On some application of algebra and geometry in coding theory

Abstract

The fundamental problem that led to the development of the theory of error correcting codes was that of having a reliable communication over unreliable channels. A commu- nication channel can be as simple as the air between the voicebox of one adressing an audience and the ears of the listeners. Copper wires connecting telephones or modems can also be considered as channels. In the case of data storage, say in a magnetic tape or disc, the magnetized eld in the magnetic tape or disc is the channel. One property of these channels is their capacity to distort the information. For instance, the copper wires connecting telephones may get heated resulting in background interrup- tions. Magnetisation on the tape may re-align over time, or the head of the drive reading the tape or disc may be ill positioned and the right magnetisation be misread! Such distotortions to our good information will be referred to as noise. There are two main ways of handling noise; physical means and system means. Under physical means, one "targets" the cause of noise and seeks to eliminate it. For instance if the drive head misreads the magnetisation, then a better drive is used as a replacement for the "now faulty drive". As for system means, one "sandwiches" the channel between two devices; an encoder and a decoder (see gure 0:1) so that any form of noise can be detected and possibly corrected. The theory of error correcting codes is involved in this. For our purpose we will consider an abstract communication channel called the Binary Symmetric Channel (BSC). In BSC, information to be transmitted is encoded as a string of 00s and 10s. An error is then considered as an interchange between the binary digits in the sent and received information symbols