Coding Theory Analysis : an Algebraic Analysis on Linear Block Codes with Application

Abstract

This paper has discussed the linear block codes from an algebraic approach. Linear block codes are created to solve the noisy channel problem in communication, and they are characterized by their specific error-correcting algorithms such as repetition, parity-check, and Hamming algorithm. Linear codes can also be seen as vector spaces defined by their generator matrices and parity-check matrices, and these matrices are tools to analyze linear code algebraically, such as proving the Plotkin bound. The algebraic properties also allow a simple algorithm for systematic encoding and decoding on a linear code, which makes linear code suitable carrying large set of data. Overall, in a communication system, an error-correcting linear code is able to protect the originality of the data with limited additional redundancy, which makes it a reliable and efficient solution to the fundamental problem in coding theory.