Coding Theory
Ian Bathelt
Math\221
02\16\2015
Cory Bennet

Coding theory is the study of codes and their properties and their propensity to work with certain types of applications. There are four general types of coding that we as humans use. The four types are: line coding, error correction or channel coding, data compression, and lastly cryptographic coding. In relation to error detecting codes, there exist error-correcting codes. The purpose of this paper is to explain the different types of coding and go over hamming distance, perfect codes, generator matrices, parity check matrices, and hamming codes. We will also be giving examples of how coding theory could be applied in a real world application; and a brief history of coding theory. According to “Wolframmathworld” (2015) “Coding theory, sometimes called algebraic coding theory, deals with the design of error-correcting codes for the reliable transmission of information across noisy channels. It makes use of classical and modern algebraic techniques involving finite fields, group theory, and polynomial algebra”(). Coding theory has roots in communication filed. Claude Shannon first published the “A Mathematical Theory of Communication” in the Bell System Technical Journal. It was a piece that featured encoding information transmitted by a send. The fundamentals of this work included probability, which he applied to his communication theory. This gave way to Shannon developing what later became known as information theory. Later, a binary code came to fruition, known as the Golay code. This was an error correcting code that could correct up to three errors in words that were 24-bits in length, and could detect the presence of a fourth. Decades later, Richard Hamming came along and began doing extensive work with numerical methods, coding systems that were automatic, and error detecting and error correcting codes. His study of these systems led to Hamming Codes, and Hamming Distance. Hamming codes use a set of mathematical parameters that detect and correct errors with up to two bits, and correct one-bit errors, and does not detect errors that are uncorrected. Most codes until the Hamming code, could not correct errors. Hamming codes are known as perfect codes, or codes that achieve the highest rate possible for codes of their block length with minimum distance. Hamming codes add little information to the data that it is correcting, therefore they are only able to detect and correct the errors in the information when errors in the information, is considerably low. This gives way to the Hamming distance. The Hamming distance is basically the distance of which two strings of information and the symbols of information that they contain, are different. So say you two strings of information with varying data inside of them, the Hamming distance is what it would take two change the two strings to be exactly alike. The differences are considered errors, where changing one is to change it to be like the other string. This all gives way to error correcting codes. Error correcting codes are techniques for helping to control errors that are made in the transmission of data over communications that are considered A: unreliable, and B: noisy communications. The basic concept is that when the sender sends an encoded message, it is done in a redundant way using an error correcting code. Redundancy in a message gives the receiver the ability to detect limited errors that could arise anywhere within the message; as well as the ability to correct any errors without having to rebroadcast the transmission of the message. This is called Forward Error Coding, which is essentially the main part of error correcting codes. According to Permutationpuzzles.org (2007), “Since a code is a finite dimensional vector space over a finite field, it only has finitely many elements. To represent a code in a computer one may of course store all the elements” (para. 1). For this type of information you would use a generator matrix. The rows of a generator matrix are a basis for linear codes. Code words in a generator matrix are linear combinations that the rows of the matrix make up, and the linear code are its row space. Row space is essentially of the possible combinations that its row vectors generate. Next we have the parity check matrix. A parity check matrix is a block of linear codes that describe linear relations where components of that code word must be satisfied. It is used in deciding whether or not a vector is a code word, and is used when algorithms are decoded. Parity check matrices takes variables that are parts of a generator matrix when they pertain to that dual code. Rows used in parity matrices or coefficients of the equations that they are checking. What that means, is that they show linear combinations of components to those code words which equal zero. The "Pulse.embs.org" (2015) website, “In 1997, a science fiction film titled Gattaca premiered in U.S. theaters, depicting a society sometime in the not-so-distant future in which people are crafted for and judged by the quality of their genetic material.” To me the most relevant real world application and implementation of coding theory, is it’s relation to DNA. If you can apply the techniques of coding theory to DNA, the Human Race may definitely one day realize the dream where no human has to suffer from inferior genetics. You take someone’s genetic code, and develop an error detecting code, where genetic mutations occur and cause disease. If you have a code that can detect that, and then correct that, I think the world would be a better place. I think the most relevant diseases where coding theory could be applied to would be auto-immune diseases. If you took someone with an auto-immune disease, and ran their genetic (DNA) code through a genetic error detecting, or Hamming code, you could pinpoint the genetic information that causes that specific disease, and replace it with different, more-healthy information. To conclude, coding theory is everywhere in today’s society. It was given birth as a child of World War II, with siblings the computer, and the nuclear bomb. Communication information gave birth to coding theory and the many different forms of it that include Error detecting and correcting, Hamming distances and codes, perfect codes, and matrices: Generator, and Parity.

References: wolframMathWorld. (2015). Retrieved from http://mathworld.wolfram.com/CodingTheory.html

permutationpuzzles.org. (2007). Retrieved from http://www.permutationpuzzles.org/AAAbook/node125.html

pulse.embs.org. (2015). Retrieved from http://pulse.embs.org/january-2015/state-art/