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A Ternary Construction of Lattices A ternary construction of lattices Hao Chen Department of Mathematics Hangzhou Dianzi University Hangzhou 310018, Zhejiang Province, China E-mail: [email protected] July, 2014 Abstract In this paper we propose a general ternary construction of lattices from three rows and ternary codes. Most laminated lattices and Kappa n lattices in R , n 24 can be recovered from our tenary construction naturally. This ternary≤ construction of lattices can be used to gener- ate many new ”sub-optimal” lattices of low dimensions. Based on this ternary construction of lattices new extremal even lattices of dimen- sions 32, 40 and 48 are also constructed. 1 Introduction How one can arrange most densely in space an infinite number of equal spheres is a classical mathematical problem and a part of Hilbert 18th problem ([11, 7]). It is deeply rooted in information theory and physics ([47, 28, 35]). The root lattices in Euclid spaces of dimensions 1, 2, 3, 4, 5, 6, 7 and 8 and the Leech lattice of dimension 24 ([11]) had been proved to be the arXiv:1306.1432v8 [math.NT] 14 Apr 2015 unique densest lattice sphere packings in these dimensions (see [11, 36, 9]). For a historic survey we refer to [11] Chapter 1 and [36]. From Voronoi’s theory ([36, 45]), there are algorithms to determine the densest lattice sphere packings for all dimensions. However the computational task for dimensions n> 9 is generally infeasible. Laminated lattices in Rn,n 24, were known ≤ from the work of A. Korkine, G.Zolotareff in 1877, T. W. Chaundy in 1946 and J . Leech in 1967 ([11], page 158-159, [36]). In 1982 J. H. Conway and N. J. A. Sloane calculated all densities of laminated lattices up to dimension 48 ([12]). The n dimensional laminated lattice, 1 n 24 and n = 11, 12, 13, ≤ ≤ 6 is the only known densest lattice sphere packing in Rn ([11], page 15). The 1 n known densest lattice in R , n = 11, 12, 13 is the Kappa lattice K11, the Coxeter-Todd lattice K12 and the Kappa lattice K13([11], page 15). The lattice L = y Rm :< y, x > Z, x L is called the dual ∗ { ∈ ∈ ∈ } lattice of the lattice L. A lattice is called integral if the inner products between lattice vectors are integers and an integral lattice is called even if the Euclid norms of all lattice vectors are even numbers. An integral lattice satisfying L = L∗ is called unimodular lattice. The minimum norm µ(L) for an unimoular even lattice satisfies µ(L) 2[ n ] + 2 and an unimodular ≤ 24 even lattice satisfying the equality is called an extremal even unimodular lattice ([11]). The most interesting cases are the jumping dimensions n 0 ≡ mod 24. In dimension 24 J. H. Conway proved that the Leech lattice is the unique extremal even unimodular lattice ([10]). In dimension 48, there are four constructed extremal even unimodular lattices ([11, 38]. It has long been an open problem if there is an extremal even extremal even unimodualr lattice in dimension 72. The recent two breakthroughs are the first consc- tructed 72 dimensional extremal even unimodular lattice ([20, 39]) and the fourth 48 dimensional extremal even unimodular lattice ([40]). We refer to [34, 2, 18, 24, 25, 26, 21] for the general bounds and the general construction of extremal even or odd unimodular or modular lattices. It is a classical re- sult of C. L. Mallows, A. M. Odlyzko and N. J. A. Sloane that there is no n dimensional extremal even unimodular lattice when n is sufficiently large ([34]). Though it is well-known that for the dimension 32, there are millions of extremal even unimodular lattices ([29]), there are only 15 known con- structed 32 dimensional extremal even unimodular lattices found by Koch and Venkov ([30, 31, 37]). We refer to [6, 42, 38], [11] page 221 for the extremal even unimodular lattices of dimension 40 and [3, 46, 50] for the four constructed 80 dimensional extremal even unimodular lattices. In this paper we propose a construction of lattices from ternary codes. The ternary construction recovers the known densest lattices including the Leech lattice Λ24, the Coxeter-Todd lattice K12, the Barnes-Wall lattice Λ16, and one of the laminated lattice Λ26 of dimension 26 naturally. Many other laminated and Kappa lattices in dimensions n 24 can also be recov- ≤ ered from our ternary construction naturally. We generate some new ”sub- optimal” low dimensional lattices from this ternary construction. Based on this ternary construction of lattices new extemal even lattices of dimensions 32, 40, 48 are also constructed. 2 n For a packing of equal non-overlapping spheres in R with centers x1, x2, ..., 1 xm, ...., the packing radius ρ is 2 mini=j xi xj . The density ∆ is n 6 V ol x R : x <t, xi, x xi <ρ {|| − ||} ∆ { ∈ || || n ∃ || − || } limt 0 V ol x R : x <t . The center density δ is Vn where Vn → { ∈ || || } n is the volume of the ball of radius 1 in R . Let b1, ..., bm be m linearly independent vectors in the Euclid space Rn of dimension n. The discrete point sets L = x1b1 + + xnbm : x1, ..., xm Z is a dimension m lattice n { · · · ∈ } 1 in R . The volume of the lattice is V ol(L) = (det(< bi, bj >)) 2 . Let λ(L) be the length of the shortest non-zero vectors in the lattice and the mini- mum norm of the lattice is just µ(L) = (λ(L))2. The set of lattice vectors with the length λ(L) is denoted by min(L) and the cardinality of min(L) is the kissing number k1(L) of the lattice L. Let µ′(L) be the minimum norm of lattices vectors in the set L 0 min(L). The number of the second \ { } \ layer lattice vectors k2(L) is the number of lattice vectors with the norm µ′(L). When the centers of the spheres are these lattice vectors in L we 1 have a lattice sphere packing for which ρ = 2 λ(L) and the center density ρn δ(L)= V ol(L) . Let q be a prime power and Fq be the finite field with q elements. A linear (non-linear ) error-correcting code C Fn is a k dimensional sub- ⊂ q space(or a subset of M vectors). For a codeword x C, the support Supp(x) ∈ of x is the set of positions of non-zero coordinates. The Hamming weight wt(x) is the number of elements in the support of x. The minimum Ham- ming weight(or distance) of the linear (or non-linear) code C is defined as d(C) = minx=y,x,y C wt(x y) . We refer to [n,k,d]q (or (n,M,d)q) 6 ∈ { − } code as linear (or non-linear)code with length n, distance d and dimen- sion k (or M codewords). For a binary code C Fn the construction A ⊂ 2 ([11]) leads to a lattice in Rn. The lattice L(C) is defined as the set of integral vectors x = (x , ..., x ) Zn satisfying x c mod 2 for some 1 n ∈ i ≡ i codeword c = (c1, ..., cn) C. This is a lattice with the center density n ∈ min √d(C),2 { } δ = 22n−k(C) . This construction A gives some best known densest lat- tice packings in low dimensions(see [11]). For a (n,M,d) non-linear binary code, the same construction gives the non-lattice packing with center den- M min √d(C),2 n · { } sity 22n . Some of them are the known best sphere packings (see [11]) or have presently known highest kissing numbers. The known dens- 5 est packing (non-lattice) in dimension 10 with center density 128 is from non-linear binary (10, 40, 4) code found by M. R. Best in 1980 ([11]). The 11 9 non-lattice packing P11a in R with the center density 256 and the kissing 3 number 566 is constructed from a non-linear binary (11, 72, 4) code ([11], page 139). 2 The main construction Lemma 2.1. Let L be a dimension r lattice in Rn with the volume vol(L) and the center density δ(L). Let E be the dimension 2r lattice in R3n de- fined by E = (x, y, z) : x L, y L, z L, x + y + z = 0 . Then the r r { ∈ 2 ∈ ∈ }2 2 volume of the E is 3 2 vol(L) and the center density of E is r δ(L) . · 3 2 Proof. It follows from Theorem 4 in page 166 of [11] directly. Let p be a prime number. For a lattice L Zn we define the the lin- ⊂ ear code CL,p over the finite field Fp as the image of the natural mapping L/pZn L Fn. −→ p T Theorem 2.2. Let L Zn be a dimension r lattice with minimum norm n ⊂ 2. Suppose 3Z L = 3L. If there exists a ternary [n,k, 6]3 code C (over F3) which is in the ternary code CL,3. Then we have a lattice sphere packing T r 2 5 k of dimension 2r with the minimum norm 36 and the volume vol(L) 3 2 − . k r 3 2 2 · Its center density is ·r δ(L) . 3 2 · Proof. Let C = (c, c, c) : c C be the [3n,k, 18] ternary code in ′ { ∈ } 3 Fn Fn Fn. It is easy to check C is in the image of the natural mapping 3 ⊕ 3 ⊕ 3 ′ E/3E Fn Fn Fn.
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