Abstract
New extremal odd unimodular lattices in dimension 36 are constructed. Some new odd unimodular lattices in dimension 36 with long shadows are also constructed.
“Dedicated to Professor Vladimir D. Tonchev on His 60th Birthday”
1. Introduction
A (Euclidean) lattice in dimension is unimodular if , where the dual lattice of is defined as for all under the standard inner product . A unimodular lattice is called even if the norm of every vector is even. A unimodular lattice, which is not even, is called odd. An even unimodular lattice in dimension exists if and only if (mod 8), while an odd unimodular lattice exists for every dimension. Two lattices and are isomorphic, denoted by , if there exists an orthogonal matrix with , where . The automorphism group of is the group of all orthogonal matrices with .
Rains and Sloane [6] showed that the minimum norm of a unimodular lattice in dimension is bounded by unless when . 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) is (see [1]). There are exactly five odd unimodular lattices in dimension having minimum norm , up to isomorphism [7]. For dimensions 33, 34, and , the minimum norm of an odd unimodular lattice is at most (see [1]). The next dimension for which there is an odd unimodular lattice with minimum norm (at least) is . Four extremal odd unimodular lattices in dimension are known, namely, Sp4D8.4 in [1], in [2, Table 2], in [3, Section 3], and in [4, Section 3]. Recently, one more lattice has been found, namely, in [5, Table ]. This situation motivates us to improve the number of known nonisomorphic extremal odd unimodular lattices in dimension . The main aim of this paper is to prove the following.
Proposition 1. There are at least 26 nonisomorphic extremal odd unimodular lattices in dimension 36.
The above proposition is established by constructing new extremal odd unimodular lattices in dimension from self-dual -codes, where is the ring of integers modulo , by using two approaches. One approach is to consider self-dual -codes. Let be a binary doubly even code of length satisfying the following conditions: Then a self-dual -code with residue code gives an extremal odd unimodular lattice in dimension by Construction A. We show that a binary doubly even [36, 7] code satisfying conditions (1) has weight enumerator (Lemma 2). It was shown in [8] that there are four codes having the weight enumerator, up to equivalence. We construct ten new extremal odd unimodular lattices in dimension from self-dual -codes whose residue codes are doubly even [36, 7] codes satisfying conditions (1) (Lemma 4). New odd unimodular lattices in dimension 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 [9]). The other approach is to consider self-dual -codes , which have generator matrices of a special form given in (14). 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 [10].
2. Preliminaries
2.1. Unimodular Lattices
Let be an odd unimodular lattice and let denote the even sublattice, that is, the sublattice of vectors of even norms. Then is a sublattice of of index [7]. The shadow of is defined to be . There are cosets of such that , where and . Shadows for odd unimodular lattices appeared in [7] and also in [11, p. 440], in order to provide restrictions on the theta series of odd unimodular lattices. Two lattices and are neighbors if both lattices contain a sublattice of index in common. If is an odd unimodular lattice in dimension divisible by , then there are two unimodular lattices containing , which are rather , namely, and . Throughout this paper, we denote the two unimodular neighbors by
The theta series of is the formal power series . The kissing number of is the second nonzero coefficient of the theta series of , that is, the number of vectors of minimum norm in . Conway and Sloane [7] gave some characterization of theta series of odd unimodular lattices and their shadows. Using [7, ], it is easy to determine the possible theta series and of an extremal odd unimodular lattice in dimension and its shadow : respectively, where is a nonnegative integer. It follows from the coefficients of and in that .
2.2. Self-Dual -Codes and Construction A
Let be the ring of integers modulo , where is a positive integer greater than . A -code of length is a -submodule of . Two -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 is self-dual if , where the dual code of is defined as for all , under the standard inner product .
If is a self-dual -code of length , then the lattice is a unimodular lattice in dimension . This construction of lattices is called Construction A.
3. From Self-Dual -Codes
From now on, we omit the term “odd” for odd unimodular lattices in dimension , since all unimodular lattices in dimension are odd. In this section, we construct ten new nonisomorphic extremal unimodular lattices in dimension from self-dual -codes by Construction A. Five new nonisomorphic unimodular lattices in dimension with minimum norm having shadows of minimum norm are also constructed.
3.1. Extremal Unimodular Lattices
Every -code of length has two binary codes and associated with : The binary codes and are called the residue and torsion codes of , respectively. If is a self-dual -code, then is a binary doubly even code with [12]. Conversely, starting from a given binary doubly even code , a method for construction of all self-dual -codes with was given in [13, Section 3].
The Euclidean weight of a codeword of is , where denotes the number of components with . The minimum Euclidean weight of is the smallest Euclidean weight among all nonzero codewords of . It is easy to see that , where denotes the minimum weight of . In addition, and has minimum norm (see, e.g., [3]). Hence, if there is a binary doubly even code of length satisfying conditions (1), then an extremal unimodular lattice in dimension is constructed as , through a self-dual -code with . If there is a binary code satisfying conditions (1), then or (see [14]).
Lemma 2. Let be a binary doubly even [36, 7] code satisfying conditions (1). Then the weight enumerator of is .
Proof. The weight enumerator of is written as
where , , , , and are nonnegative integers. By the MacWilliams identity, the weight enumerator of is given by
Since , the weight enumerator of is written using and :
Suppose that does not contain the all-one vector . Then . In this case, since the coefficients of and are and , these yield that and , respectively, which gives the contradiction. Hence, contains . Then . Since the coefficient of is , the weight enumerator of is uniquely determined as .
Remark 3. A similar approach shows that the weight enumerator of a binary doubly even code satisfying conditions (1) is uniquely determined as .
It was shown in [8] that there are four inequivalent binary codes containing . The four codes are doubly even. Hence, there are exactly four binary doubly even codes satisfying conditions (1), up to equivalence. The four codes are optimal in the sense that these codes achieve the Gray-Rankin bound, and the codewords of weight are corresponding to quasi-symmetric SDP designs [15]. Let be the binary doubly even code corresponding to the quasi-symmetric SDP design, which is the residual design of the symmetric SDP design in [8, Section 5] . As described above, all self-dual -codes with have . Hence, are extremal.
Using the method in [13, Section 3], self-dual -codes are constructed from . Then ten extremal unimodular lattices are constructed, where , , and . To distinguish between the known lattices and our lattices, we give in Table 1 the kissing numbers , , and the orders of the automorphism groups, where denotes the number of vectors of norm in defined in (2) . These have been calculated by MAGMA. Table 1 shows the following.
Lemma 4. The five known lattices and the ten extremal unimodular lattices are nonisomorphic to each other.
Remark 5. In this way, we have found two more extremal unimodular lattices , where are self-dual -codes with . However, we have verified by MAGMA that the two lattices are isomorphic to in [3] and in [4].
Remark 6. For , it follows from and that one of the two unimodular neighbors and defined in (2) is extremal. We have verified by MAGMA that the extremal one is isomorphic to .
For , the code is equivalent to some code with generator matrix of the form where , , , and are -matrices, denotes the identity matrix of order , and denotes the zero matrix. We only list in (11) the matrix to save space. Note that in (10) can be obtained from since . A generator matrix of is obtained from that of
3.2. Unimodular Lattices with Long Shadows
The possible theta series of a unimodular lattice in dimension having minimum norm and its shadow are as follows: respectively, where and are integers with [3]. Then the kissing number of is at most and . Unimodular lattices with and are often called unimodular lattices with long shadows (see [9]). Only one unimodular lattice in dimension with and was known, namely, in [3].
Let be one of , , , and . Since , one of the two unimodular neighbors and in (2) is extremal and the other is a unimodular lattice with minimum norm having shadow of minimum norm 5. We denote such lattices constructed from , , , and by , , , and , respectively. We list in Table 2 the orders of the automorphism groups, which have been calculated by MAGMA. Table 2 shows the following.
Proposition 7. There are at least 5 nonisomorphic unimodular lattices in dimension 36 with and .
4. From Self-Dual -Codes
In this section, we construct more extremal unimodular lattices in dimension from self-dual -codes .
Let denote the transpose of a matrix . An matrix is negacirculant if it has the following form: Let and be -codes of length with generator matrices of the following form: where are listed in Table 3, and and are negacirculant matrices with first rows and listed in Table 3. It is easy to see that these codes are self-dual since . Thus, and are unimodular lattices, for given in Table 3. In addition, we have verified by MAGMA that these lattices are extremal.
To distinguish between the above eleven lattices and the known lattices, in Table 1 we give , , and , which have been calculated by MAGMA. The two lattices have the identical , for each of the pairs and . However, we have verified by MAGMA that the two lattices are nonisomorphic for each pair. Therefore, we have the following.
Lemma 8. The 26 lattices in Table 1 are nonisomorphic to each other.
Lemma 8 establishes Proposition 1.
Remark 9. Similar to Remark 6, it is known [3] that the extremal neighbor is isomorphic to for the case where is in [3], and we have verified by MAGMA that the extremal neighbor is isomorphic to for the case where is in [4].
5. 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 .
5.1. Unimodular Lattices from Ternary Self-Dual Codes
Let be a ternary self-dual code of length . The two unimodular neighbors and given in (2) are described in [16] as and . In this section, we use the notation and , instead of and , since the explicit constructions and some properties of and are given in [16]. By Theorem 6 in [16] (see also Theorem 3.1 in [2]), is extremal when satisfies the following condition (a), and both and are extremal when satisfies the following condition (b):(a)extremal (minimum weight ) and admissible (the number of ’s in the components of every codeword of weight is even),(b)minimum weight and maximum weight . For each of (a) and (b), no ternary self-dual code satisfying the condition is currently known.
5.2. Condition (a)
Suppose that satisfies condition (a). Since has minimum weight , has minimum norm and kissing number . By Theorem 6 in [16], and . Hence, since the shadow of contains no vector of norm , by (3) and (4), has theta series . It follows that .
By Theorem 1 in [17], the possible complete weight enumerator of a ternary extremal self-dual code of length containing is written as using some , where , , , and and , , and . From the minimum weight, we have the following: Since contains the term , if is admissible, then Hence, the complete weight enumerator of a ternary admissible extremal self-dual code containing is uniquely determined, which is listed in (18).
5.3. Condition (b)
Suppose that satisfies condition (b). By the Gleason theorem (see Corollary 5 in [17]), the weight enumerator of is uniquely determined as By Theorem 6 in [16] (see also Theorem 3.1 in [2]), and are extremal. Hence, and .
Note that a unimodular lattice contains a -frame if and only if for some ternary self-dual code . Let be any of the five lattices given in Table 2. Let be the set . We define the simple undirected graph , whose set of vertices is the set of pairs in and two vertices are adjacent if . It follows that the -frames in are precisely the -cliques in the graph . We have verified by MAGMA that are regular graphs with valency , and the maximum sizes of cliques in are . Hence, none of these lattices is constructed from some ternary self-dual code by Construction A.
Disclosure
This work was carried out at Yamagata University.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author would like to thank Masaaki Kitazume for bringing the observation in Section 5 to the author’s attention. This work is supported by JSPS KAKENHI Grant no. 23340021.