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A Detailed Study of the Relationship Between Some Kyushu J. Math. 72 (2018), 123–141 doi:10.2206/kyushujm.72.123 A DETAILED STUDY OF THE RELATIONSHIP BETWEEN SOME OF THE ROOT LATTICES AND THE CODING THEORY Michio OZEKI∗ (Received 26 December 2016 and revised 27 April 2017) Abstract. In the present article we study the even unimodular lattice which lies between the # root lattice m · An and the dual lattice .m · An/ . Here m · An is an orthogonal sum of m # copies of the root lattice An. In the course of the study the code over the ring An=An arises in a natural way. We find that an intimate relationship between the even unimodular lattice # containing m · An as a sublattice and the error correcting code over the ring An=An exists. As a consequence we could reconstruct sixteen non-isometric Niemeier lattices out of twenty- four non-isometric lattices by using the present approach. 1. Introduction In [18] H.-V. Niemeier has succeeded in the complete classification of the 24-dimensional even unimodular lattices. There are exactly twenty-four non-isometric classes. But for the Leech lattice each isometry class of the other twenty-three classes of the lattices has the unique root sublattice of rank 24, and each root sublattice has the unique (up to isometry) 24-dimensional even unimodular overlattice. Later, Venkov [25] and Conway–Sloane [4] respectively gave further proofs of the same result. After these three groups of researchers, other researchers in combinatorics who take an interest in re-constructing the so-called twenty-four Niemeier lattices by way of the coding theory usually trace the route which starts from appropriate codes of length 24 over a certain finite ring and reaches at those lattices (cf. [1, 5, 17]). In the present article we consider the relationship between the lattice theory and the coding theory in the reverse route. More precisely, we start from an orthogonal sum of m copies of the root lattices of type An, and we explore what even unimodular lattices L lie between the dual of m · An and m · An. A closer analysis of this procedure leads to a natural derivation of the codes over peculiar finite rings. As a consequence of our present research we could show that some of the Niemeier lattices are constructed from the codes of length less than twenty-four. This fact is useful for the arithmetical studies of the Niemeier lattices, such as the computation of the Fourier coefficients of the Siegel theta series associated with the Niemeier lattices or the covering radius problem of the Niemeier lattices and the even unimodular lattices of dimension greater than 24 that are treated in the present article. 2010 Mathematics Subject Classification: Primary 11H06; Secondary 11H71. Keywords: unimodular lattice; root lattice; coding theory. ∗Emeritus Professor at the Department of Mathematical Sciences, Faculty of Science, Yamagata University. c 2018 Faculty of Mathematics, Kyushu University 124 M. Ozeki 2. Lattices 2.1. Some generalities on lattices Let Z be the ring of rational integers and let R be the field of real numbers. A finitely g generated Z-module L in R with a positive definite metric is called a positive definite quadratic lattice. Since we will only deal with the positive definite quadratic lattices, we shall henceforth refer to these simply as the lattices. A lattice L is integral if L satisfies .x; y/ 2 Z for any x; y 2 L where .·; ·/ is the bilinear form associated to the metric. Two integral lattices L1 and L2 are said to be isometric if and only if there exists a bijective linear mapping from L1 to L2 preserving the metric. The maximal number of linearly independent vectors over R in L is called the rank of L. The dual lattice L# of L is defined by # D f 2 ⊗ j 2 8 2 g L y L Z Q .x; y/ Z; x L : Here Q is the field of rational numbers. A lattice L is even if any element x of L has an even norm .x; x/. In an even lattice L, we say that x is a 2m-vector if .x; x/ D 2m holds for some natural number m. Let 32m.L/ be the set defined by 32m.L/ D fx 2 L j .x; x/ D 2mg: (1) A lattice L is said to be unimodular if L D L#. If L and M are two lattices such that M ⊂ L holds, then M is called a sublattice of L and L is called an overlattice of M. Suppose that there are two sublattices L1 6D f0g and L2 6D f0g of the lattice L satisfying the condition .y; z/ D 0; 8y 2 L1; 8z 2 L2; and that any element x 2 L can be written as x D y C z; y 2 L1; z 2 L2. Then L is called an orthogonal sum of the two sublattices L1 and L2, and we write as L D L1 ⊕ L2. An orthogonal sum of more than two sublattices are likewise defined. When the lattice L is an orthogonal sum of m sublattices each one is isometric to L1, then we simply write L D m · L1. D ⊕ # D # ⊕ # We may note that if L L1 L2 then it holds that L L1 L2. When L cannot be expressed as an orthogonal sum of two sublattices, then L is called irreducible. An integral lattice L is called a root lattice if L has a basis consisting of 2-vectors. It is known [26] that all the irreducible integral root lattices are An .n ≥ 1/, Dn .n ≥ 4/ and En .n D 6; 7; 8/. In closing this section, we give a table of the root sublattice R for each of the Niemeier lattices and the cardinalities of 32.R/. The table will be referred to later. 2.2. An-type root lattices In the present paper, we are particularly interested in the root lattices An. Let e1; e2;::: be orthonormal vectors in an appropriate Euclidean vector space. The root lattice An (n ≥ 1) is the lattice spanned by the vectors e1 − e2; e2 − e3;:::; en − enC1 over Z. # # Let An be the dual lattice of the root lattice An. In [18], the quotient module An=An is precisely described, and it is a cyclic group of order n C 1. The representative of a cyclic The root lattices and the coding theory 125 TABLE 1. The root sublattices R of the Niemeier lattice and the cardinality j32.R/j. R D24 3 · E8 E8 ⊕ D16 A24 2 · D12 .2 · E7/ ⊕ D10 j32.R/j 1104 720 720 600 528 432 R E7 ⊕ A17 D9 ⊕ A15 3 · D8 2 · A12 4 · E6 E6 ⊕ D7 ⊕ A11 j32.R/j 432 384 336 312 288 288 R 4 · D6 D6 ⊕ .2 · A9/ 3 · A8 .2 · D5/ ⊕ .2 · A7/ 4 · A6 6 · D4 j32.R/j 240 240 216 192 168 144 R D4 ⊕ .4 · A5/ 6 · A4 8 · A3 12 · A12 24 · A1 ; j32.R/j 144 120 96 72 48 0 # generator of An=An is given by n 1 X n α D et − enC : (2) 1 n C 1 n C 1 1 tD1 Note that α1 is the minimum representative in the equivalence class to which α1 belongs. Here, we state that α1 is a minimum representative if α1 satisfies the inequality .α1; α1/ ≤ .β; β/ for all β, which is equivalent to α1 modulo An. The set of all the minimum representatives of # An=An is given by nC1−` nC1 ` X n C 1 − ` X α D 0; α D et − et ; 1 ≤ ` ≤ n: (3) 0 ` n C 1 n C 1 tD1 tDnC2−` The norm .α`; α`/ of α` is computed to be `.n C 1 − `/ .α ; α / D ; 1 ≤ ` ≤ n: (4) ` ` n C 1 ≤ ≤ Further, when 1 `1 < `2 n, the inner product .α`1 ; α`2 / is computed to be ` .n C 1 − ` / .α ; α / D 1 2 : (5) `1 `2 n C 1 # Remark 1. Let α` be a minimal representative of An=An. There are other minimum representatives in the class α` C An. The number of minimum representatives in the class C nC1 α` An is easily computed to be ` . 3. Code over a finite field F p Let p be a prime number and let Fp be a finite field of p elements. An Tm; kU code over Fp is m m a vector subspace of dimension k of the vector space Fp . In Fp , the inner product is defined ? in the usual way. Let C be an Tm; kU code over a finite field Fp. The dual code C of the code C is defined by ? m C D fv 2 Fp j .u; v/c D 0; 8u 2 Cg; m where .u; v/c is the inner product on the space Fp . 126 M. Ozeki An Tm; kU code C is called self-orthogonal if C satisfies the condition C ⊂ C?, and self- dual if C satisfies the condition C D C?. A well-known method to make a new code from the two given codes C1 and C2, over Fp, is an orthogonal sum of the two codes C1 ⊕ C2 (cf. [14, Ch. 1]). When we need an orthogonal sum of k copies of a code C, we may use Ck as an abbreviation. As to the general references, the readers may read [14] or [22, Ch. 3]. 3.1. Binary linear codes D n T U Let V F2 be the vector space of dimension n over F2.
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