CHAPTER 2
Linear Block Codes
1 Error control block codes fundamentals & bounds
Let GF(q)n denote the vector space over GF(q) comprising
q n vectors of length n. A q-ary block code C of length n is
a subset of GF(q)n.
Suppose C has qk codewords for some
SPECIAL TOPIC I
Error-Correction for Channels with
Synchronization Errors
1 The Davey-MacKay (DM) Construction
Data transmitted through a channel with synchronization errors are subjected to bit insertions and deletions.
Consequently, the number of bits
Algebraic Coding
Q. 1:
2 +4+2 is a primitive polynomial in GF(5)[] of degree 2. Construct the polynomial representation
of the nonzero elements i of GF(5)[]/ 2 + 4 + 2 for 0 i 4.
Solution:
The first 5 nonzero elements of GF(25) may be obtained as follows.
CHAPTER 7
Low-Density Parity-Check Codes
1 LDPC Codes
A q-ary low-density parity-check (LDPC) code is a linear
block code over GF(q) with the following attribute:
The row and column (Hamming) weights of its parity-check
matrix H are small compared to the
SPECIAL TOPIC II
Channel Polarization & Polar Coding
1 The main idea
Given N independent copies of a binary-input discrete memoryless channel W with symmetric capacity I(W ), one can
(i)
synthesize a set of N binary-input channels cfw_WN : 1 i
N that e
CHAPTER 3
BCH Codes, Reed-Solomon Codes,
Hard and Soft Decision Decoding
1 BCH codes
In the study of BCH codes, it is convenient to express a
message vector m = (m0, m1, . . . , mk1), the corresponding codeword c = (c0, c1 , . . . , cn1 ) and received wo
CHAPTER 6
Turbo Codes, Iterative Decoding
& EXIT Charts
1 Encoder Structure
The basic encoding scheme of a turbo code is shown below.
The turbo encoder consists of two rate 1/2 recursive, systematic convolutional (RSC) encoders with memory order m and
an
CHAPTER 4
Binary Convolutional Codes
1 Convolutional encoders
A convolutional encoder of rate k/n is a k-input, n-output
time-invariant system. At time i, k bits
(0)
(1)
(k1)
(1)
(n1)
xi = (xi , xi , . . . , xi
)
enter the system and n bits
(0)
yi = (yi
CHAPTER 1
Fundamentals of Galois Fields
1 What is a Galois field?
A Galois field (GF) is a finite set of elements, F , with addition and multiplication defined on F , such that
1. F is closed under addition, i.e.,
a + b F,
a, b F
where + is the additive
CHAPTER 5
Decoding & Performance Analysis of
Convolutional Codes
1 Maximum likelihood decoding
Recall, given a received word r, a maximum likelihood (ML)
decoder finds the codeword estimate y that maximizes the
probability p(r| ). A realization of an ML
FACULTY OF COMMERCE
GROUP 4
PROGRAMME
RISK MANAGEMENT AND INSURANCE
DEPARTMENT
RISK AND ACTUARIAL SCIENCE
COURSE
APPLIED RISK (CIN 2107)
LECTURER
MR KHUMALO
DATE
18 NOVEMBER 2014
QUESTION: IDENTIFY AND DISCUSS THE KEY FACTORS THAT UNDERPIN THE
USE OF INSU