Decoding Lattice Partitions with application to Decoding Coset Codes

F.-W. Sun and Henk .C.A.van Tilborg

D e p a r t m e n t of M a t h e m a t i c s and C o m p u t i n g Science Eindhoven Universi ty of Technology P.Q.Box 513, 5600 MB Eindhoven

T h e Nether lands

Abstract-Several new algorithms for decoding lattice parti- tions are presented. They apply to Viterbi decoding of multidi- mensional trellis codes based on these partitions. In [l, 21, trellis- based algorithms were presented for decoding the lattice parti- tions. The new algorithms can achieve about 50% reduction of the complexity of decoding the lattice partitions in terms of real additions/comparisons compared with the algorithms of [l, 21. The complexity of the resulting overall Viterbi decoding algo- rithms still shows a modest improvement. An algorithm for soft decision decoding the first-order Reed-Muller code (8,4,4) or the Gosset lattice is also presented. It involves a t most 17 real operations, thus, improving the best known algorithm.

Summary A typical multidimensional trellas coded modulatzon(MTCM)

scheme can be simply described by two basic ingredients: one is the cosets of a lattice partition A/A’, where A is a lattice and A’ is a sublattice such that the order of the partition is finite; the other is a conventional binary convolutional code. The output of the binary encoder chooses the coset, and some other information bits specify an element in the coset [I, Fig.11.

The trellis diagram of the resulting multidimensional trellis code is essentially the same as that of the convolutional code. The difference is that the labels on the branches of the trel- lis disgram of the convolutional code now correspond to cosets. Thus, a trellis-searched decoding algorithm such as the Viterbi algorithm can be used to decode a multidimensional trellis code.

In a soft-decision Viterbi-decoding algorithm, the first step of the decoding requires computing the branch metrics. This step is called decoding the branches. For an MTCM based on a lattice partition, decoding the branches turns out to be equiva- lent to decoding the lattace partataon in use. This means that the closest points in each of the cosets to the received point has to determined and the associated metrics need to be calculated.

In [1, 21, Forney gave trellas-based algorzthms for decoding lattice partitions. His algorithms are optimal trellis decoding for given coordinate order and alphabet among all the trellis decoding in the sense that it uses smallest number of trellis states [3]. Therefore, the expression trellas-based algorithm will simply stand for the kind of trellis decoding algorithms described in [2].

Certainly decoding the branches is only part of the overall de- coding procedure. However, for a code whose number of states is small relative to its dimension, a considerable portion of the overall decoding work is due to decoding the branches. Further- more, in most practical implementations, the number of trellis code states used has been very low(typical1y 4 or 8, occasion- ally 16 but rarely more than 16) [4]. Therefore, by reducing the complexity of decoding the branches, it is possible to achieve a considerable amount of reduction of the overall decoding com- plexity.

In this work, we present several new algorithms for decoding the lattice partitions. Most of the algorithms can achieve about 50% reduction of the complexity of decoding the branches. They result in modest improvement of the overall decoding complexity.

Most previous known efficient algorithms for soft-decision de- coding block codes and lattices rely on decoding each coset of a subcode of certain type in combination with Wagner decoding rule. The Wagner decoding rule applies to binary codes whose check matrix consists of a single all-one row. It states that an entry-by-entry hard detection of the received word is to be fol- lowed, unless the number of 1 bits is already even, by inversion of the least reliable bit. Our algorithms also fall into this category.

The complexity of the decoding is measured by the total num- ber of real additions and comparisons. Certainly the actually running time always depend to some extent on implementation technology in use. We did try, however, to evaluate all the al- gorithms in a uniform way. In compliance with the convention estabilished in the literature, we ignore such operations as mem- ory addressing, negation, taking the absolute value, and multi- plication by 2, as well as the checking of logical conditions and modulo 2 additions.

References

G.D.Forney Jr.,“Coset codes-Part I: Introduction and geo- metrical classification,” IEEE Trans. on Inform. Theory vol. IT-34,pp. 1123-1151, Sept. 1988.

G.D.Forney Jr.. “Coset codes-Part 1I:Binary lattices and related codes,” IEEE Trans. on Inform. Theory vol. 34,pp.1152-1187, Sept. 1988.

D.J. Muder, ‘‘ Minimal Trellises for Block Codes,” IEEE Trans. on Inform. Theory vo1.34, N0.5 , pp1049-1083 Sept. 1988.

A.J.Viterbi, J.K. Wolf,E. Aehavi, R. Padovani, “ A Prag- matic Approach to Trellis-Coded Modulation,” IEEE Com Mag. pp. 11-19, July 1989.

66