Abstract — We consider the design of channel codes for improving the data rate and/or the reliability of communications over fading channels using multiple transmit antennas. Data is encoded by a channel code and the encoded data is split into � streams that are simultaneously transmitted using � transmit antennas. The received signal at each receive antenna is a linear superposition of the � transmitted signals perturbed by noise. We derive performance criteria for designing such codes under the assumption that the fading is slow and frequency nonselective. Performance is shown to be determined by matrices constructed from pairs of distinct code sequences. The minimum rank among these matrices quantifies the diversity gain, while the minimum determinant of these matrices quantifies the coding gain. The results are then extended to fast fading channels. The design criteria are used to design trellis codes for high data rate wireless communication. The encoding/decoding complexity of these codes is comparable to trellis codes employed in practice over Gaussian channels. The codes constructed here provide the best tradeoff between data rate, diversity advantage, and trellis complexity. Simulation results are provided for 4 and 8 PSK signal sets with data rates of 2 and 3 bits/symbol, demonstrating excellent performance that is within 2–3 dB of the outage capacity for these channels using only 64 state encoders.

The problem of minimizing the vertex count at a given time index in the trellis for a general (nonlinear) code is shown to be NPcomplete. Examples are provided that show that 1) the minimal trellis for a nonlinear code may not be observable, i.e., some codewords may be represented by more than one path through the trellis and 2) minimizing the vertex count at one time index may be incompatible with minimizing the vertex count at another time index. A trellis product is defined and used to construct trellises for sum codes. Minimal trellises for linear codes are obtained by forming the product of elementary trellises corresponding to the one-dimensional subcodes generated by atomic codewords. The structure of the resulting trellis is determined solely by the spans of the atomic codewords. A correspondence between minimal linear block code trellises and configurations of non-attacking rooks on a triangular chess board is established and used to show that the number of distinct minimal li...

Abstruct- In this semi-tutorial paper, we will investigate the computational complexity of an abstract version of the Viterbi algorithm on a trellis, and show that if the trellis has e edges, the complexity of the Viterbi algortithm is @(e). This result suggests that the “best ” trellis representation for a given linear block code is the one with the fewest edges. We will then show that, among all trellises that represent a given code, the original trellis introduced by Bahl, Cocke, Jelinek, and Raviv in 1974, and later rediscovered by Wolf, Massey, and Forney, uniquely minimizes the edge count, as well as several other figures of merit. Following Forney and Kschischang and Sorokine, we will also discuss “trellis-oriented ” or “minimal-span ” generator matrices, which facilitate the calculation of the size of the BCJR trellis, as well as the actual construction of it. Index Terms-Block complexity.

We show that the family of maximal fixed-cost (MFC) codes, with codeword costs defined in a right-cancellative semigroup, are rectangular, and hence admit biproper trellis presentations. Among all possible trellis presentations for a rectangular code, biproper trellises minimize a wide variety of complexity measures, including the Viterbi decoding complexity. Examples of MFC codes include such "nonlinear" codes as permutation codes, shells of constant norm in the integer lattice, and linear codes over a finite field. The intersection of two rectangular codes is another rectangular code; therefore "nonlinear" codes such as lattice shells or words of constant weight in a linear code have biproper trellis presentations. We show that every rectangular code can be interpreted as an MFC code. Applications of these results include error detection, trellis-based indexing, and soft-decision decoding.

An important problem in the theory and application of block code trellises is to find a coordinate permutation of a given code to minimize the trellis complexity. In this paper we show that the problem of finding a coordinate permutation that minimizes the number of vertices at a given depth in the minimal trellis for a binary linear block code is NP-complete. Keywords: permutation trellis complexity, NP-completeness. I. Introduction Although the codes obtained by permuting the coordinates of a linear block code are equivalent, it is well known that the minimal trellises for these equivalent codes in general are not equivalent [1--7]; in particular, different coordinate permutations may yield trellises with different state complexity profiles. As J. L. Massey pointed out some time ago [2]: "the art of trellis decoding of a linear code [is] that of re-arranging the order of digits in the code word to obtain a non-systematic code for which [the trellis complexity] is as small as possibl...

We present an efficient algorithm for computing the minimal trellis for a group code over a finite Abelian group, given a generator matrix for the code. We also show how to cornpure a succinct representation of the minimal trellis for such a code, andpresent algorithms that use this information to efficiently compute local descriptions of the minimal trellis. This extends the work of Kschischang and Sorokine, who handled the case of linear codes over fields. An important application of our algorithms is to the construction qf minireal trellises for lattices. A key step in our work is handling codes over cyclic groups C'p, where p is a prime. Such a code can be viewed as a submodule over the ring Zp. Because of the presence of zero-divisors in the ring, submodules do not share the useful properties of vector spaces. We get around this difficulty by restricting the notion of linear combination to p-linear combination, and introducing the notion of a p-generator equence, which enjoys properties similar to that of a generector matrix for a vector space.

Trellis linearity, first considered by McEliece in 1996, turns out to be crucial in the study of tail-biting trellises. In this chapter, basic structural properties of linear trellises are investigated. A rigorous definition of linearity is given for both conventional and tail-biting trellises.

In 1948 Shannon developed fundamental limits on the efficiency of communication over noisy channels. The coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero. Fifty years later, codes for the Gaussian channel have been discovered that come close to these fundamental limits. There is now a substantial algebraic theory of error-correcting codes with as many connections to mathematics as to engineering practice, and the last 20 years have seen the construction of algebraic-geometry codes that can be encoded and decoded in polynomial time, and that beat the Gilbert–Varshamov bound. Given the size of coding theory as a subject, this review is of necessity a personal perspective, and the focus is reliable communication, and not source coding or cryptography. The emphasis is on connecting coding theories for Hamming and Euclidean space and on future challenges, specifically in data networking, wireless communication, and quantum information theory.

An important problem in the theory and application of block code trellises is to find a coordinate permutation for a given code that minimizes the trellis state complexity. In this paper we show that the problem of minimizing a given component of the state complexity profile is NP-complete. We describe an algorithm, though not guaranteed to find an optimal coordinate ordering, uses a heuristic descent technique to find "good" solutions to the problem. We have applied this algorithm to various codes, and our results are tabulated.

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 fl of L, and are obtained based on a decoding algorithm which is an improved version of Kannan's method. The latter is currently the fastest known method for the decoding of a general lattice. For the decoding of a point xx x, the proposed algorithm recursively searches inside an n-dimensional rectangular parallelepiped (cube), centered at xx 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. It is shown that for any selected basis, the decoding complexity (using RCS algorithms) of any (fl 1) grows at least exponentially with n and fl. 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.