Extremal unimodular lattices in dimension 36

In this paper, new extremal odd unimodular lattices in dimension $36$ are constructed. Some new odd unimodular lattices in dimension $36$ with long shadows are also constructed.


Introduction
A (Euclidean) lattice L ⊂ R n in dimension n is unimodular if L = L * , where the dual lattice L * of L is defined as {x ∈ R n | (x, y) ∈ Z for all y ∈ L} under the standard inner product (x, y). A unimodular lattice is called even if the norm (x, x) of every vector x is even. A unimodular lattice, which is not even, is called odd. An even unimodular lattice in dimension n exists if and only if n ≡ 0 (mod 8), while an odd unimodular lattice exists for every dimension. Two lattices L and L ′ are isomorphic, denoted L ∼ = L ′ , if there exists an orthogonal matrix A with L ′ = L · A, where L · A = {xA | x ∈ L}. The automorphism group Aut(L) of L is the group of all orthogonal matrices A with L = L · A.
Rains and Sloane [17] showed that the minimum norm min(L) of a unimodular lattice L in dimension n is bounded by min(L) ≤ 2⌊n/24⌋+ 2 unless n = 23 when min(L) ≤ 3. We say that a unimodular lattice meeting the upper bound is extremal.
The smallest dimension for which there is an odd unimodular lattice with minimum norm (at least) 4 is 32 (see [13]). There are exactly five odd unimodular lattices in dimension 32 having minimum norm 4, up to isomorphism [4]. For dimensions 33, 34 and 35, the minimum norm of an odd unimodular lattice is at most 3 (see [13]). The next dimension for which there is an odd unimodular lattice with minimum norm (at least) 4 is 36. Four extremal odd unimodular lattices in dimension 36 are known, namely, Sp4(4)D8.4 in [13], G 36 in [6,Table 2], N 36 in [7, Section 3] and A 4 (C 36 ) in [8,Section 3]. Recently, one more lattice has been found, namely, A 6 (C 36,6 (D 18 )) in [9, Table II]. This situation motivates us to improve the number of known non-isomorphic extremal odd unimodular lattices in dimension 36. The main aim of this paper is to prove the following: Proposition 1. There are at least 26 non-isomorphic extremal odd unimodular lattices in dimension 36.
The above proposition is established by constructing new extremal odd unimodular lattices in dimension 36 from self-dual Z k -codes, where Z k is the ring of integers modulo k, by using two approaches. One approach is to consider self-dual Z 4 -codes. Let B be a binary doubly even code of length 36 satisfying the following conditions: the minimum weight of B is at least 16, the minimum weight of its dual code B ⊥ is at least 4.
Then a self-dual Z 4 -code with residue code B gives an extremal odd unimodular lattice in dimension 36 by Construction A. We show that a binary doubly even [36,7] code satisfying the conditions (1) and (2) has weight enumerator 1 + 63y 16 + 63y 20 + y 36 (Lemma 2). It was shown in [15] that there are four codes having the weight enumerator, up to equivalence. We construct ten new extremal odd unimodular lattices in dimension 36 from self-dual Z 4 -codes whose residue codes are doubly even [36,7] codes satisfying the conditions (1) and (2) (Lemma 4). New odd unimodular lattices in dimension 36 with minimum norm 3 having shadows of minimum norm 5 are constructed from some of the new lattices (Proposition 7). These are often called unimodular lattices with long shadows (see [14]). The other approach is to consider self-dual Z k -codes (k = 5, 6,7,9,19), which have generator matrices of a special form given in (7). Eleven more new extremal odd unimodular lattices in dimension 36 are constructed by Construction A (Lemma 8). Finally, we give a certain short observation on ternary self-dual codes related to extremal odd unimodular lattices in dimension 36. All computer calculations in this paper were done by Magma [1].

Unimodular lattices
Let L be an odd unimodular lattice and let L 0 denote the even sublattice, that is, the sublattice of vectors of even norms. Then L 0 is a sublattice of L of index 2 [4]. The shadow S(L) of L is defined to be L * 0 \ L. There are cosets Shadows for odd unimodular lattices appeared in [4] and also in [5, p. 440], in order to provide restrictions on the theta series of odd unimodular lattices. Two lattices L and L ′ are neighbors if both lattices contain a sublattice of index 2 in common. If L is an odd unimodular lattice in dimension divisible by 4, then there are two unimodular lattices containing L 0 , which are rather than L, namely, L 0 ∪ L 1 and L 0 ∪ L 3 . Throughout this paper, we denote the two unimodular neighbors by The theta series θ L (q) of L is the formal power series θ L (q) = x∈L q (x,x) . The kissing number of L is the second nonzero coefficient of the theta series of L, that is, the number of vectors of minimum norm in L. Conway and Sloane [4] gave some characterization of theta series of odd unimodular lattices and their shadows. Using [4, (2), (3)], it is easy to determine the possible theta series θ L 36 (q) and θ S(L 36 ) (q) of an extremal odd unimodular lattice L 36 in dimension 36 and its shadow S(L 36 ): respectively, where α is a nonnegative integer. It follows from the coefficients of q and q 3 in θ S(L 36 ) (q) that 0 ≤ α ≤ 16.

Self-dual Z k -codes and Construction A
Let Z k be the ring of integers modulo k, where k is a positive integer greater than 1. A Z k -code C of length n is a Z k -submodule of Z n k . Two Z k -codes are equivalent if one can be obtained from the other by permuting the coordinates and (if necessary) changing the signs of certain coordinates. A code C is selfdual if C = C ⊥ , where the dual code C ⊥ of C is defined as {x ∈ Z n k | x · y = 0 for all y ∈ C}, under the standard inner product x · y.
If C is a self-dual Z k -code of length n, then the following lattice is a unimodular lattice in dimension n. This construction of lattices is called Construction A.

From self-dual Z 4 -codes
From now on, we omit the term odd for odd unimodular lattices in dimension 36, since all unimodular lattices in dimension 36 are odd. In this section, we construct ten new non-isomorphic extremal unimodular lattices in dimension 36 from self-dual Z 4 -codes by Construction A. Five new non-isomorphic unimodular lattices in dimension 36 with minimum norm 3 having shadows of minimum norm 5 are also constructed.

Extremal unimodular lattices
Every Z 4 -code C of length n has two binary codes C (1) and C (2) associated with C: The binary codes C (1) and C (2) are called the residue and torsion codes of C, respectively. If C is a self-dual Z 4 -code, then C (1) is a binary doubly even code with C (2) = C (1) ⊥ [3]. Conversely, starting from a given binary doubly even code B, a method for construction of all self-dual Z 4 -codes C with C (1) = B was given in [16,Section 3]. (2) ) and A 4 (C) has minimum norm min{4, d E (C)/4} (see e.g. [7]). Hence, if there is a binary doubly even code B of length 36 satisfying the conditions (1) and (2), then an extremal unimodular lattice in dimension 36 is constructed as (1) and (2), then k = 7 or 8 (see [2]).
Remark 5. In this way, we have found two more extremal unimodular lattices A 4 (C), where C are self-dual Z 4 -codes with C (1) = B 36,1 . However, we have verified by Magma that the two lattices are isomorphic to N 36 in [7] and A 4 (C 36 ) in [8].
For i = 1, 2, . . . , 10, the code C 36,i is equivalent to some code C 36,i with generator matrix of the form: where A, B 1 , B 2 , D are (1, 0)-matrices, I n denotes the identity matrix of order n and O denotes the 22 × 7 zero matrix. We only list in Figure 1 the 7 × 29 . A generator matrix of A 4 (C 36,i ) is obtained from that of C 36,i .
In this section, we construct more extremal unimodular lattices in dimension 36 from self-dual Z k -codes (k ≥ 5).
Let A T denote the transpose of a matrix A. An n × n matrix is negacir-   culant if it has the following form: Let D 36,i (i = 1, 2, . . . , 9) and E 36,i (i = 1, 2) be Z k -codes of length 36 with generator matrices of the following form: where k are listed in Table 3, A and B are 9 × 9 negacirculant matrices with first rows r A and r B listed in Table 3. It is easy to see that these codes are self-dual since AA T + BB T = −I 9 . Thus, A k (D 36,i ) (i = 1, 2, . . . , 9) and A k (E 36,i ) (i = 1, 2) are unimodular lattices, for k given in Table 3. In addition, we have verified by Magma that these lattices are extremal.
Lemma 8 establishes Proposition 1.  Remark 9. Similar to Remark 6, it is known [7] that the extremal neighbor is isomorphic to L for the case where L is N 36 in [7], and we have verified by Magma that the extremal neighbor is isomorphic to L for the case where L is A 4 (C 36 ) in [8].

Related ternary self-dual codes
In this section, we give a certain short observation on ternary self-dual codes related to some extremal odd unimodular lattices in dimension 36.

Unimodular lattices from ternary self-dual codes
Let T 36 be a ternary self-dual code of length 36. The two unimodular neighbors Ne 1 (A 3 (T 36 )) and Ne 2 (A 3 (T 36 )) given in (3) are described in [10] as L S (T 36 ) and L T (T 36 ). In this section, we use the notation L S (T 36 ) and L T (T 36 ), instead of Ne 1 (A 3 (T 36 )) and Ne 2 (A 3 (T 36 )), since the explicit constructions and some properties of L S (T 36 ) and L T (T 36 ) are given in [10]. By Theorem 6 in [10] (see also Theorem 3.1 in [6]), L T (T 36 ) is extremal when T 36 satisfies the following condition (a), and both L S (T 36 ) and L T (T 36 ) are extremal when T 36 satisfies the following condition (b): (a) extremal (minimum weight 12) and admissible (the number of 1's in the components of every codeword of weight 36 is even), (b) minimum weight 9 and maximum weight 33.
For each of (a) and (b), no ternary self-dual code satisfying the condition is currently known.
By Theorem 6 in [10] (see also Theorem 3.1 in [6]), L S (T 36 ) and L T (T 36 ) are extremal. Hence, min(A 3 (T 36 )) = 3 and min(S(A 3 (T 36 ))) = 5. Note that a unimodular lattice L contains a 3-frame if and only if L ∼ = A 3 (C) for some ternary self-dual code C. Let L 36 be any of the five lattices given in Table 2. Let L 36 are adjacent if (x, y) = 0. It follows that the 3-frames in L 36 are precisely the 36-cliques in the graph Γ(L 36 ). We have verified by Magma that Γ(L 36 ) are regular graphs with valency 368, and the maximum sizes of cliques in Γ(L 36 ) are 12. Hence, none of these lattices is constructed from some ternary self-dual code by Construction A.