The MacWilliams identities for m-spotty Rosenbloom-Tsfasman weight enumerator

Error control codes have extensively been applied to semiconductor memories using high density RAM chips with wide Input/Output data. High density semiconductor RAM chips with wide Input/Output, e.g., with 8-bit or 16-bit Input/Output data, have become popular in recent years. However, these semiconductor memories are highly vulnerable to multiple random bit errors when they are exposed to strong electromagnetic waves, radio active particles, or energetic cosmic particles. These multiple random bit errors, typically 2- or 3-bit errors, are usually confined to a byte region because RAM chips, each of which corresponds to a byte having length bits, are physically independent. These errors can be effectively corrected or detected by m-spotty byte error control codes. Firstly, m-spotty byte error control codes have been characterized by the m-spotty Hamming distance. This study defines a new distance function, called m-spotty Rosenbloom-Tsfasman distance, of the m-spotty byte error control codes. Morever, this study proposes a MacWilliams type identity for m-spotty Rosenbloom-Tsfasman weight enumerators via the new distance function, which is a metric. However, it is shown that there exists a relationship between array codes and the m-spotty Rosenbloom-Tsfasman weight.This thesis consists of six chapters. In the first chapter, some basic definitions and theorems related to algebraic structures, some information about linear codes, m-spotty byte error control codes and MacWilliams identity are given.The second chapter presents a survey of the m-spotty byte error control codes over finite fields.In the third and fourth chapters, the MacWilliams identity for m-spotty Rosenbloom-Tsfasman weight enumerators over some algebraic structures is obtained.In the fifth chapter, the array codes having the m-spotty Rosenbloom-Tsfasman weight are introduced.In the sixth and the last chapter, the conclusion and possible work are given.