124
Chapter 9
Weight and Distance Enumeration
The weight and distance enumerators record the weight and distance information for the code. In turn they can be analyzed to reveal properties of the code. The most important result is MacWilliams' Theorem, wh
Notes on Coding Theory
J.I.Hall Department of Mathematics Michigan State University East Lansing, MI 48824 USA 9 September 2010
ii Copyright c 2001-2010 Jonathan I. Hall
Preface
These notes were written over a period of years as part of an advanced underg
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
88
Chapter 7
Codes over Subfields
In Chapter 6 we looked at various general methods for constructing new codes from old codes. Here we concentrate on two more specialized techniques that result from writing the field F as a vector space over its subfield
Some (nite) simple groups
MTH913-SS11
J.I.Hall
28 January 2011
1
1
CFSG
( 1.1) Theorem. A nite simple group is isomorphic to one of:
(1) a cyclic group of prime order p: Cp ;
(2) an alternating group: Alt(n);
(3) a classical group: PSLn (q ), PSpn (q ), P
Chapter 2
Sphere Packing and Shannon's Theorem
In the first section we discuss the basics of block coding on the m-ary symmetric channel. In the second section we see how the geometry of the codespace can be used to make coding judgements. This leads to t
A-152
A.3
A.3.1
Special Topics
The Euclidean algorithm
Let F be a field. In Theorem A.2.16 we gave a nonconstructive proof for the existence of the greatest common divisor of two polynomials a(x) and b(x) of F [x]. The Euclidean algorithm is an algorithm
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 ,
A-138
A.2
A.2.1
Polynomial Algebra over Fields
Polynomial rings over fields
indeterminate polynomial
We have introduced fields in order to give arithmetic structure to our alphabet F . Our next wish is then to give arithmetic structure to words formed fro
Syllabus
Mathematics 913, Spring 2011
Advanced Group Theory
MWF 9:10-10:00
C304 Wells Hall
J. Hall
D219 Wells Hall
x34653
[email protected]
Office Hours: MWF 10:20-11:10 (or by arrangement)
Course grades will be based on homework and (possibly) a project
76
Chapter 6
Modifying Codes
If one code is in some sense good, then we can hope to find from it similar and related codes that are also good. In this chapter we discuss some elementary methods for modifying a code in order to find new codes. In two furth
30
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 n over the field F is a subspace of F n . Thus the
MATH 523 LECTURE NOTES Summer 2008
These notes are intended to provide additional background from abstract algebra that is necessary to provide a good context for the study of algebraic coding theory. In particular, we need to show how to construct all ni
Chapter 1
Introduction
Claude Shannon's 1948 paper "A Mathematical Theory of Communication" gave birth to the twin disciplines of information theory and coding theory. The basic goal is efficient and reliable communication in an uncooperative (and possibl
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 913, Spring 2011 HW 2
Due Wednesday, 27 April 2011
1. Let m : V W D be a nondegenerate pairing. For nite dimensional
U0 V and Y0 W , let Y1 be a complement to (U0 Y0 ) in W and let
U1 be a complement to (Y0 U0 ) in V . Set U = U0 U1 and Y = Y0 Y1 .
P
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
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