Comparison of BCH-Codes over Galois Ring, Quasi-Galois Ring and their Residue Field

Abstract

The goal of coding theory is to successfully transmit data over a noisy channel and to fix errors in corrupted communications. For many applications in computer science and engineering, it is crucial. The main notion is that the sender should use redundant information to create an error correcting code and encrypt the message. Development of data transferring codes were started with the first article (Interlando, 1995) of Claude Shannon in 1948. In 1950, for this purpose Hamming (Hamming, 1950) and Golay (Golay, 1949) introduced cyclic block code known as binary hamming and Golay codes respectively. These classes of codes have the capability to detect up to two errors and correct one error. In 1953, Muller (Muller, 1953) introduced a multiple error correcting codes techniques and Reed (Reed, 1953) developed technique of such type of codes. In 1957, (Prange, 1959) initiated an idea of cyclic codes in two symbols. In addition, (Prange, 1959) used the coset equivalence for decoding the Group codes in 1959. The cyclic codes initially developed over binary field 2 and into its Galois field extension   2 m GF . Though it was further extended over the prime filed p and its Galois field extension   m GF p . The remarkable development in coding theory began when in 1960, Hoequenghem, Bose and Chaudhuri explained the large class of codes which correct multiple errors known as BCH-Code. In 1960, Peterson gave error correction procedure for BCH-Code over finite field. Forney gave the decoding technique of BCH-Code by using Barlekamp Massey algorithm in 1965. In 1972, (Blake I. , 1972) proposed the concept of linear codes over n , the ring of integers modulo n where n is the product of primes. However, he did not explain the codes over the local ring m , 1 p m  . In 1975, (Blake I. F.) went on to talk about linear codes across the ring n when m n p  , where p is a prime and m is a positive integer. (Spiegel, 1977) and (E. Spiegel, 1978) demonstrated in 1977 and 1978 that codes over m p can be defined in terms of codes over p . As a result, we can establish codes over n , for every positive integers n. (Shankar, 1979) created BCH-Code over m p in 1979 and she also devised BCH-Code for arbitrary integers. She created BCH-Code over the GR using the maximal cyclic Subgroup of the Group of units of GR and related these to BCH Code over the Galois filed using the reduction map. In 1999, (de Andrade, 1999) built BCH-Code over finite unitary commutative rings. The cyclic Subgroup of the Group of units of a GR was specified in both (De Andrade, 1999) and (Shankar, 1979) building procedures. In 2012, (Shah t. a., 2012) devised a decoding approach that enhances code rate. In addition, (Shah T. M., 2013) explained how to decode a lengthy binary BCH-Codes using cyclic code in 2013. In 2017, (Shah T. N., 2017) devised an approach for constructing the maximal cyclic Subgroup of any arbitrary Group of units of GRs. There are four chapters in this thesis. In chapter 1, some key concepts of abstract algebra and error correcting codes are introduced, which are crucial for understanding Subsequent chapters. DRSML QAU vii In Chapter 2, we will discuss a brief comparison between Galois and Q-GRs, we also discuss their properties and structures by an example. We will design BCH-Code over GR, Q-GR and their Residue field and compare these codes in each of these three cases in chapter 3. Chapter 4 consist of method of designing Substitution Box over Sylow p-Subgroup of Group of units of GR by using a new concept of affine map