Sinan Kahraman

Code based efficient maximum-likelihood decoding of short polar codes

Polar codes are known as the first provable code construction to achieve Shannon capacity for arbitrary symmetric binary-input channels. Although, there exist efficient sub-optimal decoders with reduced complexity for polar codes, the complexity of the optimum ML decoder increases exponentially. Hence the optimum decoder is infeasible for the practical implementation of polar coding. In this paper, our motivation is about developing efficient ML decoder with reduced complexity. In this purpose, polar code based sphere decoding algorithm is proposed with the optimal performance. Additionally, proposed technique exploits two properties of polar coding to reduce decoding complexity. By this way, the reduced complexity of optimal decoding is only cubic, not exponential.

Folded tree maximum-likelihood decoder for Kronecker product-based codes

In this paper, we propose efficient maximum-likelihood (ML) decoding for binary Kronecker product-based (KPB) codes. This class of codes, have a matrix defined by the n-fold iterated Kronecker product Gn = F⊗n of a binary upper-triangular kernel matrix F, where some columns are suppressed given a specific puncturing pattern. Polar and Reed- Muller codes are well known examples of such KPB codes. The triangular structure of Gn enables to perform ML decoding as a binary tree search for the closest codeword to the received point. We take advantage of the highly regular fractal structure of Gn and the “tree folding” technique to design an efficient ML decoder, enabling to decode relatively longer block lengths than with a standard binary tree search. The tree κ-folding operation transforms the binary tree with N levels into a non-binary tree with N=2κ levels, where the search can be significantly accelerated by a suitable ordering of the branch metrics. For a given κ we can find (n over κ) different folding which lead to decoders with different complexity, for a given code. Using the proposed folded tree decoder, we provide exact ML performances of some Reed-Muller and polar codes over a binary AWGN channel for the block length up to 256.

Dimensionality reduction for the golden code with worst-case decoding complexity of O(m 2 )

In this paper we introduce an efficient decoding method which is based on the dimensionality reduction of the sphere decoder search tree for the golden code. A codeword of the golden code has four independent m-QAM data symbols, hence, the required complexity of the exhaustive-search decoder is m4. An efficient implementation of the maximum-likelihood decoder for the golden code with a worst-case complexity is known to be proportional to m2.5. Our motivation is for an efficient decoder with a worst-case complexity of no more than m2.5. In this purpose, we show that our proposed method has m2 complexity in the worst-case with a loss of only 1 dB with respect to optimal decoding.

Folded successive cancelation decoding of polar codes

Polar codes are the first explicit class of codes that are provably capacity-achieving under the successive cancelation (SC) decoding. As a suboptimal decoder, SC has quasi-linear complexity N(1 + log N) in the code length N. In this paper, we propose a new non-binary SC decoder with reduced complexity N/2(1 + log N/2) based on the folding operation, which was first proposed in [11] to implement folded tree maximum-likelihood decoding of polar codes. Simulation results for the additive white Gaussian noise channel show that folded SC decoders can achieve the same error performance of standard SC by suitable selecting the folding of the polar code.

Multiple Folding for Successive Cancelation Decoding of Polar Codes

Polar coding is known as the first provably capacity-achieving coding scheme under low-complexity suboptimal successive cancelation decoding (SCD). The large error-correction capability of finite-length polar codes is mostly achieved with relatively long codes. SCD is the conventional decoder for polar codes and exhibits a quasi-linear complexity in terms of the code length. Practical decoder schemes with low latency are important for high-speed polar coding applications. In this letter, we propose a nonbinary multiple folded SCD scheme to reduce the decoding latency of standard binary polar codes. Multiple foldings were first proposed to improve the efficiency of folded tree maximum-likelihood decoder for Kronecker product-based codes. By successively applying the folding operation κ times on the SCD, for a code length N, the latency is reduced from 2N - 1 to (N/2κ-1) - 1 time slots, assuming full parallelization. We show that multiple folded SCD can be effectively implemented for up to κ = 3 foldings due to memory limitations. This decoder achieves exactly the same performance of the original SCD with significantly reduced latency.

Fast maximum-likelihood decoding for BLAST systems: Decomposed Matrix Structure technique

In this paper, we propose a technique which is efficient in reducing the required computational complexity of the maximum-likelihood decoding for BLAST system. Our technique is based on the determination of reducible computational complexity for any given BLAST system. In this fashion a specialized decoding can be defined. Our idea is based on the determination of zero entries of the upper-triangular matrix R which occurs in QR-decomposition step of the sphere decoding algorithm. Through the determined zero at kth row of the matrix R, at least 8 arithmetic operations can be canceled for every time of node visiting in the search tree at level k. Furthermore, properties of the matrix structure can be exploited in sphere decoding in order to reduce complexity of the metric computations for each branch. For this case, considerable saving in the computational complexity can be obtained.

A New Code Design Criterion for Linear Dispersive Space-Time Block Codes

In this study, we present a new code design criterion for space-time block codes. Using the proposed criterion, it is possible to determine linear dispersive space-time block codes that are able to work in high rates such as R = 8 or 16 bit/channel use. These new codes differ from previously proposed linear dispersed codes by the simplicity of the transmitter design, in that, every antenna transmits strictly from a QAM constellation. Furthermore, the criterion aims to maximize the channel capacity by the help of crucial equation of number of symbols = number of channel uses X min(number of transmit antennas.number of receive antennas) and to minimize the bit-error probability at the same time. The minimization of symbol-error probability is achieved by the minimization of bit-error probability. We also produced some simulation results for comparison of the performances of various linear dispersive codes to strengthen our claims.

Dimensionality reduced decoding for the golden code with the worst-case complexity of O(m 1.5 ) for low range of SNR

In this paper we introduce an efficient decoding method which is based on the dimensionality reduction of the search tree in the sphere decoder for the golden code in a low SNR regime. A codeword of the golden code has four independent m-QAM data symbols, hence, the required complexity of the exhaustive-search decoder is m4. An efficient implementation of the maximum-likelihood decoder for the golden code with a worst-case complexity is known to be proportional to m2.5. Additionally, in low range of SNR, sphere decoding has significantly high expected decoding complexity. Our motivation is for an efficient decoder with a worst-case complexity of no more than m2 for a low SNR regime. In this purpose, we show that our proposed method has m1.5 complexity in the worst-case with a loss of only 1 dB with respect to optimal decoding.

Fast decoding for silver codes

In this work, we introduce a method for fast decoding of silver codes that is dimensionality reduced method on search tree of the sphere decoder. In the recent works, it is known that the fast decoding method for the silver code has the worst-case complexity of m2. By the use of proposed method, silver code has only m1.5 decoding complexity in worst-case. In this case, the error performance of the proposed method for the silver codes has limited loss about 1 dB in the receiver unit.

On fast ML decoding for high-rate STBCs by dimensionality reduction

In this work, a new fast decoding technique that is based on dimensionality reduction for the search tree is presented for the high-rate space-time block coding. By the use of the proposed technique, the high-rate Golden codes and various WiMAX codes can be efficiently decoded with the optimal ML detection. In the sphere decoding algorithm, 8 dimensional search tree for 2 × 2 Golden code can be separated into 2 semi dependent, 6 dimensional sub trees. For the recommended X4 code in the WiMAX systems, 4 semi dependent, 5 dimensional sub trees can be obtained. By the proposed method, the computational complexity of the sphere decoding can be reduced by about %80 and the speed of the decoding can be increased about 5 times.