Upper bounds on trellis complexity of lattices

Unlike block codes, n-dimensional lattices can have minimal trellis diagrams with an arbitrarily large number of states, branches, and paths. In particular, we show by a counterexample that there is no f(n), a function of n, such that all rational lattices of dimension n have a trellis with less than f(n) states. Nevertheless, using a theorem due to Hermite, we prove that every integral lattice Λ of dimension n has a trellis T, such that the total number of paths in T is upper-bounded by P(T)⩽n!(2/√3)^n2(n-1/2)V(Λ) ^n-1 where V(n) is the volume of Λ. Furthermore, the number of states at time i in T is upper-bounded by |S_i|⩽(2/√3)^i2(n-1)V(Λ)²ⁱ² n/. Although these bounds are seldom tight, these are the first known general upper bounds on trellis complexity of lattices     

Density/length profiles and trellis complexity of lattices

The density/length profile (DCP) of a lattice Λ is analogous to the dimension/length profile of a linear code. The DLP is a geometrical invariant of Λ that includes the coding gain of Λ. Duality results analogous to those of linear block codes are derived for lattices. Bounds on the DLP may be derived from bounds on Hermite's constants; these hold with equality for many dense lattices. In turn, the DLP lowerbounds the state complexity profile of a minimal trellis diagram for Λ in any coordinate system. It is shown that this bound can be met for the E₈ lattice by a laminated lattice construction with a novel trellis diagram. Bounds and constructions for other important low-dimensional lattices are given. Two laminated lattice constructions of the Leech lattice yield trellis diagrams with maximum state space sizes 1024 and 972     

Reduced complexity decoding of convolutional codes based on the M-algorithm and the minimal trellis

In this paper, we propose three new sub-optimum, reduced complexity decoding algorithms for convolutional codes. The algorithms are based on the minimal trellis representation for the convolutional code and on the M-algorithm. Since the minimal trellis has a periodically time-varying state profile, each algorithm has a different strategy to select the number of surviving states in each trellis depth. We analyse both the computational complexity, in terms of arithmetic operations, and the bit error rate performance of the proposed algorithms over the additive white Gaussian noise channel. Results demonstrate that considerable complexity reductions can be obtained at the cost of a small loss in the performance, as compared to the Viterbi decoder.     

New trellis codes based on lattices and cosets

A new technique is proposed for constructing trellis codes. which provides an alternative to Ungerboeck's method of "set partitioning." The new codes use a signal constellation consisting of points from ann-dimensional latticeLambda, with an equal number of points from each coset of a sublatticeLambda '. One part of the input stream drives a generalized convolutional code whose outputs are cosets ofLambda ', while the other part selects points from these cosets. Several of the new codes are better than those previously known.     

关于非线性码的格子复杂度

研究了非线性码的格子复杂度.给出了非线性码的维数/长度轮廓的定义,并利用这一定义,将Forney所给出的线性码格子复杂度的新下界推广到了非线性码上去.然后推出了非线性码的Berger-Be′ery上界.     

Entropy/length profiles, bounds on the minimal covering ofbipartite graphs, and trellis complexity of nonlinear codes

The trellis representation of nonlinear codes is studied from a new perspective. We introduce the new concept of entropy/length profile (ELP). This profile can be considered as an extension of the dimension/length profile (DLP) to nonlinear codes. This elaboration of the DLP, the entropy/length profiles, appears to be suitable to the analysis of nonlinear codes. Additionally and independently, we use well-known information-theoretic measures to derive novel bounds on the minimal covering of a bipartite graph by complete subgraphs. We use these bounds in conjunction with the ELP notion to derive both lower and upper bounds on the state complexity and branch complexity profiles of (nonlinear) block codes represented by any trellis diagram. We lay down no restrictions on the trellis structure, and we do not confine the scope of our results to proper or one-to-one trellises only. The basic lower bound on the state complexity profile implies that the state complexity at any given level cannot be smaller than the mutual information between the past and the future portions of the code at this level under a uniform distribution of the codewords. We also devise a different probabilistic model to prove that the minimum achievable state complexity over all possible trellises is not larger than the maximum value of the above mutual information over all possible probability distributions of the codewords. This approach is pursued further to derive similar bounds on the branch complexity profile. To the best of our knowledge, the proposed upper bounds are the only upper bounds that address nonlinear codes. The novel lower bounds are tighter than the existing bounds. The new quantities and bounds reduce to well-known results when applied to linear codes     

Efficient complexity reduction technique in trellis decodingalgorithm

An efficient reduced search trellis decoding algorithm in which the decoder selects a part of existing paths by using a threshold value of the path metric is proposed. The threshold value at each time stage of the trellis is found by simply investigating the statistics of the path metrics, and does not require any prior knowledge such as the signal-to-noise ratio     

Dimension/length profiles and trellis complexity of linear blockcodes

This semi-tutorial paper discusses the connections between the dimension/length profile (DLP) of a linear code, which is essentially the same as its “generalized Hamming weight hierarchy”, and the complexity of its minimal trellis diagram. These connections are close and deep. DLP duality is closely related to trellis duality. The DLP of a code gives tight bounds on its state and branch complexity profiles under any coordinate ordering; these bounds can often be met. A maximum distance separable (MDS) code is characterized by a certain extremal DLP, from which the main properties of MDS codes are easily derived. The simplicity and generality of these interrelationships are emphasized     

Designing lexicographic codes with a given trellis complexity

We generalize constructions of lexicographic codes to produce locally optimal codes with a desired trellis decoding complexity. These constructions are efficient for high-rate codes and provide a means for automated code design. As a byproduct, we improve known bounds on the parameters of lexicodes     

Syndrome trellis codes based on minimal span generator matrix

To improve the embedding efficiency of steganography, syndrome coding based on the coding theory has attracted many researchers’ attentions. In this paper, we make use of the relationship between syndrome coding for minimizing additive distortion and maximum likelihood decoding for linear codes to analyze the main parameters of convolutional codes which influence the embedding efficiency. And, the new syndrome trellis codes based on minimal span generator matrix is proposed. It can be considered an alternative construction of the state-of-the-art syndrome trellis codes (STCs) proposed by Filler and Fridrich recently. Experimental results show that the proposed scheme owns the same embedding performance to STCs and achieve the reduced time complexity and storage requirement meanwhile.     

MAP algorithms for decoding linear block codes based onsectionalized trellis diagrams

The maximum a posterioriprobability (MAP) algorithm is a trellis-based MAP decoding algorithm. It is the heart of turbo (or iterative) decoding that achieves an error performance near the Shannon limit. Unfortunately, the implementation of this algorithm requires large computation and storage. Furthermore, its forward and backward recursions result in a long decoding delay. For practical applications, this decoding algorithm must be simplifled and its decoding complexity and delay must be reduced. In this paper, the MAP algorithm and its variation's, such as log-MAP and max-log-MAP algorithms, are first applied to sectionalized trellises for linear block codes and carried out as two-stage decodings. Using the structural properties of properly sectionalized trellises, the decoding complexity and delay of the MAP algorithms can be reduced. Computation-wise optimum sectionalizations of a trellis for MAP algorithms are investigated. Also presented in this paper are bidirectional and parallel MAP decodings     

MEASURING STRUCTURE COMPLEXITY OF UML CLASS DIAGRAMS

To provide system designer a valid measure to evaluate the structure complexity of class diagrams objectively,this letter first proposes a nethod to transform a class diagrams into weighted class dependence graph,then presents a structure complexity measure for class diagrams based on tntropy distance.     

基于额外列多元LDPC码网格最小最大译码算法

为了降低多元LDPC(Low Density Parity Check Code)码网格最小最大(Trellis Min-Max,T-MM)译码算法复杂度,减少译码过程中所需存储空间,提出一种基于额外列的T-MM译码算法(Extra-Column-based Trellis Min-Max,EC-T-MM).选取网格中可靠度最高的信息构造出优化的配置集,生成一列用于更新校验节点的q维额外列信息,再根据网格路径中偏移量信息,从最小值、次小值和额外列信息中得到校验节点的外在输出信息,通过网格的路径优化降低校验节点的更新复杂度.在译码过程中,用偏移量信息代替所有变量节点输入信息,减少存储空间.仿真结果表明:该算法在几乎不损失性能的前提下,降低了计算复杂度及所需的存储空间.     

A Goppa-like bound on the trellis state complexity of algebraic-geometric codes

For a linear code C of length n and dimension k, Wolf (1978) noticed that the trellis state complexity s(C) of C is upper-bounded by w(C):=min(k,n-k). We point out some new lower bounds for s(C). In particular, if C is an algebraic-geometric code, then s(C)/spl ges/w(C)-(g-a), where g is the genus of the underlying curve and a is the abundance of the code.     

The dynamics of group codes: state spaces, trellis diagrams, andcanonical encoders

A group code C over a group G is a set of sequences of group elements that itself forms a group under a component-wise group operation. A group code has a well-defined state space Σ_k at each time k. Each code sequence passes through a well-defined state sequence. The set of all state sequences is also a group code, the state code of C. The state code defines an essentially unique minimal realization of C. The trellis diagram of C is defined by the state code of C and by labels associated with each state transition. The set of all label sequences forms a group code, the label code of C, which is isomorphic to the state code of C. If C is complete and strongly controllable, then a minimal encoder in controller canonical (feedbackfree) form may be constructed from certain sets of shortest possible code sequences, called granules. The size of the state space Σ_k is equal to the size of the state space of this canonical encoder, which is given by a decomposition of the input groups of C at each time k. If C is time-invariant and ν-controllable, then |Σ_k|=Π_1⩽j⩽v|F_j/F _j-1|^j, where F₀ ⊆···⊆ Fν is a normal series, the input chain of C. A group code C has a well-defined trellis section corresponding to any finite interval, regardless of whether it is complete. For a linear time-invariant convolutional code over a field G, these results reduce to known results; however, they depend only on elementary group properties, not on the multiplicative structure of G. Moreover, time-invariance is not required. These results hold for arbitrary groups, and apply to block codes, lattices, time-varying convolutional codes, trellis codes, geometrically uniform codes and discrete-time linear systems     

关于几类线性分组码网格图复杂度的研究

本文研究了几类线性分组码C[n,k,d]的网格图复杂度s（C）。给出并证明了码长为奇数的两类线性分组码的网格图复杂度。同时得出了有关可纠t个错的本原BCH码[2^m-1,2^m-1-mt]及其扩展本在BCH码的网格图复杂度的若干结论。从而避免了必须先寻找码的直和结构才可得到码的网格图复杂度的较好上界。     

Construction of de Bruijn sequences of minimal complexity

It is well known that the linear complexity of a de Bruijn sequenceSof length2^{n}is bounded below by2^{n- 1} + nforn geq 3. It is shown that this lower bound is attainable for alln.     

Reduced complexity closest point decoding algorithms for random lattices

Abstract-Closest point algorithms find wide applications in decoding block transmissions encountered with single- or multiuser communication links relying on a single or multiple antennas. Capitalizing on the random channel and noise models typically encountered in wireless communications, the sphere decoding algorithm (SDA) and related complexity-reducing techniques are approached in this paper from a probabilistic perspective. With both theoretical analysis and simulations, combining SDA with detection ordering is justified. A novel probabilistic search algorithm examining potential candidates in a descending probability order is derived and analyzed. Based on probabilistic search and an error-performance-oriented fast stopping criterion, a computationally efficient layered search is developed. Having comparable decoding complexity to the ing-canceling (NQ algorithm with detection ordering, simulations confirm that the novel layered search achieves considerable error-performance enhancement.     

On (n, n-1) convolutional codes with low trellis complexity

We show that the state complexity profile of a convolutional code C is the same as that of the reciprocal of the dual code of C in case that minimal encoders for both codes are used. Then, we propose an optimum permutation for any given (n, n-1) binary convolutional code that will yield an equivalent code with the lowest state complexity. With this permutation, we are able to find many (n, n-1) binary convolutional codes which are better than punctured convolutional codes of the same code rate and memory size by either lower decoding complexity or better weight spectra     

On the complexity of decoding lattices using the Korkin-Zolotarevreduced basis

Upper and lower bounds are derived for the decoding complexity of a general lattice L. The bounds are in terms of the dimension n and the coding gain γ of L, and are obtained based on a decoding algorithm which is an improved version of Kannan's (1983) method. The latter is currently the fastest known method for the decoding of a general lattice. For the decoding of a point x, the proposed algorithm recursively searches inside an, n-dimensional rectangular parallelepiped (cube), centered at x, with its edges along the Gram-Schmidt vectors of a proper basis of L. We call algorithms of this type recursive cube search (RCS) algorithms. It is shown that Kannan's algorithm also belongs to this category. The complexity of RCS algorithms is measured in terms of the number of lattice points that need to be examined before a decision is made. To tighten the upper bound on the complexity, we select a lattice basis which is reduced in the sense of Korkin-Zolotarev (1873). It is shown that for any selected basis, the decoding complexity (using RCS algorithms) of any sequence of lattices with possible application in communications (γ⩾1) grows at least exponentially with n and γ. It is observed that the densest lattices, and almost all of the lattices used in communications, e.g., Barnes-Wall lattices and the Leech lattice, have equal successive minima (ESM). For the decoding complexity of ESM lattices, a tighter upper bound and a stronger lower bound result are derived