Design Synthesis and Performance Evaluation of Codes with Good Rank Distance Properties for Wireless Communications and Information Storage Systems

Abstract

Rank-metric codes, a class of subspace codes, are error control codes that can be used to correct errors in applications that require two dimensional information transmission. In these applications, errors are confined to certain rows or columns or both. This is due to the nature of perturbations introduced by the channel. When errors are confined to a few columns (error bursts), error control codes possessing burst error correction capability can be employed. However, in scenarios where errors disturb the information transmission (all the columns), such that one or few rows are corrupted, burst error correcting codes by themselves fail to detect and correct all the errors. It has been shown that if error pattern is such that it has disturbed the information transmission uniformly (error matrix having rank less than certain value), then rank metric codes are the best choice for ensuring information integrity. The design and synthesis of rank error correcting codes started with the discovery of maximum rank distance (MRD) codes and maximum rank array codes (MRA) codes. These were mainly designed to overcome rank errors or crisscross errors. The search for codes with good rank distance properties continued and many low rate codes with good rank distance properties were identified within the class of Cyclic and Abelian codes. These were used to construct non-orthogonal Space Time Block codes (STBC). The application of the rank metric codes as Space-Time Block codes for MIMO systems has the potential to improve the performance of MIMO communication systems. In literature, Space- Time Block Code designs have been extracted from (m; 1) MRD codes, MRA codes and Full rank cyclic codes over the Galois fields F qm with rate 1=n. While these full rank codes had good rank-distance properties, they suffered from low spectral efficiencies and the lack of a suitable decoding algorithm. It was then felt that if high rate full rank codes could be synthesized from the family of Cyclic or Abelian codes, and an efficient decoding algorithm could be devised, it could lead to the design of highly efficient STBCs for wireless communication, codes for correcting crisscross errors in both storage media and power line communication. This motivated us to search for the existence of high rate full rank codes from within the families of Quasi-Cyclic, Cyclic and Abelian codes (polynomial codes). We have demonstrated that full rank high rate codes ican be found within the class of polynomial codes by specifying the procedure that can be used to construct (n; k) full rank codes over Fqm. Further, we have stated and proved theorems that allow the determination of the exact rank of these codes. A decoding algorithm based on the parity check matrix representation has been devised. It determines the unique solution if rank of the error vector R q(e) ≤ bm2−1c. The use of Galois Field Fourier Transform (GFFT) description of polynomial allows the specification of a direct relationship between the choice of k free transform components and rank of the corresponding codeword vector. Additionally, the use of GFFT provides an additional degree of freedom in the choice of k− free transform components for a specified rank requirement. This freedom can be employed to construct an index key based communication scheme, which can provide an additional layer of physical layer security. We have demonstrated that the bit error rate (BER) performance of the proposed codes as STBCs in wireless applications is superior to that of codes derived from MRD and MRA constructions. Rank preserving maps such as the Gaussian Integer map or Eisenstein-Jacobi integer map have been employed to synthesize STBC designs. The BER performance of these codes has been determined in power line communication applications also. It is observed that the performance is identical to that of Low Rank Parity Check codes (derived from Gabidulin codes). In addition, the proposed constructions provide flexibility as a large number of full rank codes meeting various needs can be easily synthesized. Thus, the focus of the research work reported in this thesis is the discovery of high rate full rank codes from the families of polynomial codes and assessment of their performance in a variety of applications. The performance of these codes is broadly superior to the state of the art in most cases and comparable in some instances. Hence, we believe that these codes can be gainfully used in many applications to strengthen the process of information transfer, storage and dissemination.