1152 IEEE TRANSACTIONSON INFORMATIONTHEORY, VOL. 34, NO. 5, SEPTEMBER 1988 Coset Codes-Part 11: Binary Lattices and Related Codes G. DAVID FORNEY, JR., FELLOW, IEEE Invited Paper Abstract -The family of Barnes-Wall lattices (including D4 and E,) of increases by a factor of 21/2 (1.5 dB) for each doubling of lengths N = 2“ and their principal sublattices, which are useful in con- dimension. structing coset codes, are generated by iteration of a simple construction What may be obscured by the length of this paper is called the “squaring construction.” The closely related Reed-Muller codes are generated by the same construction. The principal properties of these that the construction of these lattices is extremely simple. codes and lattices, including distances, dimensions, partitions, generator The only building blocks needed are the set Z of ordinary matrices, and duality properties, are consequences of the general proper- integers, with its infinite chain of two-way partitions ties of iterated squaring constructions, which also exhibit the interrelation- 2/22/4Z/. , and an elementary construction that we ships between codes and lattices of different lengths. An extension called call the “squaring construction,” which produces chains of the “cubing construction” generates good codes and lattices of lengths 2N-tuples with certain guaranteed distance properties from N = 3.2”, including the Golay code and Leech lattice, with the use of special bases for 8-space. Another related construction generates the chains of N-tuples. Iteration of this construction produces Nordstrom-Robinson code and an analogous 16-dimensional nonlattice the entire family of lattices, determines their minimum packing. These constructions are represented by trellis diagrams that squared distances, shows their partition (sublattice) struc- display their structure and interrelationships and that lead to efficient ture, and gives general interrelationships between the lat- maximum likelihood decoding algorithms. General algebraic methods for determining minimal trellis diagrams of codes, lattices, and partitions are tices of different dimension. The construction also natu- given in an Appendix. rally points to structural decompositions that we illustrate by trellis diagrams and that lead to efficient maximum I. INTRODUCTION likelihood decoding algorithms. Other attributes of these lattices, such as their generator matrices, “code formulas” COMPANION PAPER [l] characterizes a large [l],and duality properties, may be easily derived from A number of the coded modulation techniques that general properties of this simple construction. have been proposed for band-limited channels as coset Actually, the development makes it clear that the most codes, i.e., as sequences of cosets of a sublattice A’ in a natural starting point for the construction is the two-di- partition A/A‘ of a binary lattice A, where the cosets are mensional lattice Z2of pairs of ordinary integers, with its selected by the outputs of a binary encoder. infinite chain of two-way partitions Z 2/RZ 2/2Z ’/ . The principal purpose of this paper is to give a unified or, equivalently, the complex lattice G of Gaussian inte- development of the family of lattices that have proved to gers [l],with its partition chain C/$JG/+~G/. , where be most useful in constructing coset codes and of the +=1+i. properties of such lattices that are most important for such These lattices are closely related to the family of applications, e.g., their minimum squared distances, their Reed-Muller codes. Indeed, the Reed-Muller codes can partitions, and aspects of their structure that are useful in be generated by the same construction, except that the decoding. starting point is the binary field GF(2), with the exhaustive This family of lattices is the sequence of 2“-dimensional two-way partition into its two elements. The two-by-two lattices called the Barnes-Wall lattices, and what we call integer matrix G(2,2)= {[10],[11]}turns out to be a key tool their “principal sublattices.” This family includes such in describing the application of the squaring construction important lattices as the Schlafli lattice D4,the Gosset to group partitions, and the rn-fold Kronecker product of lattice E,, and the infinite sequence A,,, A,,, . of this matrix with itself, Le., the N X N integer matrix Barnes-Wall lattices, whose fundamental coding gain [l] G(N,N)that contains all the generators of all the Reed-Muller codes of length N = 2”, turns out to be very helpful in characterizing the results of rn-fold iterated Manuscript received September 2, 1986; revised September 18, 1987. squaring constructions (Lemma 2). This paper was partially presented at the 1986 IEEE International Sym- posium on Information Theory, Ann Arbor, MI, October 8, and at the A construction that we call the “cubing construction,” 1986 IEEE Communications Theory Workshop, Palm Springs, CA, April whch is closely related to the twofold iteration of the 28. squaring construction, produces groups of 3N-tuples from The author is with the Codex Corporation, 7 Blue fill River Road, Canton, MA 02021. groups of N-tuples. The principal use that we make of this IEEE Log Number 8824502. construction is to construct the (24,12) Golay code and the 0018-9448/88/0900-1152$01 .OO 01988 IEEE FORNEY: COSET CODES-PART I1 1153 24-dimensional Leech lattice. In addition to the cubing and the Leech lattice. Finally, in the Appendix we show construction, we need to introduce special bases of 8-space how to determine the state spaces and trellis diagrams of to obtain the requisite distance properties. codes, lattices, and partitions algebraically, using trellis- Using these bases, we go on to construct the length-16 oriented generator matrices. nonlinear Nordstrom-Robinson code, which is better than any comparable linear code, and an analogous 16-dimen- sional nonlattice packing, which falls just short of the 11. PRELIMINARIES density of the 16-dimensional Barnes-Wall lattice A16. These constructions are closely related to each other and This paper is about discrete sets on which a distance to those already described. Furthermore, they show that metric is defined, partitions of such sets, and set construc- while coset code constructions may almost always be based tions based on such partitions. In this section we gather on partitions that result from coset decompositions of the elementary facts that we will need about such sets and groups (codes, lattices), the resulting constructions need partitions. These sets will almost always be algebraic not themselves be linear to have good distance properties groups, i.e., closed under some addition operation; how- (indeed, many of the best trellis codes are nonlinear [l]). ever, the constructions do not essentially depend on group These constructions lead directly to unexpectedly simple properties, and to emphasize this point, we defer the and highly structured trellis diagrams for these codes and discussion of group properties for as long as possible. lattices. The trellis diagrams suggest maximum likelihood decoding algorithms for both codes and lattices that gener- A. Partitions ally turn out to be improvements over the best previously known algorithms. For completeness, we give methods for Let S be any discrete set with elements s E S. An systematic algebraic determination of minimal trellis dia- M-way partition of S is specified by a set of M disjoint grams for linear codes and lattices in an Appendix. subsets T(a)whose union is S, where a is a label for the Relatively little in this paper is new. All of the codes and subset T(a).We denote such a partition by S/T, and we lattices are well-known (except for the 16-dimensional say that the order of the partition is IS/TI = M. Ordinar- nonlattice packing, which was noticed earlier by Conway ily, in this paper, the order of a partition will be finite, and Sloane but has not previously been published). Their even when the sets involved are infinite. constructions and properties have generally been derived A subset labeling is any one-to-one map between the earlier in various forms, some essentially equivalent to our subsets and a set of M labels. Examples of labels which we constructions, which we have attempted to acknowledge shall use include: a subset index i, where, for example, appropriately. While we know of no readily accessible text 0 I i IM-1; binary K-tuples a, when M = 2K; or a on lattices, the recent book by Conway and Sloane [2] is an system of subset representatives c E S, one from each sub- encyclopedic reference for practically everythmg here and set. When S contains a zero element 0, we call the subset far more. What we hope to have contributed is a unified that contains 0 the zero subset T(O),or simply T, and use 0 treatment of the lattices that are most useful in applica- as its representative. tions, with a derivation of their principal properties, at a For example, there is a two-way partition of the set of reasonably elementary mathematical level. We do believe ordinary integers 2 into the even integers, 22, and the that the structural properties exhibited in our trellis dia- odd integers, 22+ 1. We say that 2/22 is a partition of grams are generally new, as well as the decoding methods order 2. The natural labels for the subsets are {O,l}, where that they suggest. 22 is the zero subset. The paper is organized as follows. In Section I1 we An m-level partition chain S,/S,/ ’ . /Sm is obtained introduce the language and elementary results that we use by repeated partitioning of subsets; i.e., the set So is first for sets, set partitions, distance measures, and additive partitioned into IS,/S,l subsets SI(a,), then each subset groups,-particularly groups with orders equal to a power of S,(a,) of So is partitioned, and so forth.