Minimal Trellis Diagram Research Articles

Subcodes constructed from bases of Reed‐Solomon codes

AbstractThis paper proposes a method of constructing nonlinear cyclic codes that form an additive group for which the decoding complexity or error correcting performance is improved relative to RS codes, which are typical linear cyclic codes. First, a code with information digit count k = 1 is constructed by using multiple RS code basis vectors obtained from Galois field Fourier transforms. Then, for k 1, codes are constructed from this code by using idempotent elements. Also, a code with code length n = 7 was constructed as an example, and both its minimal trellis diagram complexity and bit error rate on the AWGN channel due to a computer simulation were shown to be excellent. In addition, an iterative decoder was applied to the codes with k = 4 and 5, which had peak trellis diagram complexity, and it was shown that the decoding complexity could be reduced. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 1, 86(7): 64–73, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecja.1170

The Minimal Generator Matrix of a Vector Space

The atomic vectors of a finitely generated vector space C over a field F are characterized for C a subspace of the product vector space ? = ∏ i =1 n ? i over F. For finite fields, the minimal trellis diagram for mixed-codes is determined, and this provides the L-section minimal trellis diagram for linear codes. As an example, an extremely simple yet comprehensive analysis of the trellis structure of Reed-Muller codes is given. In particular, a trellis oriented generator matrix for the 2 l -section minimal trellis diagram of a Reed-Muller code is presented.

Trellis complexity and minimal trellis diagrams of lattices

This paper presents results on trellis complexity and low-complexity trellis diagrams of lattices. We establish constructive upper bounds on the trellis complexity of lattices. These bounds both improve and generalize the similar results of Tarokh and Vardy (see ibid., vol.43, p.1294-1300, 1997). We also construct trellis diagrams with minimum number of paths for some important lattices. Such trellises are called minimal. The constructed trellises, which are novel in many cases, can be employed to efficiently decode the lattices via the Viterbi algorithm. In particular, a general structure for minimal trellis diagrams of D/sub n/ lattices is obtained. This structure corresponds to a new code formula for D/sub n/. Moreover, we develop some important duality results which are used in both deriving the upper bounds, and finding the minimal trellises. All the discussions are based on a universal approach to the construction and analysis of trellis diagrams of lattices using their bases.

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 /spl Lambda/ of dimension n has a trellis T, such that the total number of paths in T is upper-bounded by P(T)/spl les/n!(2//spl radic/3)/sup n2(n-1/2)/V(/spl Lambda/)/sup n-1/ where V(n) is the volume of /spl Lambda/. Furthermore, the number of states at time i in T is upper-bounded by |S/sub i/|/spl les/(2//spl radic/3)/sup i2(n-1)/V(/spl Lambda/)/sup 2i2/n/. Although these bounds are seldom tight, these are the first known general upper bounds on trellis complexity of lattices.

Quasi-cyclic structure of Reed-Muller codes and their smallest regular trellis diagram

The largest quasi-cyclic subcode of the Reed-Muller code R(r,m), invariant under the shift T/sup 2m-l/ is determined. This code, denoted QCR(r,m,l), is presented through its module decomposition into cyclic submodules. The smallest regular trellis diagram (SRTD) is defined for block codes. This trellis and its construction algorithm are given for the class of cyclic-form codes. Using the cyclic-form structure of QCR(r,m,l), the 2/sup l/-section SRTD of this code is determined. The eight-section SRTD is given for the Reed-Muller codes R(r,m). The quasi-cyclic subcodes of R(r,m) with regular 2/sup l/-section minimal trellis diagrams are presented.

On generalized Hamming weights of binary primitive BCH codes with minimum distance one less than a power of two

The generalized Hamming weights introduced by Wei (1991) have been shown to be fundamental descriptive parameters of a linear block code. They have been found to be useful in certain cryptographic applications and in the studies of minimal trellis diagrams of linear block codes. In this correspondence, we determine the first few and the last few generalized Hamming weights of binary primitive BCH codes with minimum distance one less than a power of two, of their extensions, and of the duals of both.

The twisted squaring construction, trellis complexity, and generalized weights of BCH and QR codes

The structure of the twisted squaring construction, a generalization of the squaring construction, is studied with respect to trellis diagrams and complexity. We show that binary affine-invariant codes, which include the extended primitive BCH codes, and the extended binary quadratic-residue codes, are equivalent to twisted squaring construction codes. In particular, a recursive symmetric reversible design of the BCH codes is derived. Using these constructions, the parameters of the minimal trellis diagram of the BCH codes are determined, including the componentwise state-space profile and trellis complexity. New designs and permutations that yield low trellis complexity for the quadratic-residue codes are presented. Generalized Hamming weights are derived from these constructions. As an example, the (48, 24, 12) quadratic-residue code is analyzed, a strictly componentwise optimal permutation is derived, and the corresponding state-space profile and complete generalized Hamming weight hierarchy are obtained.

Density/length profiles and trellis complexity of lattices

The density/length profile (DCP) of a lattice /spl Lambda/ is analogous to the dimension/length profile of a linear code. The DLP is a geometrical invariant of /spl Lambda/ that includes the coding gain of /spl Lambda/. 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 /spl Lambda/ in any coordinate system. It is shown that this bound can be met for the E/sub 8/ 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. >

Dimension/length profiles and trellis complexity of linear block codes

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.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On complexity of trellis structure of linear block codes

An upper bound on the number of states of a minimal trellis diagram for a linear block code is derived. Using this derivation a cyclic (or shortened cyclic) code or its extended code is shown to be the worst in terms of trellis state complexity among the linear codes of the same length and dimension. The complexity of the minimal trellis diagrams for linear block codes of length 2/sup m/, including the Reed-Muller codes, is analyzed. The construction of minimal trellis diagrams for some extended and permuted primitive BCH codes is presented. It is shown that these codes have considerably simpler trellis structure than the original codes in cyclic form without bit-position permutation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the optimum bit orders with respect to the state complexity of trellis diagrams for binary linear codes

It was shown earlier that for a punctured Reed-Muller (RM) code or a primitive BCH code, which contains a punctured RM code of the same minimum distance as a large subcode, the state complexity of the minimal trellis diagrams is much greater than that for an equivalent code obtained by a proper permutation of the bit positions. The problem of finding a permutation of the bit positions for a given code that minimizes the state complexity of its minimal trellis diagram is related to the generalized Hamming weight hierarchy of a code, and it is shown that, for RM codes, the standard binary order of bit positions is optimum at every bit position with respect to the state complexity of a minimal trellis diagram by using a theorem due to V.K. Wei (1991). The state complexity of the trellis diagram for the extended and permuted (64, 24) BCH code is discussed. >