Chapter 3
Linear Codes
An important class of codes are linear codes in the vector space Fqn , where Fq is a nite
eld of order q.
Denition 3.1 (Linear code). A linear code C is a code in Fqn for which, whenever
x, y C, then ax + by C for all a, b Fq . That
Chapter 5
Golay Codes
Lecture 16, March 10, 2011
We saw in the last chapter that the linear Hamming codes are nontrivial perfect codes.
Question. Are there any other nontrivial perfect codes?
Answer. Yes, two other linear perfect codes were found by M.J.E
Chapter 6
Introduction to Finite Fields (cont.)
6.1
Recall
Theorem. Zm is a eld m is a prime number.
Theorem (Subeld Isomorphic to Zp ). Every nite eld has the order of a power of
a prime number p and contains a subeld isomorphic to Zp .
Corollary (Isomor
Chapter 4
Hamming Codes
Lecture 14, March 3, 2011
4.1
Denition and Properties
A basis for a vector space V is a linearly independent set of vectors in V which spans
the space V . The space V is nite-dimensional if it has a nite basis. The dimension of
a n
Chapter 2
Introduction to Finite Field
Lecture 7, February 1, 2011
Recall:
Denition (Ring). A commutative ring (R, +, ) is a non-empty set R together with
two binary operations: addition (+) and multiplication () such that:
1). (R, +) is an abelian group,
Chapter 7
Cyclic Codes
Lecture 21, March 29, 2011
7.1
Denitions and Generator Polynomials
Cyclic codes are an important class of linear codes for which the encoding and decoding
can be eciently implemented using shift registers. Many common linear codes,
Chapter 8
BCH Codes
8.1
Denitions
We dened the least common multiple lcm(f1 (x), f2 (x) of two nonzero polynomials
f1 (x), f2 (x) Fq [x] to be the monic polynomial of the lowest degree which is a multiple
of both f1 (x) and f2 (x). Suppose we have t nonze
Chapter 1
Introduction
Lecture 1, January 11, 2011
1.1
Examples
In the modern era, digital information has become a valuable commodity. For example,
governments, corporations, and universities all exchange enormous quantities of digitized
information ever
Chapter 9
Review
The nal exam focuses on the part after midterm exam. The codes in this part are all
linear codes, so we need some facts and results for linear codes, such as:
Denition of linear codes;
the parameters of linear codes;
the minimum weight