Arvind Sridharan

LDPC block and convolutional codes based on circulant matrices

A class of algebraically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes and their convolutional counterparts is presented. The QC codes are described by sparse parity-check matrices comprised of blocks of circulant matrices. The sparse parity-check representation allows for practical graph-based iterative message-passing decoding. Based on the algebraic structure, bounds on the girth and minimum distance of the codes are found, and several possible encoding techniques are described. The performance of the QC LDPC block codes compares favorably with that of randomly constructed LDPC codes for short to moderate block lengths. The performance of the LDPC convolutional codes is superior to that of the QC codes on which they are based; this performance is the limiting performance obtained by increasing the circulant size of the base QC code. Finally, a continuous decoding procedure for the LDPC convolutional codes is described.

Iterative Decoding Threshold Analysis for LDPC Convolutional Codes

An iterative decoding threshold analysis for terminated regular LDPC convolutional (LDPCC) codes is presented. Using density evolution techniques, the convergence behavior of an iterative belief propagation decoder is analyzed for the binary erasure channel and the AWGN channel with binary inputs. It is shown that for a terminated LDPCC code ensemble, the thresholds are better than for corresponding regular and irregular LDPC block codes.

Implementation aspects of LDPC convolutional codes

Potentially large storage requirements and long initial decoding delays are two practical issues related to the decoding of low-density parity-check (LDPC) convolutional codes using a continuous pipeline decoder architecture. In this paper, we propose several reduced complexity decoding strategies to lessen the storage requirements and the initial decoding delay without significant loss in performance. We also provide bit error rate comparisons of LDPC block and LDPC convolutional codes under equal processor (hardware) complexity and equal decoding delay assumptions. A partial syndrome encoder realization for LDPC convolutional codes is also proposed and analyzed. We construct terminated LDPC convolutional codes that are suitable for block transmission over a wide range of frame lengths. Simulation results show that, for terminated LDPC convolutional codes of sufficiently large memory, performance can be improved by increasing the density of the syndrome former matrix.

Terminated LDPC convolutional codes with thresholds close to capacity

An ensemble of LDPC convolutional codes with parity-check matrices composed of permutation matrices is considered. The convergence of the iterative belief propagation based decoder for terminated convolutional codes in the ensemble is analyzed for binary-input output-symmetric memoryless channels using density evolution techniques. We observe that the structured irregularity in the Tanner graph of the codes leads to significantly better thresholds when compared to corresponding LDPC block codes

A construction for low density parity check convolutional codes based on quasi-cyclic block codes

A set of convolutional codes with low density parity check matrices is derived from a class of quasi-cyclic low density parity check block codes. Their performance when decoded using the belief propagation algorithm is investigated.

A new construction for low density parity check convolutional codes

Low density parity check (LDPC) block codes have been shown to achieve near capacity performance for binary transmission over noisy channels. Block codes, however, require splitting the data to be transmitted into frames, which can be a disadvantage in some applications. Convolutional codes, on the other hand, have no such requirement, and are well suited for continuous transmission. Felstrom and Zigangirov (1999) proposed the construction of periodic time-varying convolutional codes with LDPC matrices. A set of time-invariant LDPC convolutional codes was described by Sridharan et al. (2002). The codes of Felstrom and Zigangirov were obtained by random construction techniques whereas those of Sridharan et al. were essentially algebraic constructions. The new LDPC convolutional codes described here are obtained by introducing a degree of randomness into the latter construction.

On the minimum distance of low-density parity-check codes with parity-check matrices constructed from permutation matrices

An ensemble of codes defined by parity-check matrices composed of M × M permutation matrices is considered. This ensemble is a subensemble of the ensemble of low-density parity-check (LDPC) codes considered by Gallager [1]. We prove that, as M → ∞, the minimum distance of almost all codes in the ensemble grows linearly with M. We also show that in several cases the asymptotic minimum-distance-to-block-length ratio for almost all codes in the ensemble satisfies Gallager’s bound [1].

A construction for irregular low density parity check convolutional codes

A technique for constructing irregular low density parity check convolutional codes is described. The constructed codes exhibit lower convergence thresholds with belief propagation decoding than their regular counterparts.

On the free distance of LDPC convolutional codes

A lower bound on the free distance of LDPC convolutional codes defined by syndrome former matrices comprised of MtimesM permutation matrices is derived. We show that asymptotically, i.e., as Mrarrinfin, for almost all codes in the ensemble the free distance grows linearly with constraint length

Low Density Parity Check Convolutional Codes Derived from Quasi-Cyclic Block Codes

Using algebraic techniques, one of the co-authors has designed a [155,64, 20] low density parity check code based on permutation matrices. This code is quasi-cyclic by construction and hence admits a convolutional representation. A set of low density parity check convolutional codes is derived from this quasi-cyclic code and its generalizations. The performance of these convolutional codes is investigated when decoded using belief propagation on their corresponding graph representations.