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
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 writt
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
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 w
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:
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
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
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 t
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 i
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 gr
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 mo
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 le
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.
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
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 co
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 foll
Due Monday, 28 March 2009
Math 810, Spring 2011 HW 4
1. Consider the ternary [13, 10] Hamming code
101201201
0 0 0 0 1 1 1 1 1
011100011
with check matrix
2012
1 1 1 1.
1222
Decode the received word
(
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
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 comple
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 carefull
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(
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 re
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. Con
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 mini
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.
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 S
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
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 p
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 w