The concept of Generalized Inverse based Decoding (GID) is introduced, as an algebraic framework for the syndrome decoding problem (SDP) and low weight codeword problem (LWP). The framework has ground on two characterizations by generalized inverses (GIs), one for the null space of a matrix and the other for the solution space of a system of linear equations over a finite field. Generic GID solvers are proposed for SDP and LWP. It is shown that information set decoding (ISD) algorithms, such as Prange, Lee-Brickell, Leon, and Stern's algorithms, are particular cases of GID solvers. All of them search GIs or elements of the null space under various specific strategies. However, as the paper shows the ISD variants do not search through the entire space, while our solvers do even when they use just one Gaussian elimination. Apart from these, our GID framework clearly shows how each ISD algorithm, except for Prange's solution, can be used as an SDP or LWP solver. A tight reduction from our problems, viewed as optimization problems, to the MIN-SAT problem is also provided. Experimental results show a very good behavior of the GID solvers. The domain of easy weights can be reached by a very few iterations and even enlarged.
This paper presents a method to merge Generalized Minimum Distance decoding of Reed-Solomon codes with the extended Euclidean algorithm. By merge, we mean that the steps taken to perform the Generalized Minimum Distance decoding are similar to those performed by the extended Euclidean algorithm. The resulting algorithm has a complexity of O(n^2).
We propose a binary message passing decoding algorithm for product codes based on generalized minimum distance decoding (GMDD) of the component codes, where the last stage of the GMDD makes a decision based on the Hamming distance metric. The proposed algorithm closes half of the gap between conventional iterative bounded distance decoding (iBDD) and turbo product decoding based on the Chase--Pyndiah algorithm, at the expense of some increase in complexity. Furthermore, the proposed algorithm entails only a limited increase in data flow compared to iBDD.
The paper introduces the simultaneous partial-inverse problem (SPI) for polynomials and develops its application to decoding interleaved Reed--Solomon codes beyond half the minimum distance. While closely related both to standard key equations and to well-known Padé approximation problems, the SPI problem stands out in several respects. First, the SPI problem has a unique solution (up to a scale factor), which satisfies a natural degree bound. Second, the SPI problem can be transformed (monomialized) into an equivalent SPI problem where all moduli are monomials. Third, the SPI problem can be s…
Generalized concatenated codes were introduced in the 1970s by Zinoviev. There are many types of codes in the literature that are known by other names that can be viewed as generalized concatenated codes. Examples include matrix-product codes, multilevel codes and generalized cascade codes. Decoding algorithms for generalized concatenated codes were developed during the 1970s and 1980s. However, their use does not appear to be as widespread as it should, especially for codes that are known by other names but can be viewed as generalized concatenated codes. In this paper we review the decoding …
It is well known that for Gaussian channels, a nearest neighbor decoding rule, which seeks the minimum Euclidean distance between a codeword and the received channel output vector, is the maximum likelihood solution and hence capacity-achieving. Nearest neighbor decoding remains a convenient and yet mismatched solution for general channels, and the key message of this paper is that the performance of the nearest neighbor decoding can be improved by generalizing its decoding metric to incorporate channel state dependent output processing and codeword scaling. Using generalized mutual informatio…
Research on polar codes has been constantly gaining attention over the last decade, by academia and industry alike, thanks to their capacity-achieving error-correction performance and low-complexity decoding algorithms. Recently, they have been selected as one of the coding schemes in the $5^{th}$ generation wireless standard (5G). Over the years various polar code decoding algorithms, like SC-list (SCL), have been proposed to improve the mediocre performance of the successive cancellation (SC) decoding algorithm for finite code lengths; however, like SC, they suffer from long decoding latency…
Generalized Goppa codes are defined by a code locator set $\mathcal{L}$ of polynomials and a Goppa polynomial $G(x)$. When the degree of all code locator polynomials in $\mathcal{L}$ is one, generalized Goppa codes are classical Goppa codes. In this work, binary generalized Goppa codes are investigated. First, a parity-check matrix for these codes with code locators of any degree is derived. A careful selection of the code locators leads to a lower bound on the minimum Hamming distance of generalized Goppa codes which improves upon previously known bounds. A quadratic-time decoding algorithm i…
Information transmission over a multiple-input-multiple-output (MIMO) fading channel with imperfect channel state information (CSI) is investigated, under a new receiver architecture which combines the recently proposed generalized nearest neighbor decoding rule (GNNDR) and a successive procedure in the spirit of successive interference cancellation (SIC). Recognizing that the channel input-output relationship is a nonlinear mapping under imperfect CSI, the GNNDR is capable of extracting the information embedded in the joint observation of channel output and imperfect CSI more efficiently than…
The key step of syndrome-based decoding of Reed-Solomon codes up to half the minimum distance is to solve the so-called Key Equation. List decoding algorithms, capable of decoding beyond half the minimum distance, are based on interpolation and factorization of multivariate polynomials. This article provides a link between syndrome-based decoding approaches based on Key Equations and the interpolation-based list decoding algorithms of Guruswami and Sudan for Reed-Solomon codes. The original interpolation conditions of Guruswami and Sudan for Reed-Solomon codes are reformulated in terms of a se…
Using the EXIT chart approach, a necessary and sufficient condition is developed for the local stability of iterative decoding of multi-edge type (MET) doubly-generalized low-density parity-check (D-GLDPC) code ensembles. In such code ensembles, the use of arbitrary linear block codes as component codes is combined with the further design of local Tanner graph connectivity through the use of multiple edge types. The stability condition for these code ensembles is shown to be succinctly described in terms of the value of the spectral radius of an appropriately defined polynomial matrix.
The problem of designing good Space-Time Block Codes (STBCs) with low maximum-likelihood (ML) decoding complexity has gathered much attention in the literature. All the known low ML decoding complexity techniques utilize the same approach of exploiting either the multigroup decodable or the fast-decodable (conditionally multigroup decodable) structure of a code. We refer to this well known technique of decoding STBCs as Conditional ML (CML) decoding. In this paper we introduce a new framework to construct ML decoders for STBCs based on the Generalized Distributive Law (GDL) and the Factor-grap…
We consider the Additive White Gaussian Noise channel with Binary Phase Shift Keying modulation. Our aim is to enable an algebraic hard decision Bounded Minimum Distance decoder for a binary block code to exploit soft information obtained from the demodulator. This idea goes back to Forney and is based on treating received symbols with low reliability as erasures. This erasing at the decoder is done using a threshold, each received symbol with reliability falling below the threshold is erased. Depending on the target overall complexity of the decoder this pseudo-soft decision decoding can be e…
In this paper an interpolation-based decoding algorithm to decode Gabidulin codes, transmitted through a finely restricted channel, is proposed. The algorithm is able to decode rank errors beyond half the minimum distance by one unit. Also the existing decoding algorithms for generalized twisted Gabidulin codes and additive generalized twisted Gabidulin codes are improved.
Families of generalized spatially-coupled low-density parity-check (GSC-LDPC) code ensembles can be formed by terminating protograph-based generalized LDPC convolutional (GLDPCC) codes. It has previously been shown that ensembles of GSC-LDPC codes constructed from a protograph have better iterative decoding thresholds than their block code counterparts, and that, for large termination lengths, their thresholds coincide with the maximum a-posteriori (MAP) decoding threshold of the underlying generalized LDPC block code ensemble. Here we show that, in addition to their excellent iterative decodi…
Iterative decoding techniques have gain popularity due to their performance and their application in most communications systems. In this paper, we present a new application of our iterative decoder on the GPCB (Generalized Parallel Concatenated Block codes) which uses cyclic permutations. We introduce a new variant of the component decoder. After extensive simulation; the obtained result is very promising compared with several existing methods. We evaluate the effects of various parameters component codes, interleaver size, block size, and the number of iterations. Three interesting results a…
Topological subsystem codes can combine the advantages of both topological codes and subsystem codes. Suchara et al. proposed a framework based on hypergraphs for construction of such codes. They also studied the performance of some subsystem codes. Later Bravyi et al. proposed a subsystem surface code. Building upon these works, we propose efficient decoding algorithms for large classes of subsystem codes on hypergraphs and surfaces. We also propose a construction of the subsystem surface codes that includes the code proposed by Bravyi et al. Our simulations for the subsystem code on the squa…
In this paper, we analyze the tradeoff between coding rate and asymptotic performance of a class of generalized low-density parity-check (GLDPC) codes constructed by including a certain fraction of generalized constraint (GC) nodes in the graph. The rate of the GLDPC ensemble is bounded using classical results on linear block codes, namely Hamming bound and Varshamov bound. We also study the impact of the decoding method used at GC nodes. To incorporate both bounded-distance (BD) and Maximum Likelihood (ML) decoding at GC nodes into our analysis without resorting on multi-edge type of degree d…
Product codes (PCs) protect a two-dimensional array of bits using short component codes. Assuming transmission over the binary symmetric channel, the decoding is commonly performed by iteratively applying bounded-distance decoding to the component codes. For this coding scheme, undetected errors in the component decoding-also known as miscorrections-significantly degrade the performance. In this paper, we propose a novel iterative decoding algorithm for PCs which can detect and avoid most miscorrections. The algorithm can also be used to decode many recently proposed classes of generalized PCs…
In this paper, we propose a framework of the mutual information-maximizing (MIM) quantized decoding for low-density parity-check (LDPC) codes by using simple mappings and fixed-point additions. Our decoding method is generic in the sense that it can be applied to LDPC codes with arbitrary degree distributions, and can be implemented based on either the belief propagation (BP) algorithm or the min-sum (MS) algorithm. In particular, we propose the MIM density evolution (MIM-DE) to construct the lookup tables (LUTs) for the node updates. The computational complexity and memory requirements are di…
This paper considers shifted inverse determinant sums arising from the union bound of the pairwise error probability for space-time codes in multiple-antenna fading channels. Previous work by Vehkalahti et al. focused on the approximation of these sums for low multiplexing gains, providing a complete classification of the inverse determinant sums as a function of constellation size for the most well-known algebraic space-time codes. This work aims at building a general framework for the study of the shifted sums for all multiplexing gains. New bounds obtained using dyadic summing techniques su…
For an $n_t$ transmit, $n_r$ receive antenna system ($n_t \times n_r$ system), a {\it{full-rate}} space time block code (STBC) transmits $n_{min} = min(n_t,n_r)$ complex symbols per channel use. The well known Golden code is an example of a full-rate, full-diversity STBC for 2 transmit antennas. Its ML-decoding complexity is of the order of $M^{2.5}$ for square $M$-QAM. The Silver code for 2 transmit antennas has all the desirable properties of the Golden code except its coding gain, but offers lower ML-decoding complexity of the order of $M^2$. Importantly, the slight loss in coding gain is n…
This is the first paper of a two-long series in which we study linear generalized inverses that minimize matrix norms. Such generalized inverses are famously represented by the Moore-Penrose pseudoinverse (MPP) which happens to minimize the Frobenius norm. Freeing up the degrees of freedom associated with Frobenius optimality enables us to promote other interesting properties. In this Part I, we look at the basic properties of norm-minimizing generalized inverses, especially in terms of uniqueness and relation to the MPP. We first show that the MPP minimizes many norms beyond those unitarily i…
In this paper, we propose a novel low complexity scaling strategy of min-sum decoding algorithm for irregular LDPC codes. In the proposed method, we generalize our previously proposed simplified Variable Scaled Min-Sum (SVS-min-sum) by replacing the sub-optimal starting value and heuristic update for the scaling factor sequence by optimized values. Density evolution and Nelder-Mead optimization are used offline, prior to the decoding, to obtain the optimal starting point and per iteration updating step size for the scaling factor sequence of the proposed scaling strategy. The optimization of t…
This paper concerns itself with the question of list decoding for general adversarial channels, e.g., bit-flip ($\textsf{XOR}$) channels, erasure channels, $\textsf{AND}$ ($Z$-) channels, $\textsf{OR}$ channels, real adder channels, noisy typewriter channels, etc. We precisely characterize when exponential-sized (or positive rate) $(L-1)$-list decodable codes (where the list size $L$ is a universal constant) exist for such channels. Our criterion asserts that: "For any given general adversarial channel, it is possible to construct positive rate $(L-1)$-list decodable codes if and only if the s…
We study a two-user state-dependent generalized multiple-access channel (GMAC) with correlated states. It is assumed that each encoder has \emph{noncausal} access to channel state information (CSI). We develop an achievable rate region by employing rate-splitting, block Markov encoding, Gelfand--Pinsker multicoding, superposition coding and joint typicality decoding. In the proposed scheme, the encoders use a partial decoding strategy to collaborate in the next block, and the receiver uses a backward decoding strategy with joint unique decoding at each stage. Our achievable rate region include…
Block-fading channel (BF) is a useful model for various wireless communication channels in both indoor and outdoor environments. The design of lattices for BF channels offers a challenging problem, which differs greatly from its counterparts like AWGN channels. Recently, the original binary Construction A for lattices, due to Forney, has been generalized to a lattice construction from totally real and complex multiplication (CM) fields. This generalized algebraic Construction A of lattices provides signal space diversity, intrinsically, which is the main requirement for the signal sets designe…
We generalise Gabidulin codes to the case of infinite fields, eventually with characteristic zero. For this purpose, we consider an abstract field extension and any automorphism in the Galois group. We derive some conditions on the automorphism to be able to have a proper notion of rank metric which is in coherence with linearized polynomials. Under these conditions, we generalize Gabidulin codes and provide a decoding algorithm which decode both errors and erasures. Then, we focus on codes over integer rings and how to decode them. We are then faced with the problem of the exponential growth …
Reducing computational complexity of the modern wireless communication systems such as massive Multiple-Input Multiple-Output (MIMO) configurations is of utmost interest. In this paper, we propose new algorithm that can be used to accelerate matrix inversion in the decoding of space-time block codes (STBC) in the uplink of dynamic massive MIMO systems. A multi-user system in which the base station is equipped with a large number of antennas and each user has two antennas is considered. In addition, users can enter or exit the system dynamically. For a given space-time block coding/decoding sch…
