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Maximum Distance Separable Convolutional Codes
Abstract
. A maximum distance separable (MDS) block code is a linear code whose distance is maximal among all linear block codes of rate k=n. It is well known that MDS block codes do exist if the field size is more than n. In this paper we generalize this concept to the class of convolutional codes of a fixed rate k=n and a fixed code degree ffi. In order to achieve this result we will introduce a natural upper bound for the free distance generalizing the Singleton bound. The main result of the paper shows that this upper bound can be achieved in all cases if one allows sufficiently many field elements. 1. Introduction Let F be a finite field and let C ae F n be an [n; k] linear block code. Let d(C) be the distance of C, i.e. d(C) is equal to the minimum Hamming distance between any two different code words x; y 2 C. The main linear coding problem asks for the construction of linear [n; k] codes whose distance d(C) is maximal among all linear [n; k] codes. The distance d(C) is always upper...
... Different equivalent definitions for convolutional codes are given in the literature. The notation and definitions used here follow that given in [38,39,41]. A rate k n convolutional code with parameters (n, k, δ) over a field F is a submodule of F [z] n generated by a reduced basic matrix ...
... By Rosenthal and Smarandache [38] the maximum free distance attainable by an (n, r, δ) convolutional code is (n − r)( δ r + 1) + δ + 1. The case δ = 0, which is the case of zero memory, corresponds to the linear Singleton bound (n − r + 1). ...
... The case δ = 0, which is the case of zero memory, corresponds to the linear Singleton bound (n − r + 1). The bound (n − r)( δ r + 1) + δ + 1 is then called the generalised Singleton bound [38] (GSB), and a convolutional code attaining this bound is known as an MDS convolutional code. The papers [38] and [41] are major beautiful contributions to this area. ...
Linear block and convolutional codes are designed using unit schemes and families of these to required length, rate, distance and type are mined. Properties, such as type and distance, of the codes follow from the types of units used and thus required codes are built from specific units.
Orthogonal units, units in group rings, Fourier/Vandermonde units and related units are used to construct and analyse linear block and convolutional codes and to construct these to predefined length rate, distance and type. Series of self-dual, dual containing, quantum error-correcting and linear complementary dual codes are constructed for both linear block and convolutional codes.
Low density parity check linear block and linear convolutional codes are constructed from unit schemes.
... The class of convolutional codes is a class of codes much investigated in the literature [4,14,18,[26][27][28]. Constructions of convolutional codes with good parameters or even maximum distance separable (MDS), i.e. optimal, convolutional codes (in the sense that they attain the generalized Singleton bound [27]) have also been presented in the literature [4, 11-14, 18, 26-28]. ...
... The class of convolutional codes is a class of codes much investigated in the literature [4,14,18,[26][27][28]. Constructions of convolutional codes with good parameters or even maximum distance separable (MDS), i.e. optimal, convolutional codes (in the sense that they attain the generalized Singleton bound [27]) have also been presented in the literature [4, 11-14, 18, 26-28]. Rosenthal et al. introduced the generalized Singleton bound [27] (see also [28]) in 1999. ...
... Constructions of convolutional codes with good parameters or even maximum distance separable (MDS), i.e. optimal, convolutional codes (in the sense that they attain the generalized Singleton bound [27]) have also been presented in the literature [4, 11-14, 18, 26-28]. Rosenthal et al. introduced the generalized Singleton bound [27] (see also [28]) in 1999. ...
In this paper, we construct new families of convolutional codes. Such codes are obtained by means of algebraic geometry codes. Additionally, more families of convolutional codes are constructed by means of puncturing, extending, expanding and by the direct product code construction applied to algebraic geometry codes. The parameters of the new convolutional codes are better than or comparable to the ones available in literature. In particular, a family of almost near MDS codes is presented.
... We also wish to point out that in algebraic geometry the degree corresponds to the degree of an associated vector bundle (i.e. quotient sheaf), see [13,20,22] for more details. ...
... An (n, k, δ) convolutional code is called MDS if its free distance is maximal among all rate k/n convolutional codes of degree δ, i.e. an (n, k, δ) convolutional code is MDS if the free distance achieves the generalized Singleton bound [22]: ...
... The concept of MDS convolutional codes was introduced by the authors in [22,26]. Strongly MDS codes are going to be a subclass of MDS codes which have a remarkable decoding capability. ...
MDS convolutional codes have the property that their free distance is maximal among all codes of the same rate and the same degree. In this paper we introduce a class of MDS convolutional codes whose column distances reach the generalized Singleton bound at the earliest possible instant. We call these codes strongly MDS convolutional codes. It is shown that these codes can decode a maximum number of errors per time interval when compared with other convolutional codes of the same rate and degree. These codes have also a maximum or near maximum distance profile. A code has a maximum distance profile if and only if the dual code has this property.
... Concerning the investigation and development of theory of convolutional codes, much effort has been paid [5,31,32,33,13,8,9,14,17,28]. More specifically, constructions of convolutional codes with good or even optimal parameters (for instance, maximum-distance-separable (MDS) codes, i.e., codes attaining the generalized Singleton bound [32]) are of great interest for several researchers [32,33,13,8,18,19,20]. ...
... Concerning the investigation and development of theory of convolutional codes, much effort has been paid [5,31,32,33,13,8,9,14,17,28]. More specifically, constructions of convolutional codes with good or even optimal parameters (for instance, maximum-distance-separable (MDS) codes, i.e., codes attaining the generalized Singleton bound [32]) are of great interest for several researchers [32,33,13,8,18,19,20]. ...
... Concerning the investigation and development of theory of convolutional codes, much effort has been paid [5,31,32,33,13,8,9,14,17,28]. More specifically, constructions of convolutional codes with good or even optimal parameters (for instance, maximum-distance-separable (MDS) codes, i.e., codes attaining the generalized Singleton bound [32]) are of great interest for several researchers [32,33,13,8,18,19,20]. ...
In this paper, we construct new sequences of asymptotically good convolutional codes. These sequences are obtained from sequences of transitive, self-orthogonal and self-dual block codes that attain the Tsfasman-Vladut-Zink bound. Furthermore, by applying the techniques of expanding, extending, puncturing, direct sum, the |u|u+v| construction and the product code construction to these block codes, we construct more new sequences of asymptotically good convolutional codes. Additionally, we show that the proposed construction method presented here also works when applied for all sequences of good block codes where lim kj/nj and lim dj/nj exist.
... Rosenthal and Smarandache [18] showed that the free Hamming distance of an (n, k, δ) convolutional code is upper bounded by ...
... This bound was called the generalized Singleton bound . An (n, k, δ) convolutional code whose free Hamming distance is equal to the generalized Singleton bound is called maximum distance separable (MDS) convolutional code [18]. ...
... As explained before, Rosenthal and Smarandache [18] showed that the free distance of an (n, k, δ) convolutional code is upper bounded by the generalized Singleton bound, d free (C) ≤ (n − k) δ k + 1 + δ + 1. and the column distance, see [11], is upper bounded by d c j ≤ (n − k)(j + 1) + 1. ...
... Let us first show that the weight of v(D)| [0,5] := v 0 + v 1 D + v 2 D 2 + v 3 D 3 + v 4 D 4 + v 5 D 5 has weight greater or equal than 6 for all u 0 , u 1 , u 2 , u 3 , u 4 , u 5 ∈ F 11 Case 6.2.2.3.2. If u 1 = 2u 2 then wt(v 3 ) = 1 and v 4 = (7u 1 + 8u 4 )G 2 . ...
... In [5], Smarandache and Rosenthal obtained an upper bound for the free distance of an (n, k, δ) convolutional code C given by ...
... Thus, we have proven that wt v(D)| [0,5] ≥ 6. Note that v n =v 5 ...
Maximum distance separable convolutional codes are characterized by the property that the free distance reaches the generalized Singleton bound, which makes them optimal for error correction. However, the existing constructions of such codes are available over fields of large size. In this paper, we present the unique construction of MDS convolutional codes of rate 1/2 and degree 5 over the field 𝔽11.
... The free distance was shown in [2] to satisfy the following upper bound that generalizes the Singleton bound for block codes. ...
... We mention here several families of distance-optimal convolutional codes. Convolutional codes with a generator matrix of memory m and distance attaining the equality in (2) for j " m are called m-MDS convolutional codes. Relating the bounds (1) and (2), [21] and [11] introduced two other types of distance-optimal convolutional codes, described below. ...
... Moreover, the column distance of strongly-MDS convolutional codes attains the Singleton bound (1) at the earliest possible time instant M . But the column distance at the time instant prior to M may not attain (2) . On the contrary, the column distances of MDP convolutional codes attain (2) for the maximum number of times although the free distance of the codes may not attain (1). ...
We construct a family of (n,k) convolutional codes with degree \delta in {k,n-k} that have a maximum distance profile. The field size required for our construction is of the order n^{2\delta}, which improves upon the known constructions of convolutional codes with a maximum distance profile. Our construction is based on the theory of skew polynomials.
... The theory of convolutional codes is more involved and there are not many known constructions of MDS convolutional codes. Maximum distance separable (MDS) codes have maximum free distance in the class of convolutional codes of a certain rate k/n and a certain degree δ, i.e., are the ones with free distance equal to the Singleton bound (n − k) δ k + 1 + δ + 1 [13]. The first construction of MDS convolutional codes was obtained by Justesen in [9] for codes of rate 1/n and restricted degrees. ...
... In [13] Smarandache and Rosenthal obtained an upper bound for the free distance of a convolutional code C of rate k/n and degree δ given by ...
... A convolutional code of rate k/n and degree δ with free distance equal to the generalized Singleton bound is called Maximum Distance Separable (MDS) convolutional code. If C is such a code and G(z) ∈ F[z] k×n is a column reduced generator matrix of C, its columns degrees are equal to δ k + 1 with multiplicity t := δ − k δ k and δ k with multiplicity k − t; see [13]. ...
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.
... For these reasons, for given rate k/n and q, generally speaking, it is desirable to construct convolutional codes with relatively small degree δ and relatively large free distance d free . The generalized Singleton bound for an (n, k, δ) q convolutional code C, in its most general form, proposed and proved by Rosenthal and Smarandache [19], states that the free distance d free of C must satisfy d free ≤ (n − k) δ k + 1 + δ + 1. ...
... As in the classical case, MDS convolutional codes form an optimal family of convolutional codes, the study of which is of great importance. For any rate k/n and any degree δ, Rosenthal and Smarandache [19] established the existence of (n, k, δ) q MDS convolutional codes over some finite field F q by techniques from algebraic geometry without giving explicit constructions. Then in a beautiful follow-up paper [21], building upon ideas from Justesen [14], the authors provided an explicit construction of MDS convolutional codes for each rate k/n and each degree δ over some F q . ...
Maximum-distance separable (MDS) convolutional codes form an optimal family of convolutional codes, the study of which is of great importance. There are very few general algebraic constructions of MDS convolutional codes. In this paper, we construct a large family of unit-memory MDS convolutional codes over \F with flexible parameters. Compared with previous works, the field size q required to define these codes is much smaller. The construction also leads to many new strongly-MDS convolutional codes, an important subclass of MDS convolutional codes proposed and studied in \cite{GL2}. Many examples are presented at the end of the paper.
... In [18], it is shown that MDS convolutional codes exist for all parameters (n, k, δ) over sufficiently large finite fields; in [11], a similar result is obtained for codes having the MDP property. In [8], sMDS convolutional codes are introduced and studied, and they are shown to exist for parameters (n, k, δ) satisfying (n − k) | δ. ...
... Statement 1 is proved in [8], and statement 2 is proved in [18]. The bound in 2 is called the generalized Singleton bound. ...
It is known that maximum distance separable and maximum distance profile convolutional codes exist over large enough finite fields of any characteristic for all parameters . It has been conjectured that the same is true for convolutional codes that are strongly maximum distance separable. Using methods from linear systems theory, we resolve this conjecture by showing that, over a large enough finite field of any characteristic, codes which are simultaneously maximum distance profile and strongly maximum distance separable exist for all parameters .
... Note that for C ′ and L = 6, its line associated codes are therefore L ′ = 3. Now, it is clear that H v is a submatrix of H when looking at (14); on the other hand, H h and H d cannot be seen as submatrices of H. (14) ...
... Note that for C ′ and L = 6, its line associated codes are therefore L ′ = 3. Now, it is clear that H v is a submatrix of H when looking at (14); on the other hand, H h and H d cannot be seen as submatrices of H. (14) ...
In general, the problem of building optimal convolutional codes under a certain criteria is hard, especially when size field restrictions are applied. In this paper, we confront the challenge of constructing an optimal 2D convolutional code when communicating over an erasure channel. We propose a general construction method for these codes. Specifically, we provide an optimal construction where the decoding method presented in the bibliography is considered.
... In [5] Smarandache and Rosenthal obtained an upper bound for the free distance of an (n, k, δ) convolutional code C given by ...
... Let us first show that the weight of v(D)| [0,5] ...
Maximum distance separable convolutional codes are characterized by the property that the free distance reaches the generalized Singleton bound, which makes them optimal for error correction. However, the existing constructions of such codes are available over fields of large size. In this paper, we present the unique construction of MDS convolutional codes of rate 1/2 and degree 5 over the field .
... Within this setting, many constructions of convolutional codes with designed free distance were provided based on different classes of block codes, such as Reed-Solomon or Reed-Muller codes. Moreover, in [32], the authors adjusted the parameters of these constructions to present the first Maximum Distance Separable (MDS) convolutional code, i.e., a convolutional code whose free distance achieves the generalized Singleton bound presented in [31], provided that the field size is congruent to 1 modulo the length of the code. Later, other examples of MDS convolutional codes, also for restricted set of parameters, were presented; see [12,29]. ...
... For this reason, for any given positive integers n and k and field size q, the aim is to construct (n, k) q convolutional codes with "small" degree δ and "large" free distance d free . The parameters of an (n, k, δ) q convolutional code C are related by the generalized Singleton bound ( [31]): ...
In this paper we present a concrete algebraic construction of a novel class of convolutional codes. These codes are built upon generalized Vandermonde matrices and therefore can be seen as a natural extension of Reed-Solomon block codes to the context of convolutional codes. For this reason we call them weighted Reed-Solomon (WRS) convolutional codes. We show that under some constraints on the defining parameters these codes are Maximum Distance Profile (MDP), which means that they have the maximal possible growth in their column distance profile. We study the size of the field needed to obtain WRS convolutional codes which are MDP and compare it with the existing general constructions of MDP convolutional codes in the literature, showing that in many cases WRS convolutional codes require significantly smaller fields.
... In this case we use for C the notation (N, k, γ; m, d f ) q . For a detailed introduction on convolutional codes see [6,35] and the references therein. We apply Lemma 7.1 to one-point AG codes from the GGS curve. ...
In this paper, algebraic-geometric (AG) codes associated with the GGS maximal curve are investigated. The Weierstrass semigroup at all -rational points of the curve is determined; the Feng-Rao designed minimum distance is computed for infinite families of such codes, as well as the automorphism group. As a result, some linear codes with better relative parameters with respect to one-point Hermitian codes are discovered. Classes of quantum and convolutional codes are provided relying on the constructed AG codes.
... is basic and therefore C = im G is the smallest σ-CCC containing the word v above. This code happens to be quite a good one, since one can show that d free (C) = 9, which is the maximum value for the free distance of any one-dimensional code of length 3 and complexity 2, see [21,Thm. 2.2]. ...
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 right annihilator ideal. An algorithmic procedure is developed which produces unique reduced generator and control polynomials. We also show how basic code properties and a minimal generator matrix can be read off from these objects. A close link between polynomial and vector description of the codes is provided by certain generalized circulant matrices.
... Recently, in [30], a bound on the free distance of convolutional codes over Z p r was derived, generalizing the bound given in [31] for convolutional codes over finite fields. Codes achieving such a bound were called Maximal Distance Separable (MDS). ...
Rosenthal et al. introduced and thoroughly studied the notion of Maximum Distance Profile (MDP) convolutional codes over (non-binary) finite fields refining the classical notion of optimum distance profile, see for instance [18, p.164]. These codes have the property that their column distances are maximal among all codes of the same rate and the same degree. In this paper we aim at studying this fundamental notion in the context of convolutional codes over a finite ring. We extensively use the notion of p-encoder to present upper-bounds on the column distances which allow to introduce the notion of MDP in the context of finite rings. A constructive method for (non necessarily free) MDP convolutional codes over Z p r is presented.
... In this section, we apply the cyclic codes constructed in Section 4 to derive new families of convolutional codes with great free distance. The theory of convolutional codes is well investigated in the literature [7,21,26,11,27,28,29,9,8,16,17,18,19]. We assume the reader is familiar with the theory of convolutional codes (see [11] for more details). ...
New properties of q-ary cyclotomic cosets modulo , where is a prime power, are investigated in this paper. Based on these properties, the dimension as well as bounds for the designed distance of some families of classical cyclic codes can be computed. As an application, new families of nonbinary Calderbank-Shor-Steane (CSS) quantum codes as well as new families of convolutional codes are constructed in this work. These new CSS codes have parameters better than the ones available in the literature. The convolutional codes constructed here have free distance greater than the ones available in the literature.
... In [11,13], it was shown that, for any rate k/n and degree δ, MDS codes form a generic set in the variety parametrizing convolutional codes of rate k/n and degree δ. In [3], the existence of (n, n − 1, δ) strongly MDS codes was established. ...
Maximum distance profile codes are characterized by the property that two trajectories which start at the same state and proceed to a different state will have the maximum possible distance from each other relative to any other convolutional code of the same rate and degree. In this paper we use methods from systems theory to characterize maximum distance profile codes algebraically. Tha main result shows that maximum distance profile codes form a generic set inside the variety which parametrizes the set of convolutional codes of a fixed rate and a fixed degree.
... This variety has been of central interest in the recent algebraic geometry literature. In the context of coding theory, it has actually been used to predict the existence of maximum-distance-separable (MDS) convolutional codes [43]. ...
The article reviews different definitions for a convolutional code which can be found in the literature. The algebraic differences between the definitions are worked out in detail. It is shown that bi-infinite support systems are dual to finite-support systems under Pontryagin duality. In this duality the dual of a controllable system is observable and vice versa. Uncontrollability can occur only if there are bi-infinite support trajectories in the behavior, so finite and half-infinite-support systems must be controllable. Unobservability can occur only if there are finite support trajectories in the behavior, so bi-infinite and half-infinite-support systems must be observable. It is shown that the different definitions for convolutional codes are equivalent if one restricts attention to controllable and observable codes.
... When m = 1, C is said to be a unit-memory convolutional code. For a detailed introduction to convolutional codes, see [7,41]. . Let X be an F q -rational non-singular curve of genus g. ...
We investigate several types of linear codes constructed from two families and of maximal curves over finite fields recently constructed by Skabelund as cyclic covers of the Suzuki and Ree curves. Plane models for such curves are provided, and the Weierstrass semigroup H(P) at an -rational point P is shown to be symmetric.
... These codes are specially appealing for the performance of sequential decoding algorithms as they have the potential to have a maximum number of errors corrected per time interval. In [10] a non-constructive proof of the existence of such codes (for all transmission rates and all degrees) was given. However, the problem of how to construct MDP codes is far from being solved and very little is known about the minimum field size required for doing so. ...
This paper deals with the problem of constructing superregular matrices that lead to MDP convolutional codes. These matrices are a type of lower block triangular Toeplitz matrices with the property that all the square submatrices that can possibly be nonsingular due to the lower block triangular structure are nonsingular. We present a new class of matrices that are superregular over a suficiently large finite field F. Such construction works for any given choice of characteristic of the field F and code parameters (n; k; d) such that (n-k)|d. Finally, we discuss the size of F needed so that the proposed matrices are superregular.
... The state-space realization x t+1 = x t A + u t B, v t = x t C + u t D has been introduced in [12] and has also been discussed in [4,13]. It is different, however, from the state-space system used in [18,20,19,17]. In those papers the codeword is made up by the combined input and output, while in our case the codeword coincides with the output of the system. ...
Detailed information about the weight distribution of a convolutional code is given by the adjacency matrix of the state diagram associated with a controller canonical form of the code. We will show that this matrix is an invariant of the code. Moreover, it will be proven that codes with the same adjacency matrix have the same dimension and the same Forney indices and finally that for one-dimensional binary convolutional codes the adjacency matrix determines the code uniquely up to monomial equivalence.
... Forney [5] was the first author who introduced algebraic tools in order to describe convolutional codes. Addressing the construction of maximum-distanceseparable (MDS) convolutional codes (in the sense that the codes attain the generalized Singleton bound introduced in [28, Theorem 2.2]), there exist interesting papers in the literature [8,28,30]. Concerning the optimality with respect to other bounds we have [26,27], and in [7], strongly MDS convolutional codes were constructed. In [2,12,16,29], the authors presented constructions of convolutional BCH codes. ...
In this paper we show how to construct new convolutional codes from old ones by applying the well-known techniques: puncturing, extending, expanding, direct sum, the (u|u + v) construction and the product code construction. By applying these methods, several new families of convolutional codes can be constructed. As an example of code expansion, families of convolutional codes derived from classical Bose- Chaudhuri-Hocquenghem (BCH), character codes and Melas codes are constructed.
... Note that d S and d R can be obtained by an exhaustive computer search. In the case of a large search space, d is determined by the maximum distance separable (MDS) limit of linear block codes [39]. The ultimate P b of the proposed QC-LDPCC-SLD is calculated by substituting Eq. 35 into 28. ...
This manuscript proposes the quasi-cyclic low-density parity-check (QC-LDPC) coded-cooperative scheme based on the split labeling diversity (QC-LDPCC-SLD) with a single antenna over the Rayleigh frequency-flat fast fading channel. The two grith-4 cycle-free QC-LDPC codes are employed in the source and relay, respectively. To cancel the cycles of length-4 in the cross-layer of the joint QC-LDPC codes obtained at the destination, a simple optimized approach is developed based on the bi-layer joint Tanner graph. In addition, 16-ary quadrature amplitude modulation (16-QAM) and 16-ary phase shift keying (16-PSK) are utilized in the proposed QC-LDPCC-SLD scheme, where the two distinct optimized labeling mappers of labeling diversity (LD) are split appropriately and equipped in the relay and source, respectively, therefore lowering the error-floor (EF) region. Moreover, the novel joint Turbo-Decoding Message-Passing iterative decoding algorithm is developed to further enhance the overall BER performance and reduce the decoding complexity. Theoretical analysis and numerical results demonstrate that the proposed QC-LDPCC-SLD scheme significantly outperforms the corresponding non-cooperative schemes by more than 3.4 dB and is closer to the EF bound at the high signal-to-noise rate. Moreover, the performance of the proposed QC-LDPCC-SLD scheme is improved by at least 16 % at a BER of the order of 10-5 compared to the existing QC-LDPC counterpart systems.
... Convolutional codes are more involved than block codes and an additional parameter needs to be introduced: the degree of the code. MDS convolutional codes were introduced in [2] and thoroughly studied by many researchers in the last two decades [3,4]. These codes were mainly investigated in the context of q-ary symmetric channels and hence the Hamming distance was considered. ...
... and the weight wt(v) of v ∈ F n q is the number of nonzero components of v. In [8], the authors obtained an upper bound for the free distance of an (n, k, δ) convolutional code C given by ...
... G(d) is said to be full row rank if its projection into Z p ((D)) is full row rank. We say that C is an (n, k)-convolutional code and following [8], a matrix G(d) satisfying these conditions is called an encoder of C, see also [5,17]. ...
In this work, we analyze the problem of catastrophicity of encoders of convolutional codes over the Laurent series with coefficients in ℤpr, ℤpr((d)). Kuijper and Pinto proved in [M. Kuijper and R. Pinto, On minimality of convolutional ring encoders, IEEE Trans. Autom. Control 55(11) (2009) 4890–4897] that, contrary to what happens for codes over 𝔽((d)), where 𝔽 is a field, when dealing with ℤpr((d)) there are convolutional codes that do not admit non-catastrophic encoders. Nevertheless it was conjectured that any catastrophic convolutional code admits another type of non-catastrophic encoder called p-encoder. In this paper we solve this conjecture for a class of (2, 1) convolutional codes over ℤp2 and show that, in fact, these codes always admit a non-catastrophic p-encoder. We also describe a constructive procedure that allows us to obtain a non-catastrophic p-encoder.
... It is shown in [3] that the free distance of an (n, k, δ) convolutional code satisfies the following upper bound that generalizes the Singleton bound for block codes: ...
Convolutional codes with a maximum distance profile attain the largest possible column distances for the maximum number of time instants and thus have outstanding error-correcting capability especially for streaming applications. Explicit constructions of such codes are scarce in the literature. In particular, known constructions of convolutional codes with rate k/n and a maximum distance profile require a field of size at least exponential in n for general code parameters. At the same time, the only known lower bound on the field size is the trivial bound that is linear in n. In this paper, we show that a finite field of size is necessary for constructing convolutional codes with rate k/n and a maximum distance profile of length L. As a direct consequence, this rules out the possibility of constructing convolutional codes with a maximum distance profile of length L >= 3 over a finite field of size O(n). Additionally, we also present an explicit construction of convolutional code with rate k/n and a maximum profile of length L = 1 over a finite field of size , achieving a smaller field size than known constructions with the same profile length.
... and the weight wt(v) of v ∈ F n q is the number of nonzero components of v. In [8], the authors obtained an upper bound for the free distance of an (n, k, δ) convolutional code C given by ...
Maximum-distance separable (MDS) convolutional codes are characterized by the property that their free distance reaches the generalized Singleton bound. In this paper, new criteria to construct MDS convolutional codes are presented. Additionally, the obtained convolutional codes have optimal first (reverse) column distances and the criteria allow to relate the construction of MDS convolutional codes to the construction of reverse superregular Toeplitz matrices. Moreover, we present some construction examples for small code parameters over small finite fields.
... In [18] Smarandache and Rosenthal established an analogue of the Singleton bound for convolutional codes. Theorem 2.2 (Singleton bound). ...
In 1997 Rosenthal and York defined generalized Hamming weights for convolutional codes, by regarding a convolutional code as an infinite dimensional linear code endowed with the Hamming metric. In this paper, we propose a new definition of generalized weights of convolutional codes, that takes into account the underlying module structure of the code. We derive the basic properties of our generalized weights and discuss the relation with the previous definition. We establish upper bounds on the weight hierarchy of MDS and MDP codes and show that that, depending on the code parameters, some or all of the generalized weights of MDS codes are determined by the length, rank, and internal degree of the code. We also prove an anticode bound for convolutional codes and define optimal anticodes as the codes which meet the anticode bound. Finally, we classify optimal anticodes and compute their weight hierarchy.
... Since sum-rank weights are smaller than or equal to Hamming weights, we deduce the following bound on sum-rank column distances from the classical bound on column distances of convolutional codes [50], [151]. Proposition 6.3. ...
Hamming distance and rank metric have long been used in coding theory. The sum-rank metric naturally extends these over fields. They have attracted significant attention for their applications in distributed storage systems, multishot network coding, streaming over erasure channels, and multi-antenna wireless communication. In this monograph, the authors provide a tutorial introduction to the theory and applications of sum-rank metric codes over finite fields. At the heart of the monograph is the construction of linearized Reed–Solomon codes, a general construction of maximum sum-rank distance (MSRD) codes with polynomial field sizes. These are specialized classical Reed–Solomon and Gabidulin code constructions in the Hamming and rank metrics, respectively and produce an efficient Welch–Berlekamp decoding algorithm. The authors proceed to develop applications of these codes in distributed storage systems, network coding, and multi-antenna communication are developed before surveying other families of codes in the sum-rank metric, including convolutional codes and subfield subcodes are described, and recent results in the general theory of codes in the sum-rank metric. This tutorial on the topic provides the reader with a comprehensive introduction to both the theory and practices of this important class of codes used in many storage and communication systems. It will be a valuable resource for students, researchers and practising engineers alike.
... , p − 1}. In [9] distance properties of convolutional codes over Z p r were analyzed and a Singleton-like bound was proposed for the free distance, generalizing the one provided in [27] for convolutional codes over finite fields. Finally, in [22] the notion of column distances for convolutional codes over Z p r was investigated and MDP convolutional codes were defined as a generalization of MDP convolutional codes over finite fields. ...
In this paper, we develop the theory of convolutional codes over finite commutative chain rings. In particular, we focus on maximum distance profile (MDP) convolutional codes and we provide a characterization of these codes, generalizing the one known for fields. Moreover, we relate MDP convolutional codes over a finite chain ring with MDP convolutional codes over its residue field. Finally, we provide a construction of MDP convolutional codes over finite chain rings generalizing the notion of superregular matrices.
In this paper, we propose a new erasure decoding algorithm for convolutional codes using the generator matrix. This implies that our decoding method also applies to catastrophic convolutional codes in opposite to the classic approach using the parity-check matrix. We compare the performance of both decoding algorithms. Moreover, we enlarge the family of optimal convolutional codes (complete-MDP) based on the generator matrix.
In this paper, we propose a new erasure decoding algorithm for convolutional codes using the generator matrix. This implies that our decoding method also applies to catastrophic convolutional codes in opposite to the classic approach using the parity-check matrix. We compare the performance of both decoding algorithms. Moreover, we enlarge the family of optimal convolutional codes (complete-MDP) based on the generator matrix.
Based on known bounds for relative generalized Hamming weights of linear codes, we provide several new bounds for generalized column distances of convolutional codes, including the Griesmer-type bound for generalized column distances. Then we construct several infinite families of convolutional codes such that the (1, 1)-Griesmer defect of these convolutional codes is small compared with the length of these convolutional codes by using cyclic codes, negacyclic codes and GRS codes. In particular, we obtain some convolutional codes such that the (1, 1)-Griesmer defect of these convolutional codes is zero or one. Next we prove that the 2-generalized column distance sequence { d 2, j ( C )}∞ j =1 of any convolutional code C is increasing and bounded from above, and the limit of the sequence { d 2, j ( C )}∞ j =1 is related to the 2- generalized Hamming weight of the convolutional code C . For i ≥ 3, we prove that the i -generalized column distance sequence { d i,j ( C )}∞ j =[ i/k −1] of any convolutional code C is bounded above and below.
In this paper, we propose three Griesmer type bounds for the minimum Hamming weight of complementary codes of linear codes. Infinite families of complementary codes meeting the three Griesmer type bounds are given to show these bounds are tight. The Griesmer type bounds proposed in this paper are significantly stronger than the classical Griesmer bound for linear codes. As a by-product, we construct some optimal few-weight codes and determine their weight distributions. As an application, Griesmer type bounds for the column distance of convolutional codes are presented. These Griesmer type bounds are stronger than the Singleton bound for convolutional codes.
We revisit the notion of rth generalized column weights for the j-truncation of a convolutional code. Taking the limit as j tends to infinity, we define rth generalized column weights of a convolutional code. We introduce suitable notions of j-equivalence and equivalence, with respect to which generalized column weights are code invariants. We establish some properties of these invariants and compare them with other definitions of generalized distance and weight which appear in the literature.
In emergency scenarios, strong mobility and serious interference cause unstable transmission of on-site information such as close-up photos and high resolution videos, which requires a robust temporary communication network. In this paper, we focus on a UAV-assisted wireless cooperative communication and coded caching network, where emergency command vehicles and a UAV serve as content providers (CPs) to cache and transmit coded fragments or complete files for rescuers regarded as content requesters (CRs). The delivery success probability and content hit ratio are theoretically derived by incorporating the physical connectivity and social relationship between CPs and CRs. Aiming at maximizing the overall content hit ratio, we propose a multiagent two-timescale deep reinforcement learning (MA2T-DRL) algorithm to jointly optimize the transmission power and caching strategies for CPs. Specifically, we develop a two tier deep-Q networks (DQNs) framework integrating a slow-timescale DQN (ST-DQN) and a fast-timescale DQN (FT-DQN) for caching decision-making and power decision-making respectively, and then the QMIX framework is leveraged to aggregate all the outputs from local ST-DQNs. Considering the cooperative characteristics of coded caching, we further propose a novel clustering method for CPs such that CPs in the same cluster have the same willingness to serve CRs, and each cluster is regarded as the agent for training which further reduces the aggregation scale of the mixing network. Simulation results show that the proposed MA2T-DRL algorithm is efficient in model training, and presents the advantages in performance and complexity compared with the single-agent centralized training and the multiagent independent distributed training.
There exists a large literature of construction of convolutional codes with maximal or near maximal free distance. Much less is known about constructions of convolutional codes having optimal or near optimal column distances. In this paper, a new construction of convolutional codes over the binary field with optimal column distances is presented.
In 1997 Rosenthal and York defined generalized Hamming weights for convolutional codes, by regarding a convolutional code as an infinite dimensional linear code endowed with the Hamming metric. In this paper, we propose a new definition of generalized weights of convolutional codes, that takes into account the underlying module structure of the code. We derive the basic properties of our generalized weights and discuss the relation with the previous definition. We establish upper bounds on the weight hierarchy of MDS and MDP codes and show that, depending on the code parameters, some or all of the generalized weights of MDS codes are determined by the length, rank, and internal degree of the code. We also prove an anticode bound for convolutional codes and define optimal anticodes as the codes which meet the anticode bound. Finally, we classify optimal anticodes and compute their weight hierarchy.
In this paper we present a concrete algebraic construction of a novel class of convolutional codes. These codes are built upon generalized Vandermonde matrices and therefore can be seen as a natural extension of Reed–Solomon block codes to the context of convolutional codes. For this reason we call them weighted Reed–Solomon (WRS) convolutional codes. We show that under some constraints on the defining parameters these codes are Maximum Distance Profile (MDP), which means that they have the maximal possible growth in their column distance profile. We study the size of the field needed to obtain WRS convolutional codes which are MDP and compare it with the existing general constructions of MDP convolutional codes in the literature, showing that in many cases WRS convolutional codes require significantly smaller fields.
Convolutional codes are essential in a wide range of practical applications due to their efficient non-algebraic decoding algorithms. In this paper, we first propose a new family of matrices over finite fields by combining Vandermonde and Moore matrices. Using favourable properties of the matrices in this new family enables us to construct a new family of convolutional codes with memory 1 and maximum distance profile. It is notable that the alphabet sizes of this new family of convolutional codes with maximum distance profile can be kept significantly smaller than those in the literature. Keeping the code rate to a constant, the alphabet size is roughly the square root of the previously best-known value.
Algebraic methods for the design of series of maximum distance separable (MDS) linear block and convolutional codes to required specifications and types are presented. Algorithms are given to design codes to required rate and required error-correcting capability and required types. Infinite series of block codes with rate approaching a given rational R with and relative distance over length approaching are designed. These can be designed over fields of given characteristic p or over fields of prime order and can be specified to be of a particular type such as (i) dual-containing under Euclidean inner product, (ii) dual-containing under Hermitian inner product, (iii) quantum error-correcting, (iv) linear complementary dual (LCD). Convolutional codes to required rate and distance and infinite series of convolutional codes with rate approaching a given rational R and distance over length approaching are designed. The designs are algebraic and properties, including distances, are shown algebraically. Algebraic explicit efficient decoding methods are referenced.KeywordsCodeMDSDual-containingQECCLCDConvolutional
We construct a family of
(n,k)
convolutional codes with degree
that have a maximum distance profile. The field size required for our construction is
, which improves upon the known constructions of convolutional codes with a maximum distance profile. Our construction is based on the theory of skew polynomials.
We consider the algorithmic problem of computing a primitive idempotent of a central simple algebra over the field of rational functions over a finite field. The algebra is given by a set of structure constants. The problem is reduced to the computation of a division algebra Brauer equivalent to the central simple algebra. This division algebra is constructed as a cyclic algebra, once the Hasse invariants have been computed. We give an application to skew constacyclic convolutional codes.
In this paper, we study the conditions for a convolutional code to be MDP in terms of the size of the base field Fq as well as the openness of the MDP property in a given family of convolutional codes. Given (n,k,δ), our main result is an explicit bound depending on (n,k,δ) such that if q is greater than this bound, there exists a (n,k,δ) MDP convolutional code. A similar result is also offered for complete MDP convolutional codes. We show that these bounds are much lower than that those appeared so far in the literature. Finally, we show an explicit and simple construction procedure for MDP convolutional Goppa codes of dimension one.
We compute the degree of the generalized Pl\"ucker embedding of a
Quot scheme X over \PP^1. The space X can also be considered as a
compactification of the space of algebraic maps of a fixed degree from \PP^1
to the Grassmanian . Then the degree of the embedded variety
can be interpreted as an intersection product of pullbacks of
cohomology classes from through the map that evaluates
a map from \PP^1 at a point x\in \PP^1. We show that our formula for the
degree verifies the formula for these intersection products predicted by
physicists through Quantum cohomology~\cite{va92}~\cite{in91}~\cite{wi94}. We
arrive at the degree by proving a version of the classical Pieri's formula on
the variety X, using a cell decomposition of a space that lies in between X
and .
It is well known that a convolutional code is essentially a linear system defined over a finite field. In this paper we elaborate on this connection. We will define convolutional codes as the dual of a complete linear behavior in the sense of Willems. Using ideas from systems theory we describe a set of generalized first order descriptions for convolutional codes. As an application of these ideas, we present a new algebraic construction for convolutional codes. Index-Terms: Convolutional codes, behaviors, duality, first order representations, code constructions. I. Introduction In this paper we take a detailed look at convolutional codes from the perspective of linear systems theory with an emphasis on duality relations and on the different representations of these codes. Using these representations, we present a construction of convolutional codes with distance lower bounded by the complexity of the encoder. Throughout the relatively short history of the theory of convolutional codes...
Part 1 Rational matrices and rational vector spaces: algebraic preliminaries Euclidean domains of rational functions pole/zero structure of a rational matrix Wiener-Hopf structure of a rational matrix minimal basis of a rational vector space preliminary results for matrix pencils. Part 2 Representations of linear time-invariant systems: dynamical systems AR representations ARMA representations first-order representations systems with split external variables. Part 3 Minimality and transformation groups: minimality of a P representation minimality of a D representation minimality of a DZ representation minimality of a DP representation transformation groups. Part 4 Realization in minimal first-order form: realization in pencil form - the abstract procedure the pencil realization in terms of a discrete-time behaviour choosing bases connections with the Fuhrmann realization. Part 5 Structural invariants: observability indices controllability indices the input-output structure.
Willems' approach to dynamical systems without a priori distinguishing between inputs and outputs is accepted, and with this as a starting point, new linear dynamical systems are introduced and studied. It is proved in particular that (in the complex case) the set of isomorphism classes of completely observable (or completely reachable) linear systems with given input and output numbers and McMillan degree, has a natural structure of a compact algebraic variety. This variety is closely connected to the one constructed by Hazewinkel using the Rosenbrock linear systems {Mathematical expression}=Ax+Bu, v=Cx+D(·)u, where D is a polynomial matrix, and may be regarded as the most natural compactification of it. (The connection is very similar to that of Grassm,mx+p(ℂ) and Matm.p(ℂ). Input/output linear systems, i.e. linear systems equipped with an extra structure which distinguishes input and output signals, are also considered. It is shown that each of them may be represented by the equations K{Mathematical expression}+L{Mathematical expression}=Fx+Gu, v=Hx+Ju (det(K-sF)∈0). Such systems clearly contain the so-called generalized linear systems. They also contain the Rosenbrock linear systems mentioned above and essentially coincide with them.
It is a classical result of Clark that the space of all proper or strictly properp m transfer functions of a fixed McMillan degreed has, in a natural way, the structure of a noncompact, smooth manifold. There is a natural embedding of this space into the set of allp (m+p) autoregressive systems of degree at mostd. Extending the topology in a natural way we will show that this enlarged topological space is compact. Finally we describe a homogenization process which produces a smooth compactification.
In this paper we give a self-contained overview of known distance measures for convolutional codes and of upper and lower bounds on the free distance. The upper bounds are valid for general trellis codes and for convolutional codes, respectively. The lower bound is valid for time-varying convolutional codes. We also present a new lower bound on the distance profile for fixed convolutional codes.
In this note we show that the geometric quotient, under a natural group action, of the generalized state space systems recently considered by Schumacher, Kuijper and Geerts is algebraically isomorphic to the space of a homogeneous autoregressive system. This result essentially follows from work of Stromme published earlier in the algebraic geometry literature. In particular, these generalized state space systems represent a realization of the space of homogeneous autoregressive systems.
Maximum distance separable (MDS) convolutional codes are defined as the row space over F(D) of totally nonsingular polynomial matrices in the indeterminate D . These codes may be used to transmit information on n parallel channels when a temporary or even an infinite break can occur in some of these channels. Their algebraic properties are emphasized, and the relevant parameters are introduced. On this basis two decoding procedures are described. Both procedures correct arbitrarily long error sequences that may occur at the same time in some of the n channels. Some specific constructions of MDS convolutional codes are presented.
Electrical Engineering Fundamentals of Convolutional Coding A volume in the IEEE Press Series on Digital and Mobile Communication John B. Anderson, Series Editor Convolutional codes, among the main error control codes, are routinely used in applications for mobile telephony, satellite communications, and voice-band modems. Written by two leading authorities in coding and information theory, this book brings you a clear and comprehensive discussion of the basic principles underlying convolutional coding. Fundamentals of Convolutional Coding is unmatched in the field for its accessible analysis of the structural properties of convolutional encoders. Other essentials covered in Fundamentals of Convolutional Coding include:
Distance properties of convolutional codes
Viterbi, list, sequential, and iterative decoding
Modulation codes
Tables of good convolutional encoders
Plus an extensive set of homework problems
The authors draw on their own research and more than 20 years of teaching experience to present the fundamentals needed to understand the types of codes used in a variety of applications today. This book can be used as a textbook for graduate-level electrical engineering students. It will be of key interest to researchers and engineers of wireless and mobile communication, satellite communication, and data communication.
In this paper we provide a state space approach for constructing
convolutional codes of rate 1/n and complexity δ, whose free
distance is n(δ+1), the maximal possible free distance
If the constraint length of a convolutional code is defined suitably, it is an obvious upper bound on the free distance of the code, and it is sometimes possible to find codes that meet this bound. It is proved here that the length of a rate 1/nu q -ary code with this property is at most qnu , and we construct a class of such codes with lengths greater than qnu/3 .
We define maximum-distance-separable convolutional codes as systematic codes with (feedback) minimum distance exceeding the number of check digits in a constraint length. The maximum length of such codes is determined for certain small fields when the rate is frac{1}{2} .
A convolutional encoder is defined as any constant linear sequential circuit. The associated code is the set of all output sequences resulting from any set of input sequences beginning at any time. Encoders are called equivalent if they generate the same code. The invariant factor theorem is used to determine when a convolutional encoder has a feedback-free inverse, and the minimum delay of any inverse. All encoders are shown to be equivalent to minimal encoders, which are feedback-free encoders with feedback-free delay-free inverses, and which can be realized in the conventional manner with as few memory elements as any equivalent encoder, Minimal encoders are shown to be immune to catastrophic error propagation and, in fact, to lead in a certain sense to the shortest decoded error sequences possible per error event. In two appendices, we introduce dual codes and syndromes, and show that a minimal encoder for a dual code has exactly the complexity of the original encoder; we show that systematic encoders with feedback form a canonical class, and compare this class to the minimal class.
Using a new parity-check matrix, a class of convolutional codes
with a designed free distance is introduced. This new class of codes has
many characteristics of BCH block codes, therefore, we call these codes
BCH convolutional codes
A maximum distance separable (MDS) convolutional code is a linear code whose distance is maximal among all linear convolutional codes of rate k/n and fixed degree #. We have conjectured already in previous work that MDS convolutional codes do exist if one allows su#ciently many field elements. In this paper we will give the proof of this fact in the case # = 0 without making use of the very well known Reed-Solomon construction. This will illustrate how things work in the general case. Also we will question the nature of the dual code of a MDSconvolutional code. We will show that in general the dual of a MDS-convolutional code is not MDS. Keywords: MDS codes, convolutional codes. 1 Introduction In the literature there were already several papers [3, 8] which considered the concept of a maximum distance separable convolutional code. In each of these approaches it was necessary to restrict the total class of rate k/n convolutional codes to a suitable subclass. This is simply du...
Using a new parity check matrix, a class of convolutional codes with a designed free distance is introduced. This new class of codes has many characteristics of BCH block codes, therefore we call these codes BCH convolutional codes. Keywords: Convolutional codes, linear systems, BCH codes, cyclotomic sets. 1 Introduction Convolutional codes having a large free distance and a low degree are often found by computer searches. Several authors have extended constructions known for block codes to convolutional codes. A survey of some of this work is provided in the book of Piret [17, Section 3.5] where more complete references can be found. Most of these constructions are based on cyclic or quasi cyclic constructions of block codes. These techniques originate in work by Massey, Costello and Justesen [13] where it is shown how the free distance of a convolutional code can be lower bounded by the distance of a related cyclic code. In [7, 8] Justesen refines the method and he constructs polyn...
The algebraic theory of convolutional codes Handbook of Coding Theory Piret Ph.: Convolutional Codes, an Algebraic Approach
- R J Mceliece
McEliece R. J.: The algebraic theory of convolutional codes. In: R. Brualdi, W.C. Huffman, V. Pless (eds) Handbook of Coding Theory. Amsterdam: Elsevier 1998 11. Piret Ph.: Convolutional Codes, an Algebraic Approach. Cambridge, MA, MIT Press 1988 12. Piret Ph., Krol T.: MDS convolutional codes. IEEE Trans. Inform. Theory, 29(2):224–232 (1983)