Algebraic Design and Implementation of Protograph Codes using Non-Commuting Permutation Matrices

Random lifts of graphs, or equivalently, random permutation matrices, have been used to construct good families of codes known as protograph codes. An algebraic analog of this approach was recently presented using voltage graphs, and it was shown that many existing algebraic constructions of graph-based codes that use commuting permutation matrices may be seen as special cases of voltage graph codes. Voltage graphs are graphs that have an element of a finite group assigned to each edge, and the assignment determines a specific lift of the graph. In this paper we discuss how assignments of permutation group elements to the edges of a base graph affect the properties of the lifted graph and corresponding codes, and present a construction method of LDPC code ensembles based on noncommuting permutation matrices. We also show encoder and decoder implementations for these codes

On Code Design for Interference Channels

abstract: There has been a lot of work on the characterization of capacity and achievable rate regions, and rate region outer-bounds for various multi-user channels of interest. Parallel to the developed information theoretic results, practical codes have also been designed for some multi-user channels such as multiple access channels, broadcast channels and relay channels; however, interference channels have not received much attention and only a limited amount of work has been conducted on them. With this motivation, in this dissertation, design of practical and implementable channel codes is studied focusing on multi-user channels with special emphasis on interference channels; in particular, irregular low-density-parity-check codes are exploited for a variety of cases and trellis based codes for short block length designs are performed. Novel code design approaches are first studied for the two-user Gaussian multiple access channel. Exploiting Gaussian mixture approximation, new methods are proposed wherein the optimized codes are shown to improve upon the available designs and off-the-shelf point-to-point codes applied to the multiple access channel scenario. The code design is then examined for the two-user Gaussian interference channel implementing the Han-Kobayashi encoding and decoding strategy. Compared with the point-to-point codes, the newly designed codes consistently offer better performance. Parallel to this work, code design is explored for the discrete memoryless interference channels wherein the channel inputs and outputs are taken from a finite alphabet and it is demonstrated that the designed codes are superior to the single user codes used with time sharing. Finally, the code design principles are also investigated for the two-user Gaussian interference channel employing trellis-based codes with short block lengths for the case of strong and mixed interference levels.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201

A Tree Construction Method Of Nested Cyclic Codes

This paper presents a tree-based algebraic construction of nested cyclic codes through a tree construction method. These codes can be used to encode different data packets, producing codewords that are added for transmission. Both encoding and decoding are performed by polynomial operations with no need of side information, so the proposed scheme may be useful in applications that require low computational complexity. In cases where the number of information sources is large, it is convenient to design the system in a systematic manner. Hence, the tree construction method may contribute to simplify the code design. © 2011 IEEE.302305Heegard, C., Partitioned linear block codes for computer memory with " stuck-at" defects (1983) IEEE Trans. Inf. Theory, 54 (4), pp. 1259-1265. , NovXiao, L., Fuja, T., Kliewer, J., Costello, D., A network coding approach to cooperative diversity (2007) IEEE Trans. Inf. Theory, 53 (10), pp. 3714-3722. , OctXiao, L., Fuja, T., Kliewer, J., Costello, D., Nested codes with multiple interpretations (2006) Proc. 2006 Conf. Inf. Sci. Sys. (CISS), , Princeton, NJ, Mar. 5-9Kelley, C., Kliewer, J., Algebraic constructions of graph-based nested codes from protographs (2010) Proc. IEEE Int. Symp. on Inf. Theory (ISIT), pp. 829-833. , JuneMacWilliams, F.J., Sloane, N.J.A., (1977) The Theory of Error-correcting Codes, , North-Holland Publishing CompanyMcEliece, R., (2002) The Theory of Information and Coding, , 2nd ed. Cambridge University Pres