(A) decoding algorithm for the 1st order reed-muller codes and orthogonal latin square codes

Abstract

In this thesis, we give simple and useful decoding algorithm of two linear codes. In Chapter 1, we introduce the basic notions about coding theory and explain some properties of linear code. Particularly the syndrome plays an important role to decode an orthogonal Latin square codes. In Chapter 2, we review the well known properties of the 1st order Reed-Muller codes R(1,m) and introduce the concepts of mass, mass distance, and pattern to give a mass-decoding method for this code. Finally we propose a new mass-decoding method for R(1,m). This method is based on the form for Hadamard code of order $n=2^m$ and provides an easy decoding which can be done manually In Chapter 3, we recall the well-known definitions concerning Latin squares and summarize a construction of (p-1) mutually orthogonal Latin squares when p is an odd prime. In $L_p$, we need to find the first and the second coordinates of codeword in order to correct the errored received vector. Finally we give a decoding algorithm which is based on the syndrome decoding for linear codes.