A non-binary (2,v)-regular LDPC code is defined by a parity-check matrix with column weight 2 and row weight v. In this report, we give an ontology-based approach to the optimization for this class of codes. All possible inter-connected cycle patterns that lead to low symbol-weight codewords are identified to put together the ontology. The optimization goal is to improve the distance property of equivalent binary images. Using the proposed method, the estimation and optimization of bit-distance spectrum becomes easily handleable. Three codes in the CCSDS recommendation are analyzed and several codes with good minimum bit-distance are designed.
In this paper asymptotic methods are used to form lower bounds on the free distance to constraint length ratio of several ensembles of regular, asymptotically good, protograph-based LDPC convolutional codes. In particular, we show that the free distance to constraint length ratio of the regular LDPC convolutional codes exceeds that of the minimum distance to block length ratio of the corresponding LDPC block codes.
A solution is presented for the asymptotic growth rate of the AWGN-pseudoweight distribution of regular low-density parity-check (LDPC) code ensembles for a selected graph cover degree M >= 1. The evaluation of the growth rate requires solution of a system of 2M+1 nonlinear equations in 2M+1 unknowns. Simulation results for the pseudoweight distribution of two regular LDPC code ensembles are presented for graph covers of low degree.
We estimate the variance of weight and stopping set distribution of regular LDPC ensembles. Using this estimate and the second moment method we obtain bounds on the probability that a randomly chosen code from regular LDPC ensemble has its weight distribution and stopping set distribution close to respective ensemble averages. We are able to show that a large fraction of total number of codes have their weight and stopping set distribution close to the average.
The error performance of the ensemble of typical LDPC codes transmitted over the binary erasure channel (BEC) is analyzed. In the past, lower bounds on the error exponents were derived. In this paper a probabilistic upper bound on this error exponent is derived. This bound holds with some confidence level.
This paper is concerned with the construction of low-density parity-check (LDPC) codes with low error floors. Two main contributions are made. First, a new class of structured LDPC codes is introduced. The parity check matrices of these codes are arrays of permutation matrices which are obtained from Latin squares and form a finite field under some matrix operations. Second, a method to construct LDPC codes with low error floors on the binary symmetric channel (BSC) is presented. Codes are constructed so that their Tanner graphs are free of certain small trapping sets. These trapping sets are …
Families of "asymptotically regular" LDPC block code ensembles can be formed by terminating (J,K)-regular protograph-based LDPC convolutional codes. By varying the termination length, we obtain a large selection of LDPC block code ensembles with varying code rates and substantially better iterative decoding thresholds than those of (J,K)-regular LDPC block code ensembles, despite the fact that the terminated ensembles are almost regular. Also, by means of an asymptotic weight enumerator analysis, we show that minimum distance grows linearly with block length for all of the ensembles in these f…
Families of "asymptotically regular" LDPC block code ensembles can be formed by terminating (J,K)-regular protograph-based LDPC convolutional codes. By varying the termination length, we obtain a large selection of LDPC block code ensembles with varying code rates, minimum distance that grows linearly with block length, and capacity approaching iterative decoding thresholds, despite the fact that the terminated ensembles are almost regular. In this paper, we investigate the properties of the quasi-cyclic (QC) members of such an ensemble. We show that an upper bound on the minimum Hamming dista…
In this paper a new message passing algorithm, which takes advantage of both tree-based re-parameterization and the knowledge of short cycles, is introduced for the purpose of decoding LDPC codes with short block lengths. The proposed algorithm is called variable factor appearance probability belief propagation (VFAP-BP) algorithm and is suitable for wireless communications applications, where both good decoding performance and low-latency are expected. Our simulation results show that the VFAP-BP algorithm outperforms the standard BP algorithm and requires a significantly smaller number of it…
For a high-rate case, it is difficult to randomly construct good low-density parity-check (LDPC) codes of short and moderate lengths because their Tanner graphs are prone to making short cycles. Also, the existing high-rate quasi-cyclic (QC) LDPC codes can be constructed only for very restricted code parameters. In this paper, a new construction method of high-rate regular QC LDPC codes with parity-check matrices consisting of a single row of circulants with the column-weight 3 or 4 is proposed based on special classes of cyclic difference families. The proposed QC LDPC codes can be constructe…
In this paper we give lower bounds on the size of $(a,b)$ elementary trapping sets (ETSs) belonging to variable-regular LDPC codes with any girth, $g$, and irregular ones with girth 8, where $a$ is the size, $b$ is the number of degree-one check nodes and satisfy the inequality $\frac{b}{a}<1$. Our proposed lower bounds are analytical, rather than exhaustive search-based, and based on graph theories. The numerical results in the literarture for $g=6,8$ for variable-regular LDPC codes match our results. Some of our investigations are independent of the girth and rely on the variables $a$, $b$ a…
We study error bounds for linear programming decoding of regular LDPC codes. For memoryless binary-input output-symmetric channels, we prove bounds on the word error probability that are inverse doubly-exponential in the girth of the factor graph. For memoryless binary-input AWGN channel, we prove lower bounds on the threshold for regular LDPC codes whose factor graphs have logarithmic girth under LP-decoding. Specifically, we prove a lower bound of $σ=0.735$ (upper bound of $\frac{Eb}{N_0}=2.67$dB) on the threshold of $(3,6)$-regular LDPC codes whose factor graphs have logarithmic girth. Our …
It is shown that dominant trapping sets of regular LDPC codes, so called absorption sets, undergo a two-phased dynamic behavior in the iterative message-passing decoding algorithm. Using a linear dynamic model for the iteration behavior of these sets, it is shown that they undergo an initial geometric growth phase which stabilizes in a final bit-flipping behavior where the algorithm reaches a fixed point. This analysis is shown to lead to very accurate numerical calculations of the error floor bit error rates down to error rates that are inaccessible by simulation. The topology of the dominant…
An elementary combinatorial Tanner graph construction for a family of near-regular low density parity check codes achieving high girth is presented. The construction allows flexibility in the choice of design parameters like rate, average degree, girth and block length of the code and yields an asymptotic family. The complexity of constructing codes in the family grows only quadratically with the block length.
The covariance evolution is a system of differential equations with respect to the covariance of the number of edges connecting to the nodes of each residual degree. Solving the covariance evolution, we can derive distributions of the number of check nodes of residual degree 1, which helps us to estimate the block error probability for finite-length LDPC code. Amraoui et al.\ resorted to numerical computations to solve the covariance evolution. In this paper, we give the analytical solution of the covariance evolution.
We study LDPC codes for the channel with input ${x}\in \mathbb{F}_q^m$ and output ${y}={x}+{z}\in \mathbb{F}_q^m$. The aim of this paper is to evaluate decoding performance of $q^m$-ary non-binary LDPC codes for large $m$. We give density evolution and decoding performance evaluation for regular non-binary LDPC codes and spatially-coupled (SC) codes. We show the regular codes do not achieve the capacity of the channel while SC codes do.
We analyze the asymptotic performance of nonbinary spatially-coupled low-density parity-check (SC-LDPC) codes built on the general linear group, when the transmission takes place over the binary erasure channel. We propose an efficient method to derive an upper bound to the maximum a posteriori probability (MAP) threshold for nonbinary LDPC codes, and observe that the MAP performance of regular LDPC codes improves with the alphabet size. We then consider nonbinary SC-LDPC codes. We show that the same threshold saturation effect experienced by binary SC-LDPC codes occurs for the nonbinary codes…
In this paper, we propose a class of finite alphabet iterative decoder (FAID), called mutual information-maximizing quantized belief propagation (MIM-QBP) decoder, for decoding regular low-density parity-check (LDPC) codes. Our decoder follows the reconstruction-calculation-quantization (RCQ) decoding architecture that is widely used in FAIDs. We present the first complete and systematic design framework for the RCQ parameters, and prove that our design with sufficient precision at node update is able to maximize the mutual information between coded bits and exchanged messages. Simulation resu…
Motivated by recently derived fundamental limits on total (transmit + decoding) power for coded communication with VLSI decoders, this paper investigates the scaling behavior of the minimum total power needed to communicate over AWGN channels as the target bit-error-probability tends to zero. We focus on regular-LDPC codes and iterative message-passing decoders. We analyze scaling behavior under two VLSI complexity models of decoding. One model abstracts power consumed in processing elements ("node model"), and another abstracts power consumed in wires which connect the processing elements ("w…
In this paper, we define two matrices named as "difference matrices", denoted by $D$ and $DD$ which significantly contribute to achieve regular single-edge QC-LDPC codes with the shortest length and the certain girth as well as regular and irregular multiple-edge QC-LDPC codes. Making use of these matrices, we obtain necessary and sufficient conditions to have single-edge $(m,n)$-regular QC-LDPC codes with girth 6, 10 and 12. Additionally, for girth 6, we achieve all non-isomorphic codes with the minimum lifting degree, $N$, for $m=4$ and $5\leq n\leq 11$, and present an exponent matrix for ea…
In this paper we design non-uniform bit-wise puncturing distributions for irregular non-binary LDPC (NB-LDPC) codes. The puncturing distributions are optimized by minimizing the decoding threshold of the punctured LDPC code, the threshold being computed with a Monte-Carlo implementation of Density Evolution. First, we show that Density Evolution computed with Monte-Carlo simulations provides accurate (very close) and precise (small variance) estimates of NB-LDPC code ensemble thresholds. Based on the proposed method, we analyze several puncturing distributions for regular and semi-regular code…
In his thesis, Wiberg showed the existence of thresholds for families of regular low-density parity-check codes under min-sum algorithm decoding. He also derived analytic bounds on these thresholds. In this paper, we formulate similar results for linear programming decoding of regular low-density parity-check codes.
We adapt a construction of Guth and Lubotzky [arXiv:1310.5555] to obtain a family of quantum LDPC codes with non-vanishing rate and minimum distance scaling like $n^{0.1}$ where $n$ is the number of physical qubits. Similarly as in [arXiv:1310.5555], our homological code family stems from hyperbolic 4-manifolds equipped with tessellations. The main novelty of this work is that we consider a regular tessellation consisting of hypercubes. We exploit this strong local structure to design and analyze an efficient decoding algorithm.
In this paper, we study the graphical structure of elementary trapping sets (ETS) of variable-regular low-density parity-check (LDPC) codes. ETSs are known to be the main cause of error floor in LDPC coding schemes. For the set of LDPC codes with a given variable node degree $d_l$ and girth $g$, we identify all the non-isomorphic structures of an arbitrary class of $(a,b)$ ETSs, where $a$ is the number of variable nodes and $b$ is the number of odd-degree check nodes in the induced subgraph of the ETS. Our study leads to a simple characterization of dominant classes of ETSs (those with relativ…
We propose an approach for optimizing nonbinary (NB) quasi-cyclic (QC) LDPC codes. This approach combines constructing of base parity-check matrices by simulated annealing and labeling the obtained base matrices aimed at maximizing the so-called generalized girth of the NB LDPC code Tanner graph. Tightened random coding bounds based on the average binary spectra for ensembles of "almost regular" NB LDPC codes of finite lengths over extensions of the binary Galois field are derived. The simulated FER performance of the sum-product BP decoding of "almost regular" NB QC-LDPC block codes are prese…
In this paper we consider regular low-density parity-check codes over a binary-symmetric channel in the decoding regime. We prove that up to a certain noise threshold the bit-error probability of the bit-sampling decoder converges in mean to zero over the code ensemble and the channel realizations. To arrive at this result we show that the bit-error probability of the sampling decoder is equal to the derivative of a Bethe free entropy. The method that we developed is new and is based on convexity of the free entropy and loop calculus. Convexity is needed to exchange limit and derivative and th…
There have been lots of efforts on the construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes with large girth. However, most of them are focused on protographs with single edges and little research has been done for the construction of QC LDPC codes lifted from protographs with multiple edges. Compared to single-edge protographs, multiple-edge protographs have benefits such that QC LDPC codes lifted from them can potentially have larger minimum Hamming distance. In this paper, all subgraph patterns of multiple-edge protographs, which prevent QC LDPC codes from having large gi…
In this paper, the minimum distance distribution of irregular generalized LDPC (GLDPC) code ensembles is investigated. Two classes of GLDPC code ensembles are analyzed; in one case, the Tanner graph is regular from the variable node perspective, and in the other case the Tanner graph is completely unstructured and irregular. In particular, for the former ensemble class we determine exactly which ensembles have minimum distance growing linearly with the block length with probability approaching unity with increasing block length. This work extends previous results concerning LDPC and regular GL…
The performance of low-density parity-check (LDPC) codes in the error floor region is closely related to some combinatorial structures of the code's Tanner graph, collectively referred to as {\it trapping sets (TSs)}. In this paper, we study the asymptotic average number of different types of trapping sets such as {\em elementary TSs (ETS)}, {\em leafless ETSs (LETS)}, {\em absorbing sets (ABS)}, {\em elementary ABSs (EABS)}, and {\em stopping sets (SS)}, in random variable-regular and irregular LDPC code ensembles. We demonstrate that, regardless of the type of the TS, as the code's length te…
In this chapter, we show how the use of differential coding and the presence of phase slips in the transmission channel affect the total achievable information rates and capacity of a system. By means of the commonly used QPSK modulation, we show that the use of differential coding does not decrease the total amount of reliably conveyable information over the channel. It is a common misconception that the use of differential coding introduces an unavoidable differential loss. This perceived differential loss is rather a consequence of simplified differential detection and decoding at the recei…
A new method for low-complexity near-maximum-likelihood (ML) decoding of low-density parity-check (LDPC) codes over the additive white Gaussian noise channel is presented. The proposed method termed belief-propagation--list erasure decoding (BP-LED) is based on erasing carefully chosen unreliable bits performed in case of BP decoding failure. A strategy of introducing erasures into the received vector and a new erasure decoding algorithm are proposed. The new erasure decoding algorithm, called list erasure decoding, combines ML decoding over the BEC with list decoding applied if the ML decoder…
In this paper, the concept of the {\it broken diagonal pair} in the chess-like square board is used to define some well-structured block designs whose incidence matrices can be considered as the parity-check matrices of some high rate cycle codes with girth 12. The structure of the proposed parity-check matrices significantly reduces the complexity of encoding and decoding. Interestingly, the constructed regular cycle codes with row-weights $t$, $3\leq t \leq 20$, $t\neq 7, 15, 16$, have the best lengths among the known regular girth-12 cycle codes. In addition, the proposed cycle codes can be…
In [1] a syndrome counting based upper bound on the minimum distance of regular binary LDPC codes is given. In this paper we extend the bound to the case of irregular and generalized LDPC codes over GF(q). The comparison to the lower bound for LDPC codes over GF(q) and to the upper bound for non-binary codes is done. The new bound is shown to lie under the Gilbert-Varshamov bound at high rates.
In this paper we study the iterative decoding threshold performance of non-binary spatially-coupled low-density parity-check (NB-SC-LDPC) code ensembles for both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (BIAWGNC), with particular emphasis on windowed decoding (WD). We consider both (2,4)-regular and (3,6)-regular NB-SC-LDPC code ensembles constructed using protographs and compute their thresholds using protograph versions of NB density evolution and NB extrinsic information transfer analysis. For these code ensembles, we show that WD of NB-SC-…
We propose non-binary LDPC codes concatenated with multiplicative repetition codes. By multiplicatively repeating the (2,3)-regular non-binary LDPC mother code of rate 1/3, we construct rate-compatible codes of lower rates 1/6, 1/9, 1/12,... Surprisingly, such simple low-rate non-binary LDPC codes outperform the best low-rate binary LDPC codes so far. Moreover, we propose the decoding algorithm for the proposed codes, which can be decoded with almost the same computational complexity as that of the mother code.
An explicit construction of a family of binary LDPC codes called LU(3,q), where q is a power of a prime, was recently given. A conjecture was made for the dimensions of these codes when q is odd. The conjecture is proved in this note. The proof involves the geometry of a 4-dimensional symplectic vector space and the action of the symplectic group and its subgroups.
One of the phenomena that causes high decoding failure rates is trapping sets. Characterization of $(a,b)$ elementary trapping sets (ETSs), their graphical properties and the lower bounds on their size in variable regular LDPC codes with column weights 3, 4, 5 and 6, where $a$ is the size of the ETS and $b$ is the number of degree-one check nodes, have been an interesting subject among researchers. Although progressive-edge-growth method (PEG) to construct LDPC codes free of an specific ETS has been proposed in the literature, it is mostly applied to LDPC codes with column weight 3. In this pa…
Cages, defined as regular graphs with minimum number of nodes for a given girth, are well-studied in graph theory. Trapping sets are graphical structures responsible for error floor of low-density parity-check (LDPC) codes, and are well investigated in coding theory. In this paper, we make connections between cages and trapping sets. In particular, starting from a cage (or a modified cage), we construct a trapping set in multiple steps. Based on the connection between cages and trapping sets, we then use the available results in graph theory on cages and derive tight upper bounds on the size o…
In this paper, we derive the average weight distributions for the irregular non-binary cluster low-density parity-check (LDPC) code ensembles. Moreover, we give the exponential growth rate of the average weight distribution in the limit of large code length. We show that there exist $(2,d_c)$-regular non-binary cluster LDPC code ensembles whose normalized typical minimum distances are strictly positive.
In this paper we present regular bilayer LDPC convolutional codes for half-duplex relay channels. For the binary erasure relay channel, we prove that the proposed code construction achieves the capacities for the source-relay link and the source-destination link provided that the channel conditions are known when designing the code. Meanwhile, this code enables the highest transmission rate with decode-and-forward relaying. In addition, its regular degree distributions can easily be computed from the channel parameters, which significantly simplifies the code optimization. Numerical results ar…
Braided convolutional codes (BCCs) are a class of spatially coupled turbo-like codes that can be described by a $(2,3)$-regular compact graph. In this paper, we introduce a family of $(d_v,d_c)$-regular GLDPC codes with convolutional code constraints (CC-GLDPC codes), which form an extension of classical BCCs to arbitrary regular graphs. In order to characterize the performance in the waterfall and error floor regions, we perform an analysis of the density evolution thresholds as well as the finite-length ensemble weight enumerators and minimum distances of the ensembles. In particular, we con…
Regular spatially-Coupled LDPC (SC-LDPC) ensembles have gained significant interest since they were shown to universally achieve the capacity of binary memoryless channels under low-complexity belief-propagation decoding. In this work, we focus primarily on the performance of these ensembles over binary channels affected by bursts of erasures. We first develop an analysis of the finite length performance for a single burst per codeword and no errors otherwise. We first assume that the burst erases a complete spatial position, modeling for instance node failures in distributed storage. We provi…
The concept and existence of sphere-bound-achieving and capacity-achieving lattices has been explained on AWGN channels by Forney. LDPC lattices, introduced by Sadeghi, perform very well under iterative decoding algorithm. In this work, we focus on an ensemble of regular LDPC lattices. We produce and investigate an ensemble of LDPC lattices with known properties. It is shown that these lattices are sphere-bound-achieving and capacity-achieving. As byproducts we find the minimum distance, coding gain, kissing number and an upper bound for probability of error for this special ensemble of regula…
In this paper, we compare the finite-length performance of protograph-based spatially coupled low-density parity-check (SC-LDPC) codes and LDPC block codes (LDPC-BCs) over GF(q). In order to reduce computational complexity and latency, a sliding window decoder with a stopping rule based on a soft bit-error-rate (BER) estimate is used for the q-ary SC-LDPC codes. Two regimes are considered: one when the constraint length of q-ary SC-LDPC codes is equal to the block length of q-ary LDPC-BCs and the other when the two decoding latencies are equal. Simulation results confirm that, in both regimes,…
We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) code and belief propagation (BP) decoding. The bit error probability for infinite block length is known by density evolution and it is well known that a difference between the bit error probability at finite iteration number for finite block length $n$ and for infinite block length is asymptotically $α/n$, where $α$ is a specific constant depending on the degree distribution, the iteration number and the erasure probability. Our main result is to derive an efficient algorithm for calculating $…
The relation between parity-check matrices of quasi-cyclic (QC) low-density parity-check (LDPC) codes and biadjacency matrices of bipartite graphs supports searching for powerful LDPC block codes. Using the principle of tailbiting, compact representations of bipartite graphs based on convolutional codes can be found. Bounds on the girth and the minimum distance of LDPC block codes constructed in such a way are discussed. Algorithms for searching iteratively for LDPC block codes with large girth and for determining their minimum distance are presented. Constructions based on all-ones matrices, …
Low-density parity check (LDPC) codes are an important class of codes with many applications. Two algebraic methods for constructing regular LDPC codes are derived -- one based on nonprimitive narrow-sense BCH codes and the other directly based on cyclotomic cosets. The constructed codes have high rates and are free of cycles of length four; consequently, they can be decoded using standard iterative decoding algorithms. The exact dimension and bounds for the minimum distance and stopping distance are derived. These constructed codes can be used to derive quantum error-correcting codes.
We consider the framework of the McEliece cryptosystem based on LDPC codes, which is a promising post-quantum alternative to classical public key cryptosystems. The use of LDPC codes in this context allows to achieve good security levels with very compact keys, which is an important advantage over the classical McEliece cryptosystem based on Goppa codes. However, only regular LDPC codes have been considered up to now, while some further improvement can be achieved by using irregular LDPC codes, which are known to achieve better error correction performance than regular LDPC codes. This is show…
We present a construction of LDPC codes that have minimum pseudocodeword weight equal to the minimum distance, and perform well with iterative decoding. The construction involves enumerating a d-regular tree for a fixed number of layers and employing a connection algorithm based on mutually orthogonal Latin squares to close the tree. Methods are presented for degrees d=p^s and d = p^s+1, for p a prime, -- one of which includes the well-known finite-geometry-based LDPC codes.
In this paper, we present a construction method of non-binary low-density parity-check (LDPC) convolutional codes. Our construction method is an extension of Felstroem and Zigangirov construction for non-binary LDPC convolutional codes. The rate-compatibility of the non-binary convolutional code is also discussed. The proposed rate-compatible code is designed from one single mother (2,4)-regular non-binary LDPC convolutional code of rate 1/2. Higher-rate codes are produced by puncturing the mother code and lower-rate codes are produced by multiplicatively repeating the mother code. Simulation …
We investigate the relation between the girth and the guaranteed error correction capability of $γ$-left regular LDPC codes when decoded using the bit flipping (serial and parallel) algorithms. A lower bound on the number of variable nodes which expand by a factor of at least $3 γ/4$ is found based on the Moore bound. An upper bound on the guaranteed correction capability is established by studying the sizes of smallest possible trapping sets.
We present a tree-based construction of LDPC codes that have minimum pseudocodeword weight equal to or almost equal to the minimum distance, and perform well with iterative decoding. The construction involves enumerating a $d$-regular tree for a fixed number of layers and employing a connection algorithm based on permutations or mutually orthogonal Latin squares to close the tree. Methods are presented for degrees $d=p^s$ and $d = p^s+1$, for $p$ a prime. One class corresponds to the well-known finite-geometry and finite generalized quadrangle LDPC codes; the other codes presented are new. We …
The minimum pseudoweight is an important parameter related to the decoding performance of LDPC codes with iterative message-passing decoding. In this paper, we consider ensembles of periodically time-varying spatially coupled LDPC (SC-LDPC) codes and the pseudocodewords arising from their finite graph covers of a fixed degree. We show that for certain $(J,K)$-regular SC-LDPC code ensembles and a fixed cover degree, the typical minimum pseudoweight of the unterminated (and associated tail-biting/terminated) SC-LDPC code ensembles grows linearly with the constraint (block) length as the constrai…
In this paper, we propose a new design method of irregular spatially-coupled low-density parity-check (SC-LDPC) codes with non-uniform degree distributions by linear programming (LP). In general, irregular SC-LDPC codes with non-uniform degree distributions is difficult to design with low complexity because their density evolution equations are multi-dimensional. To solve the problem, the proposed method is based on two main ideas: A local design of the degree distributions and pre-computation of the input/output message relationship. These ideas make it possible to design the degree distribut…
Ensembles of (J,K)-regular low-density parity-check convolutional (LDPCC) codes are known to be asymptotically good, in the sense that the minimum free distance grows linearly with the constraint length. In this paper, we use a protograph-based analysis of terminated LDPCC codes to obtain an upper bound on the free distance growth rate of ensembles of periodically time-varying LDPCC codes. This bound is compared to a lower bound and evaluated numerically. It is found that, for a sufficiently large period, the bounds coincide. This approach is then extended to obtain bounds on the trapping set …
Controlling small size trapping sets and short cycles can result in LDPC codes with large minimum distance $d_{\min}$. We prove that short cycles with a chord are the root of several trapping sets and eliminating these cycles increases $d_{\min}$. We show that the lower bounds on $d_{\min}$ of an LDPC code with chordless short cycles, girths 6 (and 8), and column weights $γ$ (and 3), respectively, are $2γ$ (and 10), which is a significant improvement compared to the existing bounds $γ+1$ (and 6). Necessary and sufficient conditions for exponent matrices of protograph-based LDPC codes with chor…
