Low-density MDS codes and factors of complete graphs

We present a class of array code of size n×l, where l=2n or 2n+1, called B-Code. The distances of the B-Code and its dual are 3 and l-1, respectively. The B-Code and its dual are optimal in the sense that i) they are maximum-distance separable (MDS), ii) they have an optimal encoding property, i.e., the number of the parity bits that are affected by change of a single information bit is minimal, and iii) they have optimal length. Using a new graph description of the codes, we prove an equivalence relation between the construction of the B-Code (or its dual) and a combinatorial problem known as perfect one-factorization of complete graphs, thus obtaining constructions of two families of the B-Code and its dual, one of which is new. Efficient decoding algorithms are also given, both for erasure correcting and for error correcting. The existence of perfect one-factorizations for every complete graph with an even number of nodes is a 35 years long conjecture in graph theory. The construction of B-Codes of arbitrary odd length will provide an affirmative answer to the conjecture

Density Evolution for Deterministic Generalized Product Codes with Higher-Order Modulation

Generalized product codes (GPCs) are extensions of product codes (PCs) where coded bits are protected by two component codes but not necessarily arranged in a rectangular array. It has recently been shown that there exists a large class of deterministic GPCs (including, e.g., irregular PCs, half-product codes, staircase codes, and certain braided codes) for which the asymptotic performance under iterative bounded-distance decoding over the binary erasure channel (BEC) can be rigorously characterized in terms of a density evolution analysis. In this paper, the analysis is extended to the case where transmission takes place over parallel BECs with different erasure probabilities. We use this model to predict the code performance in a coded modulation setup with higher-order signal constellations. We also discuss the design of the bit mapper that determines the allocation of the coded bits to the modulation bits of the signal constellation.Comment: invited and accepted paper for the special session "Recent Advances in Coding for Higher Order Modulation" at the International Symposium on Turbo Codes & Iterative Information Processing, Brest, France, 201

PERFORMANCE COMPARISON OF NON-INTERLEAVED BCH CODES AND INTERLEAVED BCH CODES

This project covers the research about the BCH error correcting codes and the performance of interleaved and non-interleaved BCH codes. Both long and short BCH codes for multimedia communication are examined in an A WGN channel. Algorithm for simulating the BCH codes was also being investigated, which includes generating the parity check matrix, generating the message code in Galois array matrix, encoding the message blocks, modulation and decoding the message blocks. Algorithm for interleaving that includes interleaving message, including burst errors and deinterleaving message is combined with the BCH codes algorithm for simulating the interleaved BCH codes. The performance and feasibility of the coding structure are tested. The performance comparison between interleaved and noninterleaved BCH codes is studied in terms of error performance, channel performance and effect of data rates on the bit error rate (BER). The Berlekamp-Massey Algorithm decoding scheme was implemented. Random integers are generated and encoded with BCH encoder. Burst errors are added before the message is interleaved, then enter modulation and channel simulation. Interleaved message is then compared with noninterleaved message and the error statistics are compared. Initially, certain amount of burst errors is used. "ft is found that the graph does not agree with the theoretical bit error rate (BER) versus signal-to-noise ratio (SNR). When compared between each BCH codeword (i.e. n = 31, n = 63 and n = 127), n = 31 shows the highest BER while n = 127 shows the lowest BER. This happened because of the occurrence of error bursts and also due to error frequency. A reduced size or errors from previous is used in the algorithm. A graph similar to the theoretical BER vs SNR is obtained for both interleaved and non-interleaved BCH codes. It is found that BER of non-interleaved is higher than interleaved BCH codes as SNR increases. These observations show that size of errors influence the effect of interleaving. Simulation time is also studied in terms of block length. It is found that interleaved BCH codes consume longer simulation time compared to non-interleaved BCH codes due to additional algorithm for the interleaved BCH codes

Circulant Arrays on Cyclic Subgroups of Finite Fields: Rank Analysis and Construction of Quasi-Cyclic LDPC Codes

This paper consists of three parts. The first part presents a large class of new binary quasi-cyclic (QC)-LDPC codes with girth of at least 6 whose parity-check matrices are constructed based on cyclic subgroups of finite fields. Experimental results show that the codes constructed perform well over the binary-input AWGN channel with iterative decoding using the sum-product algorithm (SPA). The second part analyzes the ranks of the parity-check matrices of codes constructed based on finite fields with characteristic of 2 and gives combinatorial expressions for these ranks. The third part identifies a subclass of constructed QC-LDPC codes that have large minimum distances. Decoding of codes in this subclass with the SPA converges very fast.Comment: 26 pages, 6 figures, submitted to IEEE Transaction on Communication

Deterministic and Ensemble-Based Spatially-Coupled Product Codes

Several authors have proposed spatially-coupled (or convolutional-like) variants of product codes (PCs). In this paper, we focus on a parametrized family of generalized PCs that recovers some of these codes (e.g., staircase and block-wise braided codes) as special cases and study the iterative decoding performance over the binary erasure channel. Even though our code construction is deterministic (and not based on a randomized ensemble), we show that it is still possible to rigorously derive the density evolution (DE) equations that govern the asymptotic performance. The obtained DE equations are then compared to those for a related spatially-coupled PC ensemble. In particular, we show that there exists a family of (deterministic) braided codes that follows the same DE equation as the ensemble, for any spatial length and coupling width.Comment: accepted at ISIT 2016, Barcelona, Spai