod 15
x 17
mod 28.
3. Show that if m = rs with r and s coprime, then x2 1
Homework Week 11
Applications of orders
1. Suppose p > 2 is a prime number, and consider the polynomial x2 + 1 = 0 in Fp .
i) Show that x2 + 1 = 0 has a solution in Fp if and only if there is an element in Z/pZ
that has order 4.
ii) Show that Z/pZ has no
Homework Week 9
Groups
1. We know that the exponent of a group divides the order of the group. Consider the group
Z/nZ . The order of Z/nZ is given by Eulers -function, as we studied last week.
i) Compute exp(Z/nZ ) for n = 2, 3, 5, 7, 9, 11, 13.
ii) Show
Homework Week 8
Orders of elements and Eulers totient function
1. Consider the group Z/mZ of units in the commutative ring Z/mZ discussed on the last
homework.
i) In class, I dened the order of an element g in a (nite) group G, denoted ord(g ), to
be the
Homework Week 5
Polynomial arithmetic
1. Using the algorithm given in class, determine the greatest common divisor of x3 + 1 and
x4 + 1 over F2 .
More nite elds
2. Suppose n is an integer 2. Consider the collection of symbols 0, 1, . . . , n 1. Given two
Homework Week 4
Computing with the (15, 7)-code from class
1. Given the data bit (1, 1, 0, 1, 0, 0, 1), construct the sent word s(x) to which it encodes.
2. Suppose we know that at most 2 errors have occurred in transmission. How would the bit
string (1,
Homework Week 3
Irreducibility
1. Write down all the irreducible polynomials of degree 2 with coefcients in F2 .
2. Write down all the irreducible polynomials of degree 3 with coefcients in F2 .
3. Write down all the irreducible polynomials of degree 4 wi