LCD MRD Error Correcting Codes: coding schemes, properties and applications

Abstract

Algebraic coding theory deals mainly with the Hamming metric introduced by R. W. Hamming. But when the code symbols are elements of a higher dimensional Galois field, this metric is not always adequate especially for the real channels. Further the Hamming metric is inappropriate in certain situations where crisscross errors occur, it is found that the rank metric introduced by E. M. Gabidulin as an ideal metric for its capability in handling varied error patterns efficiently. The maximal size codes in rank metric known as maximum rank distance codes are relatively rich in algebraic structure. Another rather important property of error-correcting codes is complementary-duality property especially form algebraic as well as from practical utility point of view. Thus the linear codes equipped with both of these properties are the center of attention throughout this work. This thesis mainly orient around the class of LCD and MRD codes, their properties and constructions so as to incorporate their possible applications to various communication systems. Beginning with the classification of LCD codes as trivial and nontrivial LCD MRD codes, LCD MRD codes of length n>N are constructed in CHAPTER 2. CHAPTER 3 presents various constructions of Block-Matrix LCD codes by utilizing the combinations of trivial and nontrivial LCD codes. In CHAPTER 4 and CHAPTER 5, an error-correcting coding scheme capable of handling multiple q-ary user messages of variable lengths, providing multiple-rate feature is given. In CHAPTER 6, the multiple-rate coding scheme is extended further in order to incorporate multiple-length codewords along with multiple-rates. CHAPTER 7 discusses the applicability of T-Direct codes in multiple-rate codes. Lastly, the concluding remarks are made in CHAPTER 8. The thesis starts with characterization of LCD MRD codes in a more general way. The LCD codes constructed using a self-dual basis are classified as trivial and non-trivial LCD codes. The LCD MRD codes constructed using a self-dual basis have an intrinsic relationship between their generator and parity-check matrix, the formal proof for the same is provided. Further, the cartesian product LCD MRD codes C(n',k',d') of length n' greater then N over F_(q^N ) are constructed by employing the trivial and non-trivial LCD MRD codes as constituent codes. After having classified the LCD MRD codes in a more general way, this thesis presents a new class of Block-Matrix LCD codes over F_((2^t )^2k ) of length nl and dimension kl, which are constructed using the generator and parity check-matrices of LCD MRD codes over the field F_((2^t )^2k ). The conditions on constituent LCD MRD codes of Block-Matrix LCD codes