A $40$-dimensional extremal Type II lattice with no $4$-frames

Norifumi Ojiro

arXiv:1812.03197·math.NT·May 28, 2019·Discret. Math.

A $40$-dimensional extremal Type II lattice with no $4$-frames

Norifumi Ojiro

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TL;DR

This paper constructs a unique 40-dimensional extremal Type II lattice lacking 40 orthogonal minimal vectors and analyzes its automorphism group, providing a novel example beyond classical binary code-based lattices.

Contribution

It introduces a new 40-dimensional extremal Type II lattice without 4-frames and determines its automorphism group, expanding the known examples of such lattices.

Findings

01

Constructed a 40-dimensional extremal Type II lattice without 4-frames.

02

Determined the automorphism group of the lattice.

03

Provided an example different from classical binary code-based lattices.

Abstract

We construct a $40$ -dimensional extremal Type II lattice not having any subsets consisting of $40$ orthogonal minimal vectors, and determine the automorphism group. This lattice gives an example different from the $16470$ lattices constructed from binary codes by classical constructions.

Tables6

Table 1. Table 3.1. Division of S 𝑆 S by the type.

$S_{i}$	$t_{0}$	$t_{1}$	$t_{2}$	$t_{4}$	$\| S_{i} \|$
$S_{1}$	24018	7752	38	1	4
$S_{2}$	24234	7608	74	1	304
$S_{3}$	24270	7584	80	1	684
$S_{4}$	24342	7536	92	1	912
$S_{5}$	24378	7512	98	1	2736
$S_{6}$	24414	7488	104	1	5472
$S_{7}$	24450	7464	110	1	9576
$S_{8}$	24486	7440	116	1	4788
$S_{9}$	24522	7416	122	1	4788
$S_{10}$	24558	7392	128	1	5016
$S_{11}$	24594	7368	134	1	1368
$S_{12}$	24630	7344	140	1	2052
$S_{13}$	24666	7320	146	1	228
$S_{14}$	24738	7272	158	1	684
$S_{15}$	24810	7224	170	1	152
$S_{16}$	24918	7152	188	1	684
$S_{17}$	24990	7104	200	1	76
$S_{18}$	25206	6960	236	1	76

Table 2. Table 3.2. Irreducible subsets and types of S 𝑆 S .

$S_{i, j}$	$t_{0}^{i, j}$	$t_{1}^{i, j}$	$t_{2}^{i, j}$	$t_{4}^{i, j}$	$\| S_{i, j} \|$
$S_{1, 1}$	2	0	0	1	4
$S_{2, 1}$	38	18	0	1	76
$S_{2, 2}$	162	32	0	1	228
$S_{3, 1}$	434	120	4	1	684
$S_{4, 1}$	114	56	0	1	228
$S_{4, 2}$	422	128	2	1	684
$S_{5, 1}$	362	154	6	1	684
$S_{5, 2}$	402	138	2	1	684
$S_{5, 3}$	414	130	4	1	684
$S_{5, 4}$	438	120	2	1	684
$S_{6, 1}$	330	168	8	1	684
$S_{6, 2}$	422	128	2	1	684
$S_{6, 3}$	422	128	2	1	684
$S_{6, 4}$	422	128	2	1	684
$S_{6, 5}$	446	114	4	1	684
$S_{6, 6}$	434	120	4	1	684
$S_{6, 7}$	434	122	2	1	684
$S_{6, 8}$	446	112	6	1	684
$S_{7, 1}$	330	168	8	1	684
$S_{7, 2}$	302	178	12	1	684
$S_{7, 3}$	362	160	0	1	684
$S_{7, 4}$	374	152	2	1	684
$S_{7, 5}$	374	152	2	1	684
$S_{7, 6}$	438	120	2	1	684
$S_{7, 7}$	850	252	6	1	1368
$S_{7, 8}$	426	128	0	1	684
$S_{7, 9}$	840	260	3	1	1368
$S_{7, 10}$	434	122	2	1	684
$S_{7, 11}$	402	138	2	1	684
$S_{7, 12}$	390	146	0	1	684
$S_{8, 1}$	362	154	6	1	684
$S_{8, 2}$	446	114	4	1	684

S_{i, j}

t_{0}^{i, j}

t_{1}^{i, j}

t_{2}^{i, j}

t_{4}^{i, j}

| S_{i, j} |

S_{8, 3}

434 120 4 1 684

S_{8, 4}

446 114 4 1 684

S_{8, 5}

434 120 4 1 684

S_{8, 6}

398 136 6 1 684

S_{8, 7}

446 112 6 1 684

S_{9, 1}

302 178 12 1 684

S_{9, 2}

422 130 0 1 684

S_{9, 3}

470 98 8 1 684

S_{9, 4}

434 122 2 1 684

S_{9, 5}

386 144 4 1 684

S_{9, 6}

426 128 0 1 684

S_{9, 7}

450 112 4 1 684

S_{10, 1}

434 120 4 1 684

S_{10, 2}

422 128 2 1 684

S_{10, 3}

422 128 2 1 684

S_{10, 4}

446 114 4 1 684

S_{10, 5}

422 128 2 1 684

S_{10, 6}

398 136 6 1 684

S_{10, 7}

114 56 0 1 228

S_{10, 8}

362 154 6 1 684

S_{11, 1}

422 130 0 1 684

S_{11, 2}

434 122 2 1 684

S_{12, 1}

330 168 8 1 684

S_{12, 2}

446 114 4 1 684

S_{12, 3}

422 128 2 1 684

S_{13, 1}

90 62 6 1 228

S_{14, 1}

450 112 4 1 684

S_{15, 1}

38 0 18 1 76

S_{15, 2}

38 18 0 1 76

S_{16, 1}

614 0 34 1 684

S_{17, 1}

38 0 18 1 76

S_{18, 1}

38 0 18 1 76

Table 3. Table 4.1. Orthogonal 4 4 4 -vectors of O 40 subscript 𝑂 40 O_{40} .

$m_{𝒗}$	19	20	21	22	23	24	25	26	28	32	total
$n_{𝒗}$	6	15	30	12	12	12	15	8	21	1	132

Table 4. Table 6.1. A Gram matrix of O 40 subscript 𝑂 40 O_{40} .

(\begin{matrix} 4 & -1 & -2 & 2 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & -1 & 1 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & -1 & 1 & 0 & 0 & -1 & -1 \\ -1 & 4 & 0 & 0 & 0 & -1 & 0 & -1 & 0 & -1 & -1 & -1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & -1 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & -1 & 1 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & -1 \\ -2 & 0 & 4 & -2 & 0 & 2 & -1 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 2 & 0 & 0 & 0 & 2 & 2 \\ 2 & 0 & -2 & 4 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & -1 & -1 \\ 0 & 0 & 0 & 0 & 4 & 1 & 0 & 1 & 0 & 1 & 0 & -1 & 0 & 1 & 0 & -1 & -1 & 0 & -1 & 1 & -1 & -1 & 1 & 0 & 1 & 0 & 0 & -1 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & -1 & -1 & 1 & 0 & 0 \\ 0 & -1 & 2 & 0 & 1 & 4 & -2 & 2 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & -2 & 1 & 0 & -2 & 2 & -1 & 0 & -2 & 2 & 0 & 0 & 2 & 1 & 1 & 0 & 0 & 1 & -1 & 0 & 0 & 1 & 1 \\ 1 & 0 & -1 & 0 & 0 & -2 & 4 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 0 & 1 & 1 & -1 & 1 & 1 & -1 & 0 & 0 & 1 & -2 & 1 & -1 & -2 & 0 & -1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 1 & 2 & -1 & 4 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -2 & 1 & -1 & -2 & 2 & -1 & 0 & -2 & 2 & -1 & 0 & 2 & -1 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 4 & 0 & 1 & 1 & 1 & -1 & 1 & 1 & 2 & 1 & 0 & -1 & 1 & 1 & 0 & -1 & -2 & -1 & -1 & 0 & -2 & 0 & 0 & -1 & 2 & -1 & 0 & 1 & 1 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 4 & 0 & 1 & 1 & 1 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 1 & 1 & -1 & 0 & 0 & -1 & 1 & 1 & -1 & 1 & 0 & 1 & -1 & 2 & 0 & 0 \\ 1 & -1 & -1 & 0 & 0 & -1 & 1 & -1 & 1 & 0 & 4 & 1 & -1 & -2 & 1 & 1 & 2 & 0 & 0 & -1 & 1 & 1 & 0 & 0 & -1 & -1 & -1 & 0 & -2 & 0 & 1 & -1 & 1 & -1 & 0 & 1 & 2 & -1 & 0 & 0 \\ 1 & -1 & -1 & 1 & -1 & -1 & 1 & 0 & 1 & 1 & 1 & 4 & 0 & -1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 1 & -1 & 0 & 4 & 2 & -1 & 0 & 0 & -1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & -1 & -1 \\ 0 & 1 & 0 & 0 & 1 & 0 & -1 & 0 & -1 & 1 & -2 & -1 & 2 & 4 & -1 & -1 & -2 & -1 & 1 & 0 & -2 & 0 & 0 & 1 & 2 & 2 & 0 & 0 & 2 & 0 & 0 & 1 & -2 & 1 & -1 & 0 & -2 & 2 & -1 & -1 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & 4 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & -1 & -1 & 0 & 0 & 0 & -1 & -1 & 1 & 0 & 0 & -1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & -1 & 1 & -1 & -1 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & -1 & 1 & 4 & 1 & -1 & 0 & -1 & 1 & 1 & -1 & 0 & -1 & -1 & 0 & 0 & -1 & -1 & 0 & 0 & 2 & 0 & -1 & 1 & 2 & -1 & -1 & -1 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 2 & -1 & 2 & 1 & 0 & -2 & 0 & 1 & 4 & 0 & 0 & -1 & 1 & 0 & 0 & -1 & -2 & -2 & 0 & 0 & -2 & 0 & 0 & 0 & 2 & -1 & 0 & 0 & 2 & -2 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & -1 & -1 & 0 & -1 & 0 & 4 & 0 & 1 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 1 & -1 & -1 & 0 & 1 & 0 \\ 1 & 1 & -1 & 0 & -1 & -2 & 1 & -2 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 4 & -1 & 0 & 1 & -1 & 0 & 1 & 2 & -1 & 0 & 1 & -2 & 0 & -1 & -1 & 0 & -1 & 1 & -1 & 0 & -1 & -1 \\ -1 & 0 & 0 & 0 & 1 & 1 & -1 & 1 & -1 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & -1 & 1 & -1 & 4 & -1 & -1 & 1 & -1 & 1 & -1 & 2 & -1 & 0 & 2 & -1 & 1 & -1 & 0 & 0 & -1 & -1 & 1 & 0 & 0 \\ 1 & -1 & 0 & 0 & -1 & 0 & 1 & -1 & 1 & 0 & 1 & 1 & 0 & -2 & 1 & 1 & 1 & 0 & 0 & -1 & 4 & 1 & 0 & 0 & -2 & -1 & 0 & 1 & -1 & -1 & 1 & -1 & 2 & 0 & 0 & 1 & 2 & -1 & 0 & 0 \\ 1 & 0 & -1 & 0 & -1 & -2 & 1 & -2 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & -1 & 1 & 4 & -2 & 0 & -1 & 1 & -1 & 0 & -1 & -1 & 1 & 0 & 1 & 0 & -1 & 1 & 0 & 0 & -1 & -1 \\ 0 & -1 & 0 & 0 & 1 & 2 & -1 & 2 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 1 & 0 & -2 & 4 & 0 & 0 & -2 & 2 & -1 & 0 & 2 & 0 & -1 & 0 & -1 & 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & -1 & 0 & -1 & 0 & -1 & -1 & -1 & 0 & -1 & 0 & 1 & -1 & 0 & -1 & -2 & 0 & -1 & 0 & 0 & 0 & 4 & 0 & 1 & -1 & 1 & 1 & 0 & 0 & -1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -2 & 1 & -1 & 0 & 0 & 2 & -1 & -1 & -2 & 0 & 1 & 1 & -2 & -1 & 0 & 0 & 4 & 2 & 0 & 0 & 2 & 0 & 0 & 1 & -2 & 1 & 0 & 0 & -2 & 2 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & -2 & 1 & -2 & -1 & 1 & -1 & 0 & 0 & 2 & 0 & -1 & -2 & 0 & 2 & -1 & -1 & 1 & -2 & 1 & 2 & 4 & -2 & 1 & 2 & -2 & 0 & 0 & -2 & 1 & 0 & 1 & -2 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & -2 & 2 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 & -1 & 2 & -1 & 0 & -2 & 4 & -1 & 0 & 2 & 0 & 1 & 0 & 0 & 0 & -2 & 0 & 0 & -1 & -1 \\ 0 & 0 & 1 & 0 & -1 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & -1 & 1 & 0 & 1 & -1 & 4 & 0 & -1 & 1 & 0 & 0 & 0 & 1 & 2 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -2 & 0 & -2 & 0 & 0 & 2 & -1 & -1 & -2 & -1 & 1 & 0 & -1 & -1 & 0 & 1 & 2 & 2 & 0 & 0 & 4 & 0 & 0 & 0 & -2 & 1 & 0 & 0 & -2 & 2 & 0 & 0 \\ 0 & -1 & 0 & 0 & 1 & 2 & -2 & 2 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & -2 & 2 & -1 & -1 & 2 & 0 & 0 & -2 & 2 & -1 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 1 & -1 & 1 & -1 & 0 & 1 & 0 & -1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 4 & 0 & 0 & -1 & 1 & 1 & 0 & 0 & 1 & 0 \\ -1 & 1 & 1 & 0 & 0 & 1 & -1 & 0 & -1 & 1 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 1 & -1 & 0 & -1 & -1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 4 & 0 & 1 & 0 & -1 & -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 2 & -1 & 1 & 1 & 0 & -2 & 0 & 2 & 2 & 0 & -1 & -1 & 2 & 1 & 0 & 0 & -2 & -2 & 0 & 0 & -2 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 2 & -2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & -1 & 0 & 0 & 1 & -1 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & -1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & -1 & 1 & 0 & 4 & -1 & 0 & 0 & 1 & -1 & 0 \\ -1 & 0 & 2 & -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & -1 & 0 & 1 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & -1 & 4 & 0 & 0 & 0 & 2 & 1 \\ 1 & -1 & 0 & 1 & -1 & -1 & 1 & -2 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & 0 & 1 & -2 & 2 & 0 & -1 & 1 & -1 & 0 & 0 & 0 & 4 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 1 & -1 & 2 & 0 & 0 & -2 & 0 & 2 & 2 & -1 & -1 & -1 & 2 & 0 & 0 & 1 & -2 & -2 & 0 & 1 & -2 & 0 & 0 & -1 & 2 & 0 & 0 & 1 & 4 & -2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 2 & -1 & 1 & 0 & 2 & 1 & -1 & -2 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 2 & 0 & 0 & 1 & -2 & 1 & 0 & 0 & -2 & 4 & 0 & 0 \\ -1 & 0 & 2 & -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & -1 & 0 & 1 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 1 & 0 & 0 & -1 & 2 & 0 & 0 & 0 & 4 & 2 \\ -1 & -1 & 2 & -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 4 \end{matrix}) .

Table 5. Table 6.2. A generator g 1 subscript 𝑔 1 g_{1} of Aut ( O 40 ) Aut subscript 𝑂 40 {\rm Aut}(O_{40}) .

(\begin{matrix} 1 & 3 & 4 & 1 & -1 & 2 & 1 & -3 & 5 & 4 & 0 & 3 & -5 & -5 & -9 & -4 & -7 & -5 & 2 & 1 & -4 & -2 & -1 & 0 & -3 & -1 & 3 & -1 & -1 & -3 & 2 & 0 & 1 & -5 & -4 & 0 & 4 & 4 & -2 & -2 \\ -5 & 0 & -5 & -3 & -4 & 4 & 1 & -8 & 9 & 8 & 4 & -2 & -5 & 5 & -2 & 3 & -1 & -5 & -5 & 0 & 3 & -1 & 0 & -3 & 3 & 4 & 7 & 2 & 3 & 2 & -5 & -4 & -2 & -2 & -1 & -3 & -3 & -6 & 3 & -1 \\ -2 & -1 & -2 & -1 & 0 & 1 & 0 & -1 & 1 & -1 & 1 & 0 & 0 & 2 & 1 & 1 & 0 & 0 & -1 & -1 & 1 & 0 & -1 & -1 & 1 & -1 & 1 & 1 & 0 & 0 & -1 & -1 & -1 & 1 & 0 & 0 & -2 & -1 & 1 & 0 \\ 0 & 1 & 1 & 0 & -2 & 2 & 1 & -3 & 4 & 4 & 1 & 0 & -3 & -1 & -4 & -1 & -2 & -3 & -1 & 1 & -1 & 0 & 0 & 0 & 0 & 1 & 3 & 0 & 1 & -1 & -1 & -1 & 0 & -3 & -2 & -1 & 1 & 0 & 0 & -1 \\ 4 & -1 & 2 & 2 & 4 & -3 & -1 & 10 & -12 & -10 & -4 & 1 & 7 & 0 & 8 & 0 & 6 & 8 & 4 & 0 & 1 & 3 & 0 & 4 & 0 & -3 & -10 & -2 & -2 & 1 & 3 & 4 & 0 & 4 & 4 & 2 & 1 & 2 & -2 & 2 \\ 3 & -2 & 2 & 1 & 3 & -3 & -2 & 6 & -8 & -7 & -2 & 1 & 5 & 0 & 6 & 0 & 3 & 6 & 3 & -1 & 0 & 1 & -1 & 3 & 0 & -4 & -6 & 0 & -2 & 0 & 1 & 2 & 1 & 3 & 2 & 1 & -1 & 2 & -1 & 1 \\ -2 & 2 & 0 & 0 & -2 & 2 & 2 & -5 & 6 & 6 & 1 & 0 & -4 & -2 & -6 & -1 & -4 & -5 & -1 & 0 & -1 & -2 & 0 & -2 & -1 & 3 & 5 & 0 & 1 & 1 & 0 & -1 & 0 & -4 & -3 & -1 & 2 & 0 & 0 & -1 \\ 2 & -2 & 2 & 2 & 2 & -4 & -2 & 5 & -6 & -4 & -2 & 0 & 3 & 0 & 4 & 0 & 2 & 4 & 4 & 0 & 0 & 0 & 0 & 2 & -1 & -2 & -5 & 0 & -2 & 1 & 2 & 2 & 2 & 2 & 2 & 0 & 0 & 2 & -1 & 1 \\ 2 & -1 & 2 & 1 & 2 & -2 & 0 & 3 & -4 & -4 & -2 & 0 & 3 & -2 & 2 & 0 & 2 & 3 & 1 & -1 & 0 & 1 & 0 & 2 & 0 & -2 & -3 & -1 & -1 & 0 & 1 & 2 & -1 & 1 & 1 & 1 & 0 & 2 & -1 & 0 \\ 2 & 1 & 2 & 1 & 1 & 0 & 1 & 4 & -3 & -4 & -1 & 2 & 2 & -2 & 0 & -2 & -1 & 1 & 2 & 1 & -1 & 0 & 0 & 1 & -2 & 0 & -4 & -2 & -1 & -1 & 3 & 3 & 0 & 0 & 0 & 1 & 3 & 2 & -2 & 0 \\ 6 & 1 & 6 & 4 & 5 & -6 & -1 & 11 & -11 & -10 & -5 & 1 & 6 & -6 & 3 & -3 & 3 & 6 & 5 & 0 & -2 & 3 & 1 & 4 & -2 & -5 & -9 & -3 & -3 & -2 & 6 & 5 & 1 & 3 & 2 & 4 & 3 & 7 & -3 & 1 \\ -1 & 2 & 1 & 0 & -2 & 3 & 2 & -5 & 7 & 5 & 1 & 1 & -5 & -4 & -9 & -2 & -6 & -6 & 0 & 1 & -3 & -2 & 1 & -2 & -3 & 3 & 5 & -1 & 0 & -2 & 1 & 0 & 0 & -4 & -4 & 0 & 3 & 2 & 0 & -2 \\ 2 & -2 & -1 & -1 & 2 & -1 & -1 & 5 & -7 & -7 & -2 & 0 & 5 & 2 & 7 & 2 & 5 & 5 & 0 & 0 & 2 & 2 & 1 & 1 & 1 & 0 & -7 & -1 & -1 & 1 & 0 & 2 & -1 & 4 & 4 & 1 & -1 & -1 & -1 & 1 \\ -1 & 0 & -3 & -2 & -1 & 3 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 3 & 2 & 2 & 1 & 0 & -1 & 1 & 2 & 1 & 1 & -1 & 1 & 2 & -3 & 0 & 0 & 1 & -1 & 0 & -2 & 1 & 2 & -1 & 0 & -3 & 0 & 0 \\ -2 & 3 & 1 & 0 & -3 & 3 & 2 & -6 & 10 & 7 & 2 & 1 & -6 & -4 & -9 & -3 & -7 & -7 & -1 & 0 & -2 & -2 & -1 & -1 & 0 & 0 & 7 & 0 & 1 & -2 & 0 & -1 & -1 & -5 & -5 & -1 & 2 & 1 & 0 & -2 \\ -10 & 5 & -4 & -3 & -11 & 10 & 4 & -23 & 30 & 25 & 9 & -2 & -19 & -2 & -23 & -1 & -16 & -22 & -6 & 2 & -2 & -7 & 0 & -9 & -1 & 8 & 22 & 4 & 4 & -1 & -5 & -8 & -1 & -12 & -10 & -6 & 0 & -3 & 3 & -5 \\ 3 & -2 & 2 & 1 & 3 & -4 & -1 & 5 & -7 & -6 & -2 & -1 & 5 & -1 & 5 & 0 & 4 & 5 & 1 & -1 & 0 & 1 & 0 & 2 & 0 & -2 & -4 & -1 & -1 & 0 & 1 & 2 & 1 & 3 & 2 & 2 & -1 & 2 & 0 & 0 \\ 1 & -1 & 1 & 1 & 2 & -2 & 0 & 4 & -5 & -3 & -1 & 0 & 3 & 1 & 4 & 0 & 3 & 4 & 1 & -1 & 1 & 1 & -1 & 2 & 0 & -1 & -2 & -1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ -3 & 2 & -2 & -2 & -3 & 5 & 2 & -8 & 9 & 7 & 2 & 1 & -6 & 0 & -7 & -1 & -5 & -7 & -3 & 0 & -1 & -2 & -1 & -3 & 0 & 2 & 7 & 0 & 1 & -1 & -2 & -3 & -1 & -4 & -3 & -1 & 1 & -1 & 1 & -2 \\ -1 & -1 & -1 & 0 & 0 & 1 & 1 & 2 & -3 & -1 & 1 & -1 & 2 & 3 & 4 & 1 & 3 & 3 & 0 & -1 & 2 & 1 & -1 & 1 & 1 & 1 & 0 & -1 & 1 & 2 & -2 & 0 & 0 & 1 & 1 & 1 & -1 & -2 & 1 & 1 \\ -1 & 1 & -1 & -2 & -1 & 2 & 0 & -5 & 6 & 5 & 1 & 0 & -4 & -2 & -5 & -1 & -4 & -4 & -1 & 0 & -2 & -2 & 0 & -2 & -1 & 2 & 5 & 1 & 1 & -1 & -1 & -1 & 1 & -3 & -3 & 0 & 1 & 0 & 0 & -1 \\ 0 & 2 & 1 & 0 & -1 & 2 & 1 & -2 & 4 & 2 & 1 & 1 & -2 & -3 & -5 & -1 & -3 & -3 & -1 & 1 & -1 & -1 & 0 & 0 & -1 & 1 & 2 & -1 & 0 & -2 & 1 & 0 & -1 & -2 & -2 & 0 & 1 & 1 & -1 & -1 \\ 5 & -2 & 3 & 2 & 5 & -6 & -3 & 10 & -13 & -10 & -6 & 1 & 7 & -1 & 9 & 0 & 6 & 9 & 5 & -1 & 0 & 3 & 1 & 4 & 0 & -5 & -11 & -1 & -3 & 1 & 3 & 4 & 1 & 5 & 4 & 2 & 1 & 3 & -2 & 2 \\ 0 & 0 & -2 & -1 & 0 & -1 & -1 & 0 & -1 & -1 & -1 & -1 & 1 & 1 & 2 & 2 & 2 & 1 & -1 & 0 & 1 & 1 & 2 & -1 & 1 & 0 & -2 & 1 & 0 & 1 & 0 & 0 & -1 & 2 & 2 & 0 & -1 & -1 & 0 & 1 \\ -3 & 2 & -2 & -1 & -3 & 5 & 2 & -5 & 6 & 5 & 2 & 1 & -5 & 1 & -6 & 0 & -4 & -6 & -1 & 1 & 0 & -1 & 0 & -3 & -1 & 3 & 4 & 0 & 0 & 0 & -1 & -2 & -1 & -3 & -2 & -1 & 1 & -1 & 1 & -1 \\ -3 & 2 & -3 & -2 & -3 & 5 & 2 & -6 & 8 & 5 & 2 & 1 & -5 & 1 & -6 & 0 & -4 & -6 & -3 & 1 & 0 & -1 & 0 & -3 & 0 & 3 & 5 & 0 & 1 & -1 & -1 & -2 & -2 & -3 & -2 & -1 & 1 & -2 & 1 & -1 \\ -2 & 0 & -1 & -1 & -2 & 2 & -1 & -4 & 5 & 5 & 2 & 0 & -4 & 1 & -3 & 0 & -3 & -3 & 0 & 0 & 0 & -2 & -1 & -1 & 0 & 1 & 4 & 1 & 0 & 0 & -2 & -2 & 1 & -2 & -2 & -1 & -1 & -1 & 1 & -1 \\ 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ -5 & 2 & -4 & -3 & -5 & 7 & 2 & -12 & 14 & 11 & 4 & 1 & -9 & 1 & -10 & 0 & -8 & -10 & -3 & 1 & -2 & -3 & 0 & -5 & -1 & 4 & 10 & 2 & 2 & -1 & -3 & -4 & -1 & -5 & -4 & -3 & 1 & -3 & 2 & -2 \\ 5 & -2 & 4 & 3 & 5 & -5 & -2 & 11 & -14 & -11 & -4 & 1 & 8 & -1 & 8 & 0 & 6 & 10 & 5 & -1 & 0 & 3 & 0 & 5 & -1 & -5 & -10 & -2 & -3 & 0 & 3 & 4 & 1 & 5 & 4 & 3 & 0 & 4 & -2 & 2 \\ 2 & 2 & 3 & 1 & 1 & 0 & 0 & 1 & 0 & -2 & -1 & 3 & 0 & -5 & -4 & -3 & -4 & -1 & 2 & 0 & -3 & -1 & -1 & 1 & -2 & -3 & -1 & -1 & -2 & -3 & 3 & 1 & 0 & -1 & -2 & 1 & 2 & 4 & -2 & -1 \\ -3 & 0 & -3 & -2 & -3 & 4 & 1 & -4 & 5 & 4 & 4 & -1 & -2 & 3 & -1 & 2 & -1 & -3 & -3 & 1 & 2 & -1 & 0 & -2 & 1 & 4 & 3 & 1 & 2 & 1 & -3 & -2 & -1 & -1 & -1 & -2 & -2 & -4 & 2 & -1 \\ -2 & 0 & -1 & -1 & -2 & 1 & 0 & -6 & 6 & 6 & 2 & -2 & -3 & -1 & -4 & 2 & -2 & -4 & -2 & 0 & 0 & -2 & 1 & -2 & 0 & 3 & 5 & 2 & 1 & 1 & -2 & -2 & 0 & -2 & -2 & -2 & -2 & -1 & 1 & -1 \\ -3 & 1 & -3 & -2 & -3 & 5 & 1 & -6 & 7 & 6 & 3 & 0 & -5 & 2 & -5 & 1 & -4 & -5 & -2 & 1 & 0 & -2 & 0 & -3 & -1 & 4 & 5 & 1 & 1 & 0 & -2 & -2 & 0 & -3 & -2 & -1 & 0 & -2 & 1 & -1 \\ 1 & -1 & 1 & 1 & 2 & -3 & -1 & 3 & -4 & -4 & -1 & 0 & 3 & 0 & 3 & 0 & 2 & 3 & 1 & -1 & 0 & 1 & 0 & 1 & 0 & -2 & -2 & 0 & -1 & 0 & 1 & 1 & 0 & 2 & 1 & 1 & -1 & 1 & 0 & 1 \\ 0 & 2 & 1 & 0 & -1 & 2 & 2 & -2 & 3 & 1 & 0 & 1 & -2 & -3 & -5 & -2 & -3 & -3 & -1 & 0 & -2 & 0 & 0 & -1 & -1 & 0 & 2 & -1 & 0 & -2 & 1 & 0 & -1 & -2 & -2 & 1 & 2 & 2 & -1 & -1 \\ 0 & 0 & 0 & 0 & 0 & -2 & -1 & 0 & 1 & 1 & 0 & -2 & -1 & -1 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & -1 & 1 & 0 & 0 \\ -3 & 2 & -3 & -2 & -3 & 5 & 2 & -5 & 8 & 5 & 2 & 1 & -5 & 0 & -6 & 0 & -5 & -6 & -1 & 1 & 0 & -1 & 1 & -3 & -1 & 4 & 4 & 0 & 1 & 0 & -1 & 0 & -2 & -3 & -2 & -1 & 2 & -2 & 1 & -1 \\ 1 & -1 & 1 & 1 & 2 & -3 & 0 & 3 & -4 & -4 & -1 & 0 & 3 & 0 & 3 & 0 & 2 & 3 & 1 & -1 & 0 & 1 & 0 & 1 & 0 & -3 & -2 & 0 & 0 & 0 & 1 & 1 & 0 & 2 & 1 & 1 & -1 & 1 & 0 & 1 \\ 1 & -1 & 1 & 1 & 2 & -2 & 0 & 3 & -4 & -4 & -1 & 0 & 3 & 0 & 3 & 0 & 2 & 3 & 1 & -1 & 0 & 1 & 0 & 1 & 0 & -2 & -2 & 0 & 0 & 0 & 1 & 1 & 0 & 2 & 1 & 1 & 0 & 1 & 0 & 1 \end{matrix}) .

Table 6. Table 6.3. A generator g 2 subscript 𝑔 2 g_{2} of Aut ( O 40 ) Aut subscript 𝑂 40 {\rm Aut}(O_{40}) .

(\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -2 & 1 & -1 & -1 & -2 & 2 & 1 & -6 & 7 & 6 & 2 & 0 & -4 & -2 & -6 & -1 & -5 & -5 & -1 & 0 & -2 & -2 & 0 & -2 & -1 & 1 & 6 & 1 & 1 & -1 & -1 & -1 & 0 & -3 & -3 & -1 & 1 & 0 & 1 & -1 \\ 0 & 0 & 1 & 0 & 0 & -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -3 & 1 & -2 & -1 & -2 & 3 & 1 & -5 & 6 & 6 & 2 & 0 & -5 & 1 & -5 & 0 & -4 & -5 & 0 & 1 & -1 & -2 & 0 & -3 & -2 & 3 & 5 & 1 & 1 & 0 & -1 & -2 & 1 & -3 & -2 & -1 & 1 & -1 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & -1 & 0 & -1 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 1 & -1 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -2 & 0 & -2 & -1 & -1 & 1 & 0 & -3 & 3 & 3 & 1 & -1 & -2 & 2 & -1 & 1 & -1 & -2 & -1 & 0 & 1 & 0 & 1 & -2 & 0 & 2 & 2 & 1 & 1 & 1 & -2 & -1 & 0 & -1 & 0 & -1 & 0 & -2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 1 & 1 & 2 & -3 & -1 & 5 & -5 & -5 & -2 & 0 & 3 & -1 & 3 & 0 & 2 & 3 & 2 & 0 & 0 & 2 & 1 & 1 & 0 & -2 & -5 & -1 & -1 & 0 & 2 & 2 & 0 & 2 & 2 & 1 & 1 & 2 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 1 & 1 & 0 & 0 & -1 & 0 & -1 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -5 & 2 & -3 & -2 & -5 & 5 & 2 & -11 & 14 & 11 & 4 & -1 & -9 & 1 & -10 & 0 & -7 & -10 & -3 & 1 & 0 & -3 & 0 & -5 & -1 & 4 & 10 & 2 & 2 & 0 & -3 & -4 & -1 & -5 & -4 & -3 & 0 & -2 & 2 & -2 \\ 0 & 0 & -1 & -1 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & -1 & 0 & 2 & -1 & 0 & 0 & 1 & 0 & 1 & -1 & 1 & 1 & 0 & 0 & -1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & -2 & -1 & 3 & -3 & -2 & -1 & 0 & 2 & -1 & 2 & 0 & 1 & 2 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & -1 & -3 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -4 & 1 & -2 & -1 & -3 & 3 & 1 & -8 & 9 & 8 & 3 & -1 & -6 & 1 & -6 & 1 & -3 & -6 & -3 & 0 & 0 & -1 & 0 & -3 & 1 & 2 & 8 & 2 & 2 & 0 & -3 & -4 & -1 & -3 & -3 & -2 & -1 & -2 & 2 & -1 \\ 5 & -2 & 3 & 1 & 4 & -5 & -2 & 8 & -10 & -9 & -4 & 1 & 7 & -2 & 7 & 0 & 4 & 7 & 3 & 0 & -1 & 1 & 1 & 3 & -1 & -2 & -9 & -1 & -2 & 0 & 3 & 4 & 1 & 4 & 3 & 1 & 1 & 2 & -2 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & -1 & -1 & 1 & -1 & 0 & 4 & -4 & -4 & -1 & -1 & 3 & 1 & 4 & 1 & 3 & 3 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 2 & -3 & -1 & 0 & 1 & 0 & 2 & 0 & 2 & 2 & 2 & 0 & -1 & 0 & 1 \\ 1 & -1 & 0 & 0 & 1 & -2 & -1 & 3 & -3 & -3 & -1 & -1 & 2 & 1 & 3 & 1 & 3 & 2 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & -1 & -2 & 0 & -1 & 0 & 0 & 0 & 0 & 2 & 2 & 1 & -2 & 0 & 0 & 1 \end{matrix}) .

Equations78

L \times L ∋ (x, y) \mapsto x \cdot y \in R .

L \times L ∋ (x, y) \mapsto x \cdot y \in R .

L^{\lor} := {x \in L \otimes R ∣ x \cdot y \in Z for all y \in L} .

L^{\lor} := {x \in L \otimes R ∣ x \cdot y \in Z for all y \in L} .

min (L) \leq 2 (1 + ⌊ n /24 ⌋),

min (L) \leq 2 (1 + ⌊ n /24 ⌋),

v_{i}^{t} v_{j} = {40 for 1 \leq i = j \leq n, for 1 \leq i \neq = j \leq n .

v_{i}^{t} v_{j} = {40 for 1 \leq i = j \leq n, for 1 \leq i \neq = j \leq n .

(6 I_{20} 3 I_{20} 3 I_{20} 12 I_{20}),

(6 I_{20} 3 I_{20} 3 I_{20} 12 I_{20}),

ψ : Z /36 Z ∋ b \mapsto (c \mapsto 3^{b} c) \in Aut (Z /19 Z) .

ψ : Z /36 Z ∋ b \mapsto (c \mapsto 3^{b} c) \in Aut (Z /19 Z) .

M := A_{1} (2) \oplus A_{1} (2) \oplus A_{19} (2) \oplus A_{19} (2)

M := A_{1} (2) \oplus A_{1} (2) \oplus A_{19} (2) \oplus A_{19} (2)

(63312),

(63312),

b_{L} (u + L, v + L) := u^{t} v + Z \in Q / Z

b_{L} (u + L, v + L) := u^{t} v + Z \in Q / Z

q_{L} (u + L) := u^{t} u + 2 Z \in Q /2 Z .

q_{L} (u + L) := u^{t} u + 2 Z \in Q /2 Z .

\frac{1}{21} (28 I_{20} 7 I_{20} 7 I_{20} 2 I_{20}) .

\frac{1}{21} (28 I_{20} 7 I_{20} 7 I_{20} 2 I_{20}) .

Z /3 Z e_{1} + \dots + Z /3 Z e_{20} + Z /21 Z e_{21} + \dots + Z /21 Z e_{40} .

Z /3 Z e_{1} + \dots + Z /3 Z e_{20} + Z /21 Z e_{21} + \dots + Z /21 Z e_{40} .

u_{\infty} (j)

u_{\infty} (j)

u_{i} (j)

(u_{i}, w_{i}) (28 I_{20} 7 I_{20} 7 I_{20} 2 I_{20})^{t} (u_{j}, w_{j}) \equiv 0 {mod (21) mod (42) (i \neq = j) (i = j) .

(u_{i}, w_{i}) (28 I_{20} 7 I_{20} 7 I_{20} 2 I_{20})^{t} (u_{j}, w_{j}) \equiv 0 {mod (21) mod (42) (i \neq = j) (i = j) .

v_{1} ⋮ v_{40} = B e_{1} ⋮ e_{40} .

v_{1} ⋮ v_{40} = B e_{1} ⋮ e_{40} .

(1, 0, 0, 0, 1, 1) \in (Z /3 Z)^{6} and (0, 7, 1, 0, 4, 17) \in (Z /21 Z)^{6} .

(1, 0, 0, 0, 1, 1) \in (Z /3 Z)^{6} and (0, 7, 1, 0, 4, 17) \in (Z /21 Z)^{6} .

B := (I_{20} O B_{1} B_{2}), B_{2} := (3 I_{10} O B_{3} 21 I_{10}),

B := (I_{20} O B_{1} B_{2}), B_{2} := (3 I_{10} O B_{3} 21 I_{10}),

{\left(\begin{tabular}[]{cccccccccccccccccccc}0&2&1&1&2&2&2&2&1&2&4&20&16&10&4&10&2&8&16&7\\ 1&0&2&1&1&2&2&2&2&1&17&10&11&19&4&7&10&11&20&13\\ 2&1&0&2&1&1&2&2&2&2&19&20&16&2&16&16&7&4&17&16\\ 2&2&1&0&2&1&1&2&2&2&17&4&11&19&17&19&16&16&16&1\\ 1&2&2&1&0&2&1&1&2&2&17&2&16&14&13&20&19&4&7&7\\ 1&1&2&2&1&0&2&1&1&2&8&11&11&13&20&10&20&10&13&1\\ 1&1&1&2&2&1&0&2&1&1&14&17&8&5&4&14&10&2&7&10\\ 1&1&1&1&2&2&1&0&2&1&7&8&5&5&11&1&14&1&11&4\\ 2&1&1&1&1&2&2&1&0&2&10&19&11&11&14&17&1&11&4&16\\ 1&2&1&1&1&1&2&2&1&0&5&7&13&20&14&2&17&7&5&13\\ 2&1&2&1&1&1&1&2&2&1&3&2&1&1&2&2&2&2&1&10\\ 1&2&1&2&1&1&1&1&2&2&1&3&2&1&1&2&2&2&2&10\\ 2&1&2&1&2&1&1&1&1&2&2&1&3&2&1&1&2&2&2&10\\ 2&2&1&2&1&2&1&1&1&1&2&2&1&3&2&1&1&2&2&10\\ 2&2&2&1&2&1&2&1&1&1&1&2&2&1&3&2&1&1&2&10\\ 2&2&2&2&1&2&1&2&1&1&1&1&2&2&1&3&2&1&1&10\\ 1&2&2&2&2&1&2&1&2&1&1&1&1&2&2&1&3&2&1&10\\ 1&1&2&2&2&2&1&2&1&2&1&1&1&1&2&2&1&3&2&10\\ 2&1&1&2&2&2&2&1&2&1&2&1&1&1&1&2&2&1&3&10\\ 1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&11\end{tabular}\right),\ \left(\begin{tabular}[]{cccccccccc}18&3&6&12&18&12&0&15&6&3\\ 6&12&12&3&18&15&12&12&3&18\\ 3&3&6&0&6&6&15&18&6&15\\ 6&18&12&3&6&3&6&6&6&9\\ 6&0&6&9&9&3&3&18&15&3\\ 15&12&12&9&3&12&3&12&9&9\\ 9&6&15&18&18&9&12&0&15&0\\ 15&15&18&18&12&0&9&0&12&6\\ 12&3&12&12&9&6&0&12&18&15\\ 18&15&9&3&9&0&6&15&18&18\end{tabular}\right).}

{\left(\begin{tabular}[]{cccccccccccccccccccc}0&2&1&1&2&2&2&2&1&2&4&20&16&10&4&10&2&8&16&7\\ 1&0&2&1&1&2&2&2&2&1&17&10&11&19&4&7&10&11&20&13\\ 2&1&0&2&1&1&2&2&2&2&19&20&16&2&16&16&7&4&17&16\\ 2&2&1&0&2&1&1&2&2&2&17&4&11&19&17&19&16&16&16&1\\ 1&2&2&1&0&2&1&1&2&2&17&2&16&14&13&20&19&4&7&7\\ 1&1&2&2&1&0&2&1&1&2&8&11&11&13&20&10&20&10&13&1\\ 1&1&1&2&2&1&0&2&1&1&14&17&8&5&4&14&10&2&7&10\\ 1&1&1&1&2&2&1&0&2&1&7&8&5&5&11&1&14&1&11&4\\ 2&1&1&1&1&2&2&1&0&2&10&19&11&11&14&17&1&11&4&16\\ 1&2&1&1&1&1&2&2&1&0&5&7&13&20&14&2&17&7&5&13\\ 2&1&2&1&1&1&1&2&2&1&3&2&1&1&2&2&2&2&1&10\\ 1&2&1&2&1&1&1&1&2&2&1&3&2&1&1&2&2&2&2&10\\ 2&1&2&1&2&1&1&1&1&2&2&1&3&2&1&1&2&2&2&10\\ 2&2&1&2&1&2&1&1&1&1&2&2&1&3&2&1&1&2&2&10\\ 2&2&2&1&2&1&2&1&1&1&1&2&2&1&3&2&1&1&2&10\\ 2&2&2&2&1&2&1&2&1&1&1&1&2&2&1&3&2&1&1&10\\ 1&2&2&2&2&1&2&1&2&1&1&1&1&2&2&1&3&2&1&10\\ 1&1&2&2&2&2&1&2&1&2&1&1&1&1&2&2&1&3&2&10\\ 2&1&1&2&2&2&2&1&2&1&2&1&1&1&1&2&2&1&3&10\\ 1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&11\end{tabular}\right),\ \left(\begin{tabular}[]{cccccccccc}18&3&6&12&18&12&0&15&6&3\\ 6&12&12&3&18&15&12&12&3&18\\ 3&3&6&0&6&6&15&18&6&15\\ 6&18&12&3&6&3&6&6&6&9\\ 6&0&6&9&9&3&3&18&15&3\\ 15&12&12&9&3&12&3&12&9&9\\ 9&6&15&18&18&9&12&0&15&0\\ 15&15&18&18&12&0&9&0&12&6\\ 12&3&12&12&9&6&0&12&18&15\\ 18&15&9&3&9&0&6&15&18&18\end{tabular}\right).}

u^{t} v \in {0, \pm 1, \pm 2, \pm 4} for all u, v \in S .

u^{t} v \in {0, \pm 1, \pm 2, \pm 4} for all u, v \in S .

t(\bm{v}):=[\,t_{0}(\bm{v}),\ t_{1}(\bm{v}),\ t_{2}(\bm{v}),\ t_{4}(\bm{v})\,],\ \ t_{n}(\bm{v}):=\big{|}\{\bm{u}\in S\ |\ \bm{u}\,{}^{\mathrm{t}}\bm{v}=n\}\big{|}.

t(\bm{v}):=[\,t_{0}(\bm{v}),\ t_{1}(\bm{v}),\ t_{2}(\bm{v}),\ t_{4}(\bm{v})\,],\ \ t_{n}(\bm{v}):=\big{|}\{\bm{u}\in S\ |\ \bm{u}\,{}^{\mathrm{t}}\bm{v}=n\}\big{|}.

t^{i}(\bm{v}):=[\,t^{i}_{0}(\bm{v}),\ t^{i}_{1}(\bm{v}),\ t^{i}_{2}(\bm{v}),\ t^{i}_{4}(\bm{v})\,],\ \ t^{i}_{n}(\bm{v}):=\big{|}\{\bm{u}\in S_{i}\ |\ \bm{u}\,{}^{\mathrm{t}}\bm{v}=n\}\big{|}.

t^{i}(\bm{v}):=[\,t^{i}_{0}(\bm{v}),\ t^{i}_{1}(\bm{v}),\ t^{i}_{2}(\bm{v}),\ t^{i}_{4}(\bm{v})\,],\ \ t^{i}_{n}(\bm{v}):=\big{|}\{\bm{u}\in S_{i}\ |\ \bm{u}\,{}^{\mathrm{t}}\bm{v}=n\}\big{|}.

t^{i,j}(\bm{v}):=[\,t^{i,j}_{0}(\bm{v}),\ t^{i,j}_{1}(\bm{v}),\ t^{i,j}_{2}(\bm{v}),\ t^{i,j}_{4}(\bm{v})\,],\ \ t^{i,j}_{n}(\bm{v}):=\big{|}\{\bm{u}\in S_{i,j}\ |\ \bm{u}\,{}^{\mathrm{t}}\bm{v}=n\}\big{|}.

t^{i,j}(\bm{v}):=[\,t^{i,j}_{0}(\bm{v}),\ t^{i,j}_{1}(\bm{v}),\ t^{i,j}_{2}(\bm{v}),\ t^{i,j}_{4}(\bm{v})\,],\ \ t^{i,j}_{n}(\bm{v}):=\big{|}\{\bm{u}\in S_{i,j}\ |\ \bm{u}\,{}^{\mathrm{t}}\bm{v}=n\}\big{|}.

m_{1} ⋮ m_{40} = B_{M} B e_{1} ⋮ e_{40},

m_{1} ⋮ m_{40} = B_{M} B e_{1} ⋮ e_{40},

l_{1} ⋮ l_{40} = (3 I_{20} O O \vspace 2 mm 21 I_{20}) e_{1} ⋮ e_{40} .

l_{1} ⋮ l_{40} = (3 I_{20} O O \vspace 2 mm 21 I_{20}) e_{1} ⋮ e_{40} .

U\left(\begin{array}[]{cc}B_{M}&B\vspace{2mm}\\ \hdashline\\ 3I_{20}&O\vspace{2mm}\\ O&21I_{20}\end{array}\right)=\begin{pmatrix}\hskip 9.95845ptB&\vspace{1.5mm}\\ \hdashline\\ \\ \hskip 9.95845ptO&\vspace{1mm}\\ &\end{pmatrix}.

U\left(\begin{array}[]{cc}B_{M}&B\vspace{2mm}\\ \hdashline\\ 3I_{20}&O\vspace{2mm}\\ O&21I_{20}\end{array}\right)=\begin{pmatrix}\hskip 9.95845ptB&\vspace{1.5mm}\\ \hdashline\\ \\ \hskip 9.95845ptO&\vspace{1mm}\\ &\end{pmatrix}.

(4004), (4116), (4226), (6116),

(4004), (4116), (4226), (6116),

n (O_{40}) = max_{v \in S} (∣ S (v) ∣) .

n (O_{40}) = max_{v \in S} (∣ S (v) ∣) .

max_{v \in S} (∣ S (v) ∣) = max_{v \in S / Γ} (∣ S (v) ∣),

max_{v \in S} (∣ S (v) ∣) = max_{v \in S / Γ} (∣ S (v) ∣),

S_{2} (v) := S_{1} (v) \cap S_{1} (u), for u \in S_{1} (v)

S_{2} (v) := S_{1} (v) \cap S_{1} (u), for u \in S_{1} (v)

∣ S_{1} (v) ∣ \geq ∣ S_{2} (v) ∣ \geq ∣ S_{1} (v) \cap S_{1} (u^{'}) ∣ \geq 2 for all u^{'} \in S_{1} (v) .

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Full text

A $40$ -dimensional extremal Type II lattice with no $4$ -frames

Norifumi Ojiro

Department of Mathematics, Graduate School of Science, Hiroshima University

1-3-1, Kagamiyama, Higashi-Hiroshima, Hiroshima, 739-8526, Japan

[email protected]

Abstract.

We construct a $40$ -dimensional extremal Type II lattice not having any subsets consisting of $40$ orthogonal minimal vectors, and determine the automorphism group. This lattice gives an example different from the $16470$ lattices constructed from binary codes by classical constructions.

Key words and phrases:

$40$ -dimensional extremal Type II lattices, $4$ -frames

2010 Mathematics Subject Classification:

11H31, 11H56, 94B05

1. Introduction

Let $L$ be a lattice, which is a free $\mathbb{Z}$ -module of finite rank with a positive definite symmetric bilinear form:

[TABLE]

The lattice $L$ is integral if $\bm{x}\cdot\bm{y}\in\mathbb{Z}$ for all $\bm{x},\bm{y}\in L$ , and furthermore $L$ is said to be even if $\bm{x}\cdot\bm{x}\in 2\mathbb{Z}$ for all $\bm{x}\in L$ . We denote by $L^{\vee}$ the dual lattice of $L$ :

[TABLE]

It is obvious by definition that $L\subset L^{\vee}$ if and only if $L$ is integral. The lattice $L$ is said to be unimodular when $L=L^{\vee}$ . An even unimodular lattice is also called a Type II lattice. It is well-known that the rank of a Type II lattice is a multiple of $8$ . We denote by ${\rm min}(L)$ the minimal norm of $L$ which is the minimum of norms $\bm{x}\cdot\bm{x}$ for all $\bm{x}\in L\setminus\{\bm{0}\}$ . By the theory of modular forms, the minimal norm of an $n$ -dimensional Type II lattice $L$ satisfies

[TABLE]

and the lattice $L$ is said to be extremal when the equality holds. We regard an $n$ -dimensional lattice $L$ as being in $\mathbb{R}^{n}$ with the canonical inner product, and write an element of $\mathbb{R}^{n}$ as a row vector. Then $\bm{x}\cdot\bm{y}=\bm{x}\,{}^{\mathrm{t}}\bm{y}$ for $\bm{x}$ , $\bm{y}\in\mathbb{R}^{n}$ . A $4$ -frame is a subset of $\mathbb{R}^{n}$ consisting of $n$ orthogonal $4$ -vectors:

[TABLE]

According to Theorem $1$ of [6], an even lattice with a $4$ -frame is constructed from a doubly even binary code, and by Theorem $3$ of [6] we see that an extremal doubly even self-dual binary code of length $40$ corresponds bijectively to a $40$ -dimensional extremal Type II lattice with a $4$ -frame. Then it was shown in [1] that the number of extremal doubly even self-dual binary codes of length $40$ is $16470$ . Further by [5] all extremal doubly even self-dual binary codes of length $40$ are obtained as the residue codes of some extremal Type II codes over $\mathbb{Z}/4\mathbb{Z}$ of the same length. As a result, we know that there are exactly $16470$ extremal Type II lattices of rank $40$ with a $4$ -frame and they are constructed from extremal doubly even self-dual binary codes or some extremal Type II codes over $\mathbb{Z}/4\mathbb{Z}$ of length $40$ . On the other hand, by [10] there are more than $8.45\times 10^{51}$ extremal Type II lattices of rank $40$ , whereas so many examples do not seem to be known: Mckay’s $M_{40}$ , the lattice $(U_{5}(2)\times 2^{1+4}_{-}.Alt_{5}).2$ in the website [7] (we simply denote this lattice by $N_{40}$ in this paper), and furthermore the $16470$ lattices with a $4$ -frame e.g. three lattices $\mathfrak{L}_{1}$ , $\mathfrak{L}_{2}$ , $\mathfrak{L}_{3}$ in [8], the lattice constructed in [2] from a double circulant code over $\mathbb{Z}/4\mathbb{Z}$ . Note that perhaps $M_{40}$ , $N_{40}$ may belong to the $16470$ examples.

Recently in [12] an extremal Type II lattice of rank $64$ was constructed from a generalized quadratic residue code of length $32$ and the discriminant group of an even lattice of rank $2$ . In the same way, we shall construct a certain $40$ -dimensional lattice denoted by $O_{40}$ as an overlattice of the lattice with a Gram matrix

[TABLE]

where $I_{n}$ denotes the unit matrix of rank $n$ . Our results are as follows:

Theorem 1.1.

The lattice $O_{40}$ is a $40$ -dimensional extremal Type II lattice with no $4$ -frames.

Theorem 1.2.

The automorphism group of $O_{40}$ is of order $684$ , which is isomorphic to the semidirect product $\mathbb{Z}/19\mathbb{Z}\rtimes_{\psi}\mathbb{Z}/36\mathbb{Z}$ of additive groups $\mathbb{Z}/19\mathbb{Z}$ and $\mathbb{Z}/36\mathbb{Z}$ , where $\psi$ is the group homomorphism defined by

[TABLE]

The automorphism group of $M_{40}$ contains a subgroup of order $13680$ (cf. [3, § $5$ of Chapter $8$ ]). The automorphism group of $N_{40}$ is of order $2^{18}\times 3^{6}\times 5^{2}\times 11$ , see [7]. Hence we have the following corollary:

Corollary 1.3.

The lattice $O_{40}$ is not isometric to any lattice of $M_{40}$ , $N_{40}$ and the $16470$ extremal Type II lattices with a $4$ -frame.

In particular Corollary 1.3 implies that $O_{40}$ cannot be obtained from binary and some $\mathbb{Z}_{4}$ -codes by classical constructions, as mentioned above. We also have the following by examining the $4$ -vectors of $O_{40}$ :

Lemma 1.4.

The lattice $O_{40}$ contains

[TABLE]

as a sublattice, where $A_{n}(2)$ denotes the lattice arising from scaling up the irreducible root lattice $A_{n}$ by $\sqrt{2}$ and $\oplus$ denotes orthogonal sum.

Let $L$ be the lattice with a Gram matrix given by (1). We denote by $L+M$ the lattice consisting of the vectors $\bm{v}+\bm{u}$ for all $\bm{v}\in L$ , $\bm{u}\in M$ . Then one has the following:

Theorem 1.5.

$O_{40}=L+M$ .

This implies that $O_{40}$ can be obtained by gluing $M$ to $L$ .

In the next section we construct $O_{40}$ from two generalized quadratic residue codes of length $20$ . In Section $3$ we investigate the types of the $4$ -vectors of $O_{40}$ and give a proof of Theorem 1.5. In Section $4$ we show that $O_{40}$ does not contain any $4$ -frames, proving that $O_{40}$ contains at most $32$ orthogonal $4$ -vectors (see Lemma 4.1). In Section $5$ we determine the automorphism group of $O_{40}$ by utilizing that $O_{40}$ has a basis consisting of $4$ -vectors. A Gram matrix of $O_{40}$ is presented as an appendix in Section $6$ . Results in this paper are based on calculations done by using GAP [13] and PARI/GP [14]. Those computational data files are available upon request addressed to the author.

The author would like to express his gratitude to Professor Ichiro Shimada who informed him of References [1], [5], [6] and [12], also advised him on this manuscript.

2. Construction

We construct a $40$ -dimensional extremal Type II lattice as an even overlattice of an even lattice using two generalized quadratic residue codes. As references on this section, the readers are recommended to see [4, in particular, Chapter $3.3$ ] and [12].

Let $R$ be the $2$ -dimensional even lattice with a Gram matrix

[TABLE]

and let $L$ be the $40$ -dimensional even lattice which is the orthogonal sum of $20$ copies of $R$ . Then a Gram matrix of $L$ is written as in (1). The discriminant group of $L$ , namely $D_{L}:=L^{\vee}/L$ is an abelian group of order $63^{20}$ and which has a natural bilinear form

[TABLE]

and a quadratic form

[TABLE]

Let $\{\bm{e}_{1},\dots,\bm{e}_{40}\}$ be a basis of $L^{\vee}$ with a Gram matrix

[TABLE]

Then $D_{L}$ can be written as

[TABLE]

We consider a subgroup $C$ of $D_{L}$ satisfying the conditions

(i)

$C$ is totally isotropic, i.e. for all $\bm{u}+L,\bm{v}+L\in C$ ,

$b_{L}(\bm{u}+L,\bm{v}+L)=\mathbb{Z}$ and $q_{L}(\bm{u}+L)=2\mathbb{Z}$ . 2. (ii)

$|C|=63^{10}$ , where $|C|$ denotes the cardinality of $C$ .

For $p:L^{\vee}\rightarrow D_{L}$ , put $L_{C}:=p^{-1}(C)$ . Then $L\subset L_{C}\subset L^{\vee}$ . It is easy to see that $L_{C}$ is even by (i) and also unimodular by (ii). We will give $C$ by utilizing a direct product of two generalized quadratic residue codes of length $20$ over $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/21\mathbb{Z}$ .

Let $A$ be a commutative ring with unity and let $A^{n}$ be the free $A$ -module which is the direct product of $n$ copies of $A$ . We give a numbering $(1,2,3,\dots,18,0,\infty)$ to the coordinates of $A^{20}$ , and identify a finite field $\mathbb{F}_{19}$ with $\{1,2,\dots,18,0\}$ . Then a generalized quadratic residue code of length $20$ over $A$ with a parameter $(a,b,d,s,t,e)\in A^{6}$ is a $A$ -submodule of $A^{20}$ generated by the vectors $\bm{u}_{1},\dots,\bm{u}_{18},\bm{u}_{0},\bm{u}_{\infty}$ which are defined as follows:

[TABLE]

where $\chi_{19}$ denotes the Legendre character modulo $19$ .

Let $\mathscr{C}_{n}$ be a generalized quadratic residue code of length $20$ over $\mathbb{Z}/n\mathbb{Z}$ with some parameter, and let $\bm{u}_{i}$ and $\bm{w}_{i}$ be the row vectors of $20$ components which are the generators of $\mathscr{C}_{3}$ and $\mathscr{C}_{21}$ respectively, defined as above. We denote by $C_{3}\times C_{21}$ the subgroup of $D_{L}$ generated by the direct product $\mathscr{C}_{3}\times\mathscr{C}_{21}\subset(\mathbb{Z}/3\mathbb{Z})^{20}\times(\mathbb{Z}/21\mathbb{Z})^{20}$ . Then one can replace (i) by

[TABLE]

Further (ii) is replaced by

(ii’)

For a basis $\{\bm{v}_{1},\dots,\bm{v}_{40}\}$ of $L_{C_{3}\times C_{21}}$ , let $B$ be a $40$ -by- $40$ integral matrix such that

[TABLE]

Then $\det(B)=63^{10}$ , where $\det(B)$ is the determinant of $B$ .

We search a code $\mathscr{C}_{3}\times\mathscr{C}_{21}$ satisfying (i’) and (ii’). Exploring by a computer one finds the code satisfying the conditions, which has a pair of parameters

[TABLE]

Then the matrix $B$ can be written as the form

[TABLE]

where $B_{1}$ and $B_{3}$ are $20$ -by- $20$ and $10$ -by- $10$ matrices respectively as follows:

[TABLE]

Applying for example ShortestVectors of GAP to $L_{C_{3}\times C_{21}}$ , one can confirm within one minute that $L_{C_{3}\times C_{21}}$ is extremal, i.e. ${\rm min}(L_{C_{3}\times C_{21}})=4$ . We denote $L_{C_{3}\times C_{21}}$ by $O_{40}$ .

Remark 2.1.

Since a generalized quadratic residue code of length $20$ is invariant under the permutations of the coordinates $p_{c}:x_{t}\mapsto x_{t+1}$ of order $19$ and $p_{s}:x_{t}\mapsto x_{4t}$ of order $9$ , the automorphism group ${\rm Aut}(O_{40})$ has a subgroup $\varGamma$ of order $342$ which arises from $p_{c}$ , $p_{s}$ and the change of sign (cf. [12, Section $3$ ]).

3. Types of $4$ -vectors

Let $S$ be the set of the $4$ -vectors of $O_{40}$ . By the theory of modular forms (cf. e.g. [11, Chapter $7$ ]), one has $|S|=39600$ . It follows immediately from ${\rm min}(O_{40})=4$ that

[TABLE]

The type of $\bm{v}\in S$ is defined by

[TABLE]

By definition $t_{0}(\bm{v})+2(t_{1}(\bm{v})+t_{2}(\bm{v})+t_{4}(\bm{v}))=39600$ and $t_{4}(\bm{v})=1$ for all $\bm{v}\in S$ . Using the function qfminim of PARI/GP, we obtain the list of the elements of $S$ . It turns out by direct calculation that $18$ types appear for the vectors of $S$ , and then we divide $S$ into subsets $S_{1},\dots,S_{18}$ which each consist of vectors with the same type, see Table 3.1.

Similarly, for $1\leq i\leq 18$ the $S_{i}$ -type of $\bm{v}\in S_{i}$ is defined by

[TABLE]

Actually the vectors of $S_{i}$ have $m_{i}$ possible $S_{i}$ -types where $m_{i}\geq 1$ , and we divide $S_{i}$ into subsets $S_{i,1},\dots,S_{i,m_{i}}$ which each consist of vectors with the same $S_{i}$ -type.

Further, for $1\leq j\leq m_{i}$ we compute the $S_{i,j}$ -type of $\bm{v}\in S_{i,j}$ defined by

[TABLE]

It turns out that the vectors of $S_{i,j}$ have the only $S_{i,j}$ -type, see Table 3.2.

This implies that $S$ is ultimately divided into $64$ subsets $S_{i,j}$ ’s by the types. We call $S_{i,j}$ an irreducible subset and $t^{i,j}$ an irreducible type from the fact. If $S$ has an irreducible subset with an irreducible type $[\,78,0,0,1\,]$ then $O_{40}$ contains a $4$ -frame. But we do not know whether the converse holds true. Therefore we count directly the numbers of orthogonal $4$ -vectors contained in $O_{40}$ in the next section.

Here $40$ vectors chosen appropriately from $S_{1,1}\cup S_{18,1}$ form a basis of the lattice $M$ described in Lemma 1.4, and thus $M\subset O_{40}$ .

Proof of Theorem 1.5.

Let $\{\bm{m}_{1},\dots,\bm{m}_{40}\}$ be a basis of $M$ and let $B_{M}$ be the matrix $B_{M}$ such that

[TABLE]

where $B$ is the matrix given by (2). Further we take a basis $\bm{l}_{1},\dots,\bm{l}_{40}$ of $L$ such that

[TABLE]

Then by computing one can easily verify that there is a unimodular matrix $U$ of rank $80$ such that

[TABLE]

This means that $U\,{}^{\mathrm{t}}({}^{\mathrm{t}}\bm{m}_{1},\dots,{}^{\mathrm{t}}\bm{m}_{40},{}^{\mathrm{t}}\bm{l}_{1},\dots,{}^{\mathrm{t}}\bm{l}_{40})={}^{\mathrm{t}}({}^{\mathrm{t}}\bm{v}_{1},\dots,{}^{\mathrm{t}}\bm{v}_{40},{}^{\mathrm{t}}\bm{0},\dots,{}^{\mathrm{t}}\bm{0})$ , where $\{\bm{v}_{1},\dots,\bm{v}_{40}\}$ is a basis of $O_{40}$ . This completes the proof. ∎

Remark 3.1.

From the case where $R$ is of the lattices with Gram matrices

[TABLE]

we can actually obtain at least two extremal Type II lattices of rank $40$ . All those lattices, however, contain a $4$ -frame because the lattices have an irreducible subset with an irreducible type $[\,78,0,0,1\,]$ , and thus belong to the $16470$ examples. One of those lattices has the $4$ -vectors of three types and the automorphism group of order at least $27360$ .

4. Orthogonal $4$ -vectors

Let $n(O_{40})$ be the maximum of the numbers of orthogonal $4$ -vectors contained in $O_{40}$ . Then we prove the following lemma:

Lemma 4.1.

$n(O_{40})=32$ .

Proof.

For each $\bm{v}\in S$ let $S(\bm{v})$ be a set of orthogonal $4$ -vectors containing $\bm{v}$ such that $|S(\bm{v})|$ is largest. By definition it is obvious that

[TABLE]

Since every automorphism $g$ of $O_{40}$ acts on $S$ and $|S(\bm{v}g)|=|S(\bm{v})|$ , one has

[TABLE]

where $\varGamma$ is the subgroup of ${\rm Aut}(O_{40})$ described in Remark 2.1. By computing we have $|S/\varGamma|=132$ . Then $S(\bm{v})$ is calculated as follows. For every $\bm{v}\in S$ we compute the set $S_{1}(\bm{v})$ consisting of $\bm{v}$ and all $4$ -vectors perpendicular to $\bm{v}$ . We put $S(\bm{v}):=S_{1}(\bm{v})$ if $|S_{1}(\bm{v})|=1$ , otherwise put

[TABLE]

such that $|S_{1}(\bm{v})\cap S_{1}(\bm{u})|$ is largest. Then $S_{2}(\bm{v})$ is the set consisting of $2$ orthogonal vectors $\bm{v}$ , $\bm{u}$ and all $4$ -vectors perpendicular to these $2$ vectors, and by definition

[TABLE]

If $|S_{2}(\bm{v})|=2$ then we put $S(\bm{v}):=S_{2}(\bm{v})$ . If not, repeating this procedure while $|S_{n}(\bm{v})|>n$ one has $|S_{m_{\bm{v}}}(\bm{v})|={m_{\bm{v}}}$ for some ${m_{\bm{v}}}\leq 40$ , and then put $S(\bm{v}):=S_{m_{\bm{v}}}(\bm{v})$ . In Table 4.1 we give the number $m_{\bm{v}}$ for $\bm{v}\in S/\varGamma$ and the number $n_{\bm{v}}$ of $\bm{v}$ such that $|S(\bm{v})|=m_{\bm{v}}$ .

One has ${\rm max}_{\bm{v}\in S/\varGamma}(|S(\bm{v})|)=32$ by Table 4.1, and thus $n(O_{40})=32$ . ∎

By Lemma 4.1, $O_{40}$ does not contain any $4$ -frames, and thus a proof of Theorem 1.1 has been completed.

5. The automorphism group

A $40$ -dimensional extremal Type II lattice $L_{40}$ has a basis consisting of $4$ -vectors and $6$ -vectors, while there is the case where $L_{40}$ cannot have a basis consisting of only $4$ -vectors, see Theorem 1 and Remark 1 in [9]. Fortunately $O_{40}$ is not of the case:

Lemma 5.1.

The lattice $O_{40}$ has a basis consisting of $4$ -vectors.

Proof.

Let $\{\bm{v}_{1},\dots,\bm{v}_{40}\}$ be the basis of $O_{40}$ given by the matrix $B$ of (2) and let $G(\bm{v}_{1},\dots,\bm{v}_{40})$ be the Gram matrix with respect to $\{\bm{v}_{1},\dots,\bm{v}_{40}\}$ . Applying the function LLLReducedGramMat of GAP to $G(\bm{v}_{1},\dots,\bm{v}_{40})^{-1}$ , one has a basis consisting of $34$ vectors of norm $4$ and $6$ vectors of norm $6$ . Then one can easily find by computing a basis consisting of $4$ -vectors containing those $34$ vectors. ∎

Actually we can take a basis $\{\bm{v}^{\prime}_{1},\dots,\bm{v}^{\prime}_{40}\}$ of $O_{40}$ such that

[TABLE]

The Gram matrix with respect to $\{\bm{v}^{\prime}_{1},\dots,\bm{v}^{\prime}_{40}\}$ is given in Section $6$ .

Let $B^{\prime}$ be the matrix such that

[TABLE]

Then one has

[TABLE]

Since each automorphism of $O_{40}$ is regarded as an element $g$ of ${\rm GL}_{40}(\mathbb{Z})$ fixing $G(\bm{v}_{1},\dots,\bm{v}_{40})$ , one has

[TABLE]

Note that $g$ preserves the type of $S$ . Hence putting $B^{\prime}:={}^{\mathrm{t}}({}^{\mathrm{t}}\bm{b}^{\prime}_{1},\dots,{}^{\mathrm{t}}\bm{b}^{\prime}_{40})$ and $V:={}^{\mathrm{t}}({}^{\mathrm{t}}\bm{v}_{1},\dots,{}^{\mathrm{t}}\bm{v}_{40})$ , we see that $\bm{b}^{\prime}_{1}gV,\dots,\bm{b}^{\prime}_{40}gV$ each belong to the same irreducible subset with $\bm{v}^{\prime}_{1},\dots,\bm{v}^{\prime}_{40}$ . Then we enumerate all the matrices $X:={}^{\mathrm{t}}({}^{\mathrm{t}}\bm{x}_{1},\dots,{}^{\mathrm{t}}\bm{x}_{40})$ satisfying the following conditions:

•

Each row vector $\bm{x}_{i}$ belongs to the same irreducible subset with $\bm{v}^{\prime}_{i}$ and particularly $\bm{x}_{2}$ belongs to a fixed complete set of representatives for $S_{7,7}/\varGamma$ , where $|S_{7,7}/\varGamma|=4$ by computing.

•

$G(\bm{x}_{1},\dots,\bm{x}_{40})=G(\bm{v}^{\prime}_{1},\dots,\bm{v}^{\prime}_{40})$ .

Exploring on a computer, it is shown that there are exactly $2$ matrices $X_{1}$ , $X_{2}$ with these properties. Since $\varGamma$ acts transitively on each $\varGamma$ -orbit of $S_{7,7}$ , for every $g\in{\rm Aut}(O_{40})$ there is an element $\gamma$ of $\varGamma$ such that $B^{\prime}g\gamma V=X_{1}$ or $B^{\prime}g\gamma V=X_{2}$ . Hence ${\rm Aut}(O_{40})$ is obtained by adding $B^{\prime-1}X_{1}V^{-1}$ and $B^{\prime-1}X_{2}V^{-1}$ to $\varGamma$ . As a result, one has

[TABLE]

By direct calculation, one can verify that ${\rm Aut}(O_{40})$ has the generators $g_{1}$ , $g_{2}$ of orders $36$ , $19$ respectively such that $g_{1}g_{2}g_{1}^{-1}=g_{2}^{3}$ (cf. Section $6$ ). Hence ${\rm Aut}(O_{40})$ contains the cyclic group $\langle g_{2}\rangle$ of order $19$ as a normal subgroup. Further one has

[TABLE]

Therefore ${\rm Aut}(O_{40})$ is isomorphic to the semidirect product of $\langle g_{2}\rangle$ and $\langle g_{1}\rangle$ . Regarding $\mathbb{Z}/19\mathbb{Z}$ and $\mathbb{Z}/36\mathbb{Z}$ as additive groups, this implies that there is the following isomorphism:

[TABLE]

where $\psi$ is the homomorphism given in Theorem 1.2, and thus the proof has been completed.

6. Appendix

In Tables $6.1$ - $6.3$ we give examples of a Gram matrix $G(\bm{v}^{\prime}_{1},\dots,\bm{v}^{\prime}_{40})$ of $O_{40}$ and the generators $g_{1}$ , $g_{2}$ of ${\rm Aut}(O_{40})$ with respect to the basis $\{\bm{v}^{\prime}_{1},\dots,\bm{v}^{\prime}_{40}\}$ , which satisfy the conditions described in the previous section:

Bibliography14

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] K. Betsumiya, M. Harada and A. Munemasa, A complete classification of doubly even self-dual codes of length 40 40 40 , Electron. J. Combin., 19(3):P 18(12pp.), 2012.
2[2] A. R. Calderbank and N. J. A. Sloane, Double circulant codes over ℤ 4 subscript ℤ 4 \mathbb{Z}_{4} and even unimodular lattices , J. Algebraic Combin., 6:119-131, 1997.
3[3] J. H. Conway and N. J. A. Sloane, Sphere packing, lattices and groups , Grundlehren der Mathematishen Wissenschaften, volume 290, third edition, Springer-Verlag, New York, 1999.
4[4] W. Ebeling, Lattices and codes , Advanced Lectures in Mathematics, Second revised edition, Vieweg, Braunschweig, 2002.
5[5] M. Harada, Extremal Type II ℤ 4 subscript ℤ 4 \mathbb{Z}_{4} -codes constructed from binary doubly even self-dual codes of length 40 40 40 , Discrete Math., 340(10):2466-2468, 2017.
6[6] M. Kitazume, T. Kondo and I. Miyamoto, Even lattices and doubly even codes , J. Math. Soc. Japan, 43(1):67-87, 1991.
7[7] G. Nebe and N. J. A. Sloane, A Catalogue of Lattices , published electronically at http://www.math.rwth-aachen.de/ ∼ similar-to \sim Gabriele.Nebe/LATTICES/.
8[8] M. Ozeki, Examples of even unimodular extremal lattices of rank 40 40 40 and their Siegel theta-series of degree 2 2 2 , J. Number Theory, 28:119-131, 1988.

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A 404040-dimensional extremal Type II lattice with no 444-frames

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

Theorem 1.1**.**

Theorem 1.2**.**

Corollary 1.3**.**

Lemma 1.4**.**

Theorem 1.5**.**

2. Construction

Remark 2.1**.**

3. Types of 444-vectors

Proof of Theorem 1.5.

Remark 3.1**.**

4. Orthogonal 444-vectors

Lemma 4.1**.**

Proof.

5. The automorphism group

Lemma 5.1**.**

Proof.

6. Appendix

A $40$ -dimensional extremal Type II lattice with no $4$ -frames

Theorem 1.1.

Theorem 1.2.

Corollary 1.3.

Lemma 1.4.

Theorem 1.5.

Remark 2.1.

3. Types of $4$ -vectors

Remark 3.1.

4. Orthogonal $4$ -vectors

Lemma 4.1.

Lemma 5.1.