Coding theory is the study of error control codes that are used to detect and correct errors that occur during data transmission or storage. It draws from mathematics, computer science, electrical engineering, and other fields. Encoding functions map data to codewords that can detect or correct errors, while decoding functions recover the original data. A generator matrix defines the structure of a code and is used to generate codewords from a message by matrix multiplication. Error correction codes allow for detection and correction of errors up to a certain threshold by adding redundant bits and using techniques like parity checks and Hamming distance calculations.

1. Coding Theory
Nithin Koshy
14CS40128

2. Code Theory
•Coding Theory is the study of error control codes
•Error control codes are used to detect and correct
the codes that occur when a data are transferred or
stored
Code Theory concept is a mix of Mathematics,
computer science, electrical engineering,
telecommunications
Linear Algebra, Abstract Algebra(groups, rings,fields)
Probability& Statistics, Signals &System etc…

3. Encoding and Decoding
•An Encoding function is an one-to-one function which
provides a means to detect or correct Errors occurred
during transmission of signals(0’s and 1’s)
•A Decoding function is an onto function which
provides a means to recapture the transmitted word

4. • One to one function:
• f(x1)=f(x2) =>x1=x2 or f(x1)!=f(x2) =>x1!=x2
• Onto function: only if for every element in set ‘B’
there exists pre image in ’A’
• Generator Matrix
• What is a generator Matrix?
• Let us consider a encoding function say E: Z2(pow
m ->z2 (pow n)
• So this means that we are encoding M bit
information into an ‘n’ bit code

5. • Matrix form is followed for E ie I MxM/Am(n-m) such
type of Matrix is known as Generator Matrix for the
given function E
• How to generate a code word ?
• C=M.G
• M=Message in row matrix form , G=Generator
Matrix

6. • An error-correcting code is an algorithm for
expressing a sequence of numbers such that any
errors which are introduced can be detected and
corrected (within certain limitations) based on the
remaining numbers.
• MAXIMUM LIKELIHOOD TECHNIQUE :
• Given an (m,n) encoding function e : Bm ->Bn , we
often need to
• determine an (n,m) decoding function d : Bn->Bm
associated with e.

7. Parity Check
• Suppose that a parity check bit is added to a bit
string before it is transmitted. What can you
conclude if you receive the bit strings 1110011
• and 10111101 as messages?
• Answer
• Since the string 1110011 contains an odd number of
1s, it cannot be a valid codeword therefore,
contain an odd number of errors.
• On the other hand, the string 10111101 contains an
even number of 1s. Hence it is either a valid
codeword or contains an even number of errors.

8. Hamming Distance
• There is a simple way to measure the distance between two
bit strings. We look at the number of positions in which these
bit strings differ
• Definition : The Hamming distance d(x, y) between the bit
strings x = x1x2 . . . xn and y = y1y2 . . . yn is the number of
positions in which these strings differ, that is, the number of i (i
= 1, 2, . . . , n) for which xi != yi..
• ex: 01110 and 11011
• Since 01110 and 11011 differ in their first, third, and fifth bits,
d(01110, 11011) = 3

9. Thanks

10. Thanks