Application of the Discrete Mathematics

¶ … Coding Theory for Discrete Mathematics

In the contemporary IT (information technology) environment, increasing number of organizations are using large computer to transmit data over a long distance and some of these data are transmitted across billions of kilometers. During data transmission, some data can be degraded; the coding theory is an effective strategy to recover degraded data in order to guarantee reliable data transmission. Coding theory also assists in recovering and detecting errors, which assists in enhancing efficient data storage and data communications.

Objective of this paper is to discuss the coding theory and its real world application. The paper discusses the errors detecting codes and its application in the next section.

Error Detecting Codes

A simple strategy to detect errors is to add parity in order to check bit. To detect errors, a bit strong will be transmitted in order to add a parity bit. Typically, when a bit string has an even number of 1s, then we put 0 when reaching the end of the string. However, when a bit string has an odd number of 1s, then we put 1 when reaching the end of the string. When the parity check bit has been summed up with bit string and errors are detected with odd number, however errors are also detected with even numbers. For example, when receiving the following strings 1110011 and 10111101 as messages, the strings cannot be a valid codeword since we get odd numbers of 1s from the string 1110011. However, when the string of 10111101 has even numbers of 1s, the system will detect errors.

The errors can also be detected by repeating a bit within a message twice. For example, when the bit string of 011001 is repeated twice, the codeword of 001111000011 is repeated twice.

Error Correcting Codes

Some powerful codes are difficult to be detected, thus, the error correcting codes are used.

In the error correcting codes, it is easy to correct errors by including redundancy. It is also possible to correct errors by including redundancy. For example, the paper uses the triple code and repeat message three times. When the message is x1x2x3, the message is presented as x1x2x3 x4x5x6 x7x8x9

"Where x1 = x4 = x7, x2 = x5 = x8, and x3 = x6 = x9."

"The valid code words are 000000000, 001001001, 010010010, 011011011, 100100100, 101101101, 110110110, and 111111111." (Rosen 2012 p 75).

Hamming Distance

Hamming distance is the strategy to measure distance between two sets of bit strings. Hamming distance also reveals the number of positions where the bit strings differ. In essence, the hamming distance uses the coding theory to derive the fundamental work. For example, when there is Hamming distance between 11011 and 01110, the bit strings and the Hamming distance between 11111 and 0000 are provided. From example, the 01110 and 11011 is differed in the 1st, 3rd and 5th bits of d (01110, 11011) = 3. Since 00000 and 11111 is differed in all the five bits, the d (00000, 11111) = 5

Perfect Codes

Perfect codes allow error correction in order to derive maximum bits between two codewords. For example, the two codewords of 00000 and 11111 can be converted into perfect binary code. The minimum distance of the stated codewords is 5.

="25/[C (5, 0) + C (5, 1) + C (5, 2)]"

= 32/16 = 2

Since the solution is 2, there are 2 codewords in the code, then, the solution is a perfect binary code.

Generator Matrices

The parity check bit is used to encode message to generate matrices.

Typically, "matrices are used to express relationships between elements in sets. Matrices are used for the communications networks and transportation systems. (Rosen, 2012 p 177).

A k-bit message of x1x2 xk

Where a 1 x k matrix x.

For example, it is possible to encoding message by adding the parity check bit where three bits E (x) = xG are and Thus

Parity Check Matrices

The parity check matrix is derived by generating matrix using a standard form.

Example of binary code to generate parity check matrices is

Meanwhile, the parity check of G. is

Hamming Codes

Hamming codes are derived using parity check matrices. The Hamming codes can also be derived using the parity-check matrix {H} and code generator matrix of {G} and derived with:

And

"The applications of Error Correcting Codes and Hamming Codes demonstrate the utility of discrete mathematics in the solution of real-world problems." (Rosen, 2012 p xi).Their applications are in wide variety of areas that include data networking, computer science, internet, business engineering, psychology, linguistics, and biology. Increasing number of computer systems have been built with the error detecting codes and errors correcting codes in them. Essentially, computer technology has become essential tool for business, thus, error detecting and error correcting tools have become utmost important for the practicability of the computer applications. Many computers have error-correcting capabilities in their random access memories, which are 100% reliable. For example, Hamming codes are built in disk storage in order to correct errors.