Single Parity Check Product Codes Research Articles

Successive Cancellation Decoding of Single Parity-Check Product Codes: Analysis and Improved Decoding

A product code with single parity-check component codes can be described via the tools of a multi-kernel polar code, where the rows of the generator matrix are chosen according to the constraints imposed by the product code construction. Following this observation, successive cancellation decoding of such codes is introduced. In particular, the error probability of single parity-check product codes over binary memoryless symmetric channels under successive cancellation decoding is characterized. A bridge with the analysis of product codes introduced by Elias is also established for the binary erasure channel. Successive cancellation list decoding of single parity-check product codes is then described. For the provided example, simulations over the binary input additive white Gaussian channel show that successive cancellation list decoding outperforms belief propagation decoding applied to the code graph. Finally, the performance of the concatenation of a product code with a high-rate outer code is investigated via distance spectrum analysis. Examples of concatenations performing within 0.7 dB from the random coding union bound are provided.

On quantum SPC product codes

Some methods have been proposed to construct quantum codes out of classical codes. There is a relationship between quantum codes and codes on the quaternary field. The structure of CSS codes has been raised to construct quantum codes. Most of them are based on the fact that a classical code is self-orthogonal. In this paper, a quantum code is constructed using a suitable permutation-based technique on the parity check matrix of classical codes. Several examples of the new quantum error correcting codes are provided. Single parity check (SPC) product codes have simple decoding algorithms, as well as good decoding efficiency. With this technique, the SPC product codes are used to construct quantum codes.

Close-to-capacity performance of extended single parity check product codes

The empirical performances of extended single parity check product codes, which are very simple and of low complexity, on a binary input additive white Gaussian noise channel by a belief propagation (BP) decoding algorithm, are presented. Simulation results show that the codes can achieve close-to-capacity performance in high code rate. The codes approach the Shannon limit according to the increase of the code length. The code of length 97751 and of rate 0.985 is only 0.44 dB away from the Shannon limit for a BER of 10−5.

Extended Single Parity Check Product Codes that Achieve Close-to-Capacity Performance in High Coding Rate

This letter proposes extended single parity check product codes and presents their empirical performances on a Gaussian channel by belief propagation (BP) decoding algorithm. The simulation results show that the codes can achieve close-to-capacity performance in high coding rate. The code of length 9603 and of rate 0.96 is only 0.77dB away from the Shannon limit for a BER of 10-5.

Enhanced Upper Bound for Erasure Recovery in SPC Product Codes

Single parity check (SPC) product codes are simple yet powerful codes that are used to correct errors and/or recover erasures. The focus of this paper is to evaluate the performance of such codes under erasure scenarios and to develop a closed-form tight upper bound for the post-decoding erasure rate. Closed-form exact expressions are derived for up to seven erasures. Previously published closed-form bounds assumed that all unrecoverable patterns should contain four erasures in a square. Additional non-square patterns are accounted for in the proposed expressions. The derived expressions are verified using exhaustive search. Eight or more erasures are accounted for by using a bound. The developed expressions improve the evaluation of the recoverability of SPC product codes without the need for simulation or search algorithms, whether exhaustive or novel.

A New Linear Programming Approach to Decoding Linear Block Codes

In this paper, we propose a new linear programming formulation for the decoding of general linear block codes. Different from the original formulation given by Feldman, the number of total variables to characterize a parity-check constraint in our formulation is less than twice the degree of the corresponding check node. The equivalence between our new formulation and the original formulation is proven. The new formulation facilitates to characterize the structure of linear block codes, and leads to new decoding algorithms. In particular, we show that any fundamental polytope is simply the intersection of a group of the so-called minimum polytopes, and this simplified formulation allows us to formulate the problem of calculating the minimum Hamming distance of any linear block code as a simple linear integer programming problem with much less auxiliary variables. We then propose a branch-and-bound method to compute a lower bound to the minimum distance of any linear code by solving a corresponding linear integer programming problem. In addition, we prove that, for the family of single parity-check (SPC) product codes, the fractional distance and the pseudodistance are both equal to the minimum distance. Finally, we propose an efficient algorithm for decoding SPC product codes with low complexity and maximum-likelihood (ML) decoding performance.

KINERJA PERPADUAN DETEKSI DATA DAN ESTIMASI KANAL ITERATIF PADA SISTEM KOMUNIKASI NIRKABEL ANTENA JAMAK

In iterative data-detection and channel-estimation algorithms, the channel estimator and the data detector recursively exchange the information order to improve the system performance. In this paper, maximum a posteriori based iterative data detection and pilot symbol assisted channel estimation of the single parity check product code for multiple antenna wireless communication is studied. Results show the algorithm can converge at a few iteration numbers and improve error probability performance of the system.

Performance analysis of fiber-optic BPPM CDMA systems with single parity-check product codes

The application of a family of single parity-check (SPC) product codes in the fiber-optic code-division multiple-access (CDMA) systems with binary pulse position modulation (BPPM) is considered in this paper. We evaluate the bit-error probabilities of the coded systems, taking into account the photodetector noise, thermal noise, and the multiple-access interference. In this paper, the performance of the employed coding schemes over a binary-input output-symmetric channel is analyzed, based on the density evolution approach. The validity of the analytically obtained bit-error rates (BERs) over binary symmetric channels (BSCs) for each of the decoding iterations is verified by the system simulation. It is then shown that the application of these coding schemes in the optical BPPM-CDMA communication system, which can be modeled as a BSC, significantly improves the system performance. Particularly, for a certain BER (less than 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-7</sup> ) and bandwidth, the coded systems not only permit a higher number (more than twice) of active users, compared with the uncoded systems, but also can operate at higher channel bit rate (up to four times) with more than 10% energy saving. Furthermore, by increasing the number of code dimensions, less power (less than 40%) is required to achieve the desired performance

An improved new structure of single parity check product codes

This paper proposes an improved new structure of Single Parity Check (SPC) product codes with an embedded interleaver. The complexity of decoding the new SPC product codes is the same as the one of the existing single parity check product codes by use of the message passing algorithm. Simulation results and bound analysis demonstrate a significant bite error rate (BER) performance improvement compared to the existing single parity check product codes for high code rates at medium signal-to-noise ratios (SNRs).

Constant amplitude multi-code CDMA with built-in single parity check product code

We show that a constant amplitude multi-code (CAMC) CDMA in a recursive structure can be reconfigured into a single parity check, product code (SPCPC). CAMC can benefit from error correction by parity decoding. We present an algorithm which reconfigures CAMC into an SPCPC structure and then refines the log likelihood ratio of the received bits in an iterative way. Computer simulation results show that CAMC has a superior performance of error correction than a generic SPCPC. Hence, CAMC provides both a low PAPR and robustness to bit errors.

Iterative decoding of parallel and serial concatenated single parity check product codes

An efficient, iterative soft-in-soft-out decoding scheme is employed for the parallel and serially concatenated single parity check (SPC) product codes, which has very low complexity, requiring only two addition-equivalent-operations per information bit. For a rate 0.8637 of parallel concatenated SPC product code, a performance of BER=10−5 at Eb/N0=3.66 dB can be achieved using this decoding scheme, which is within 1 dB from the Shannon limit.

Parallel and Serial Concatenated Single Parity Check Product Codes

The parallel and serial concatenation of codes is well established as a practical means of achieving excellent performance. In this paper, we introduce the parallel and serial concatenation of single parity check (SPC) product codes. The weight distribution of these codes is analyzed and the performance is bounded. Simulation results confirm these bounds at high signal-to-noise ratios. The performance of these codes (and some variants) is shown to be quite good given the low decoding complexity and reasonably short blocklengths.

Recoverability study of SPC product codes under erasure decoding

Product codes are powerful codes that can be used to correct errors or recover erasures. The simplest form of a product code is that where every row and every column is terminated by a single parity bit referred to as single parity check (SPC) product code. This code has a minimum distance of four and is thus guaranteed to recover all single, double, and triple erasure patterns. Judging the code performance based on its minimum distance is very pessimistic because the code is actually capable of recovering many higher erasure patterns. Kousa [IEEE Trans. Commun. 50 (Jan) (2002) 7] develops a novel approach for deriving an upper bound on the post-decoding erasure rate for the SPC product code with iterative decoding. We derive a formula for finding the number of unrecoverable basic pattern and the number of recoverable pattern generated from the unrecoverable basic pattern. The results are compared with that of Kousa [IEEE Trans. Commun. 50 (Jan) (2002) 7].

Asymptotic performance of single parity-check product codes

This article investigates the asymptotic performance of single parity-check (SPC) product (PCs) from a decoding point of view. Specifically, the probability of bit error is bounded before and after the decoding of each dimension, similar to the analysis of iterated codes by Elias (1954). It is shown that the asymptotic probability of bit error can be driven to zero as the number of dimensions, and hence the block length, increases at signal-to-noise ratios (SNRs) within 2 dB of capacity over the additive white Gaussian noise (AWGN) channel.

A novel approach for evaluating the performance of SPC product codes under erasure decoding

Product codes are powerful codes that can be used to correct errors or recover erasures. The simplest form of a product code is that where every row and every column is terminated by a single parity bit, referred to as single parity check (SPC) product code. This code has a minimum distance of four and is thus guaranteed to recover all single, double, and triple erasure patterns. Judging the code performance based on its minimum distance is very pessimistic because the code is actually capable of recovering many higher erasure patterns. This paper develops a novel approach for deriving an upper bound on the post-decoding erasure rate for the SPC product code with iterative decoding. Simulation shows that the derived bound is very tight.

Single parity check product codes

This paper considers the performance of iteratively decoded single parity check (SPC) multidimensional product codes in an additive white Gaussian noise channel. Asymptotic performance bounds are compared to simulation results. A new code structure based on SPC product codes is introduced. This structure involves interleaving between the encoding of each dimension in the product code. An analysis of the weight distribution is used to explain the good performance results for these randomly interleaved SPC product codes.

Evaluation of the Undetected Error Probability of Single Parity-Check Product Codes

In this paper, an expression for the undetected error probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(P\epsilon)</tex> of single parity-check product (SPCP) codes used for error detection over a binary symmetric channel is derived. It is shown that square SPCP codes need not obey a certain commonly used bound. Approximate expressions for the maximum <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(P\epsilon)</tex> and the corresponding maximizing ε are given.