In this paper, we propose a systematic low density generator matrix (LDGM) code ensemble, which is defined by the Bernoulli process. We prove that, under maximum likelihood (ML) decoding, the proposed ensemble can achieve the capacity of binary-input output symmetric (BIOS) memoryless channels in terms of bit error rate (BER). The proof technique reveals a new mechanism, different from lowering down frame error rate (FER), that the BER can be lowered down by assigning light codeword vectors to light information vectors. The finite length performance is analyzed by deriving an upper bound and a lower bound, both of which are shown to be tight in the high signal-to-noise ratio (SNR) region. To improve the waterfall performance, we construct the systematic convolutional LDGM (SC-LDGM) codes by a random splitting process. The SC-LDGM codes are easily configurable in the sense that any rational code rate can be realized without complex optimization. As a universal construction, the main advantage of the SC-LDGM codes is their near-capacity performance in the waterfall region and predictable performance in the error-floor region that can be lowered down to any target as required by increasing the density of the uncoupled LDGM codes. Numerical results are also provided to verify our analysis.
Basic properties of a characteristic matrix for a tail-biting convolutional code are investigated. A tail-biting convolutional code can be regarded as a linear block code. Since the corresponding scalar generator matrix Gt has a kind of cyclic structure, an associated characteristic matrix also has a cyclic structure, from which basic properties of a characteristic matrix are obtained. Next, using the derived results, we discuss the possibility of trellis reduction for a given tail-biting convolutional code. There are cases where we can find a scalar generator matrix Gs equivalent to Gt based …
We present a general theory of entanglement-assisted quantum convolutional coding. The codes have a convolutional or memory structure, they assume that the sender and receiver share noiseless entanglement prior to quantum communication, and they are not restricted to possess the Calderbank-Shor-Steane structure as in previous work. We provide two significant advances for quantum convolutional coding theory. We first show how to "expand" a given set of quantum convolutional generators. This expansion step acts as a preprocessor for a polynomial symplectic Gram-Schmidt orthogonalization procedur…
Technical University of Munich（慕尼黑工业大学）；University of Southern California（南加州大学）
Convolutional Neural Networks (CNNs) have emerged as highly successful tools for image generation, recovery, and restoration. A major contributing factor to this success is that convolutional networks impose strong prior assumptions about natural images. A surprising experiment that highlights this architectural bias towards natural images is that one can remove noise and corruptions from a natural image without using any training data, by simply fitting (via gradient descent) a randomly initialized, over-parameterized convolutional generator to the corrupted image. While this over-parameteriz…
Polarization-adjusted convolutional (PAC) codes were recently proposed and arouse the interest of the channel coding community because they were shown to approach theoretical bounds for the (128,64) code size. In this letter, we propose systematic PAC codes. Thanks to the systematic property, improvement in the bit-error rate of up to 0.2 dB is observed, while preserving the frame-error rate performance. Moreover, a genetic-algorithm based construction method targeted to approach the theoretical bound is provided. It is then shown that using the proposed construction method systematic and non-…
Maximum distance separable convolutional codes are the codes that present best performance in error correction among all convolutional codes with certain rate and degree. In this paper, we show that taking the constant matrix coefficients of a polynomial matrix as submatrices of a superregular matrix, we obtain a column reduced generator matrix of an MDS convolutional code with a certain rate and a certain degree. We then present two novel constructions that fulfill these conditions by considering two types of superregular matrices.
We investigate the notion of cyclicity for convolutional codes as it has been introduced by Piret and Roos in the seventies. Codes of this type are described as submodules of the module of all vector polynomials in one variable with some additional generalized cyclic structure but also as specific left ideals in a skew polynomial ring. Extending a result of Piret, we show in a purely algebraic setting that these ideals are always principal. This leads to the notion of a generator polynomial just like for cyclic block codes. Similarly a control polynomial can be introduced by considering the ri…
New families of unit memory as well as multi-memory convolutional codes are constructed algebraically in this paper. These convolutional codes are derived from the class of group character codes. The proposed codes have basic generator matrices, consequently, they are non catastrophic. Additionally, the new code parameters are better than the ones available in the literature.
A pragmatic approach for the construction of space-time codes over block fading channels is investigated. The approach consists in using common convolutional encoders and Viterbi decoders with suitable generators and rates, thus greatly simplifying the implementation of space-time codes. For the design of pragmatic space-time codes a methodology is proposed and applied, based on the extension of the concept of generalized transfer function for convolutional codes over block fading channels. Our search algorithm produces the convolutional encoder generators of pragmatic space-time codes for var…
In this paper we obtain the [60,30,12], [64,32,12], [68,34,12], [72,36,12] self-dual codes as tailbitting convolutional codes with the smallest constraint length K=9. In this construction one information bit is modulo two added to the one of the encoder outputs and the first row in the quasi-cyclic generator matrix is replaced by the obtained row. The pure quasi-cyclic construction with K=10 is also available for [68,34,12] and [72,36,12] codes. The new [72,36,12] singly even self-dual code with parameters Beta=483 Gamma=0 was obtained.
In this paper, we provide a performance analysis of a new class of serial concatenated convolutional codes (SCCC) where the inner encoder can be punctured beyond the unitary rate. The puncturing of the inner encoder is not limited to inner coded bits, but extended to systematic bits. Moreover, it is split into two different puncturings, in correspondence with inner code systematic bits and parity bits. We derive the analytical upper bounds to the error probability of this particular code structure and address suitable design guidelines for the inner code puncturing patterns. We show that the p…
We propose in this paper to exploit convolutional low density generator matrix (LDGM) codes for transmission of Bernoulli sources over binary-input output-symmetric (BIOS) channels. To this end, we present a new framework to prove the coding theorems for linear codes, which unifies the channel coding theorem, the source coding theorem and the joint source-channel coding (JSCC) theorem. In the presented framework, the systematic bits and the corresponding parity-check bits play different roles. Precisely, the noisy systematic bits are used to limit the list size of typical codewords, while the …
A systematic convolutional encoder of rate $(n-1)/n$ and maximum degree $D$ generates a code of free distance at most ${\cal D} = D+2$ and, at best, a column distance profile (CDP) of $[2,3,\ldots,{\cal D}]$. A code is \emph{Maximum Distance Separable} (MDS) if it possesses this CDP. Applied on a communication channel over which packets are transmitted sequentially and which loses (erases) packets randomly, such a code allows the recovery from any pattern of $j$ erasures in the first $j$ $n$-packet blocks for $j<{\cal D}$, with a delay of at most $j$ blocks counting from the first erasure. Thi…
In 1995, Best et al. published a formula for the exact bit error probability for Viterbi decoding of the rate R=1/2, memory m=1 (2-state) convolutional encoder with generator matrix G(D)=(1 1+D) when used to communicate over the binary symmetric channel. Their formula was later extended to the rate R=1/2, memory m=2 (4-state) convolutional encoder with generator matrix G(D)=(1+D^2 1+D+D^2) by Lentmaier et al. In this paper, a different approach to derive the exact bit error probability is described. A general recurrent matrix equation, connecting the average information weight at the current a…
We characterize the generator matrix in standard form of generalized Gabidulin codes. The parametrization we get for the non-systematic part of this matrix coincides with the $q$-analogue of generalized Cauchy matrices, leading to the definition of $q$-Cauchy matrices. These matrices can be represented very conveniently and their representation allows to define new interesting subfamilies of generalized Gabidulin codes whose generator matrix is a structured matrix. In particular, as an application, we construct Gabidulin codes whose generator matrix is the concatenation of an identity block an…
Maximum distance profile (MDP) convolutional codes have the property that their column distances are as large as possible for given rate and degree. There exists a well-known criterion to check whether a code is MDP using the generator or the parity-check matrix of the code. In this paper, we show that under the assumption that $n-k$ divides $δ$ or $k$ divides $δ$, a polynomial matrix that fulfills the MDP criterion is actually always left prime. In particular, when $k$ divides $δ$, this implies that each MDP convolutional code is noncatastrophic. Moreover, when $n-k$ and $k$ do not divide $δ$…
Random network coding recently attracts attention as a technique to disseminate information in a network. This paper considers a non-coherent multi-shot network, where the unknown and time-variant network is used several times. In order to create dependencies between the different shots, particular convolutional codes in rank metric are used. These codes are so-called (partial) unit memory ((P)UM) codes, i.e., convolutional codes with memory one. First, distance measures for convolutional codes in rank metric are shown and two constructions of (P)UM codes in rank metric based on the generator …
In this paper, we construct the first families of asymmetric quantum convolutional codes (AQCC)'s. These new AQCC's are constructed by means of the CSS-type construction applied to suitable families of classical convolutional codes, which are also constructed here. The new codes have noncatastrophic generator matrices and they present great asymmetry. Since our constructions are performed algebraically, it is possible to derive several families of such codes and not only codes with specific parameters. Additionally, several different types of such codes are obtained.
In this paper, we show that the code-trellis and the error-trellis for a convolutional code can be reduced simultaneously, if reduction is possible. Assume that the error-trellis can be reduced using shifted error-subsequences. In this case, if the identical shifts occur in the subsequences of each code path, then the code-trellis can also be reduced. First, we obtain pairs of transformations which generate the identical shifts both in the subsequences of the code-path and in those of the error-path. Next, by applying these transformations to the generator matrix and the parity-check matrix, w…
A class of one-dimensional convolutional codes will be presented. They are all MDS codes, i. e., have the largest distance among all one-dimensional codes of the same length n and overall constraint length delta. Furthermore, their extended row distances are computed, and they increase with slope n-delta. In certain cases of the algebraic parameters, we will also derive parity check matrices of Vandermonde type for these codes. Finally, cyclicity in the convolutional sense will be discussed for our class of codes. It will turn out that they are cyclic if and only if the field element used in t…
This paper proposes a new family of recursive systematic convolutional codes, defined in the non-binary domain over different Galois fields GF(q) and intended to be used as component codes for the design of non-binary turbo codes. A general framework for the design of the best codes over different GF(q) is described. The designed codes offer better performance than the non-binary convolutional codes found in the literature. They also outperform their binary counterparts when combined with their corresponding QAM modulation or with lower order modulations.
Designs and methods for nested lattice codes using Construction D' lattices for coding and convolutional code lattices for shaping are described. Two encoding methods and a decoding algorithm for Construction D' coding lattices that can be used with shaping lattices for power-constrained channels are given. We construct nested lattice codes with good coding properties, a high shaping gain, and low-complexity encoding and decoding. Convolutional code generator polynomials for Construction A lattices with the greatest shaping gain are given, as a result of an extensive search. It is shown that r…
A new kind of Convolutional Codes generalizing Goppa Codes is proposed. This provides a systematic method for constructing convolutional codes with prefixed properties. In particular, examples of Maximum-Distance Separable (MDS) convolutional codes are obtained.
In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Max…
Noncatastrophic encoders are an important class of polynomial generator matrices of convolutional codes. When these polynomials have coefficients in a finite field, these encoders have been characterized are being polynomial left prime matrices. In this paper we study the notion of noncatastrophicity in the context of convolutional codes when the polynomial matrices have entries in a finite ring. In particular, we need to introduce two different notion of primeness in order to fully characterize noncatastrophic encoders over the finite ring Z_{p^r}. The second part of the paper is devoted to i…
In this paper, we provide a new approach to the analytical estimation of the bit-error rate (BER) for convolutional codes for Viterbi decoding in the binary symmetric channel (BSC). The expressions we obtained for lower and upper BER bounds are based on the active distances of the code and their distance spectrum. The estimates are derived for convolutional codes with the rate $R=\frac{1}{2}$ but can be easily generalized for any convolutional code with rate $R=\frac 1n$ and systematic encoder. The suggested approach is not computationally expensive for any crossover probability of BSC channel…
Polar codes are a new class of error correcting linear block codes, whose generator matrix is specified by the knowledge of transmission channel parameters, code length and code dimension. Moreover, regarding computational security, it is assumed that an attacker with a restricted processing power has unlimited access to the transmission media. Therefore, the attacker can construct the generator matrix of polar codes, especially in the case of Binary Erasure Channels, on which this matrix can be easily constructed. In this paper, we introduce a novel method to keep the generator matrix of pola…
The equivalence of a systematic convolutional encoder as linear state-space control system is first realized and presented through an example. Then, utilizing this structure, a new optimal state-sequence estimator is derived, in the spirit of the Viterbi algorithm. Afterwords, a novel way to perform optimal decoding is achieved, named the Bowyer Decoder, which is a fully deterministic decoder in that the full FSM is known to the decoding algorithm.
