# 4D Moser brute force search

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The brute force search program requires first building a preliminary lookup table, and then a refined lookup table, to determine the Pareto-optimal statistics for all forbidden Level 2 sets. Details of the lookup table construction are here.

The basic idea is to run over pairs of Level 1 slices and Level 3 slices, which are 3D Moser sets. For each such pair, compute the forbidden Level 2 set, then use the lookup table to find the optimal statistics for that pair, add that to a global table of feasible (a,b,c,d,e) statistics for 4D Moser sets, and iterate. However, the total number of such pairs is $3813884 \times 3813884 \sim 1.4 \times 10^{13}$, which is computationally infeasible.

However, one can use symmetries to cut this number down. Look at the "a" corners of the Level 1 and Level 3 sets; these are two 8-bit strings (which we call the "a-signatures" of the Level1 and Level3 sets), so there are $2^{16}$ possible choices for these. Actually we can eliminate those choices for which a=1 and a=2, because if (0,b,c,d,e) is a feasible statistic then (1,b,c,d,e) and (2,b,c,d,e) is feasible also (just pick a "b", "c", "d", or "e" point which is not in the set, and then pick a pair of "a" points with that midpoint). In fact (3,b,c,d,e) is also feasible, see Lemma below.

For the remaining configurations, one exploits the symmetry group of $[3]^4$, which has order $4! \times 2^4 = 384$. There are 391 remaining equivalence classes; for each equivalence class, we pick a representative which minimises the number of Level1 x Level3 pairs. A table of these representatives can be found here. A table of how many Moser sets there are for each a-signature can be found here.

With these reductions, the number of pairs to check drops to 62 billion (or more precisely, 62009590818).

The code to scan the pairs is here. When compiled (say, as scan.exe), the format is

 scan x y


which will scan the signature-pair classes from x to y, where $0 \leq x \leq y \leq 390$, and output the relevant feasible statistics to stdout and to an output file. If instead one types

 scan x y count


then this will indicate what percentage of the scan range is covered by x to y.

In the case x=372 and y=390 (which only takes about a minute to run), here is the stdout output and file output (for testing purposes). The full file output can be found here.

Lemma Suppose that (0,b,c,d,e) is a feasible statistic for a 4D Moser set. Then (3,b,c,d,e) is also feasible.

Proof Let A be a 4D Moser set with statistics (0,b,c,d,e). It suffices to show that we can add three "a" points to A and still have a Moser set, i.e. one can find three "a" points whose three midpoints are omitted by A. We assume for contradiction that this is not possible.

Suppose first that A contains a "d" point, such as 2221. Then A must omit either 2211 or 2231; without loss of generality we may assume that it omits 2211. Similarly we may assume it omits 2121 and 1221. Then we can add 1131, 1311, 3111 to A, a contradiction. Thus we may assume that A contains no "d" points.

Now suppose that A omits a "c" point, such as 2211. Then one can add 3333, 3111, 1311 to A, a contradiction. Thus we may assume that A contains all "c" points, which in particular implies that A omits the "e" point 2222.

Since A contains all the "c" points, it must omit a "b" point, such as 2111. But then 3111, 1111, 3333 can be added to the set, a contradiction.

## Pareto Maxima and Extremal Points

This routine was run on a Linux cluster, taking around two hours. The output files were collated, there were 390 Pareto maxima:

    3    16    24     0     0
4    14    19     2     0
4    15    24     0     0
4    16     8     4     1
4    16    14     4     0
4    16    23     0     0
4    17    21     0     0
4    18    19     0     0
5    12    12     4     1
5    12    13     6     0
5    12    15     5     0
5    12    19     2     0
5    13    10     4     1
5    13    14     5     0
5    13    21     1     0
5    15     9     4     1
5    15    12     3     1
5    15    13     5     0
5    15    18     3     0
5    15    20     1     0
5    15    22     0     0
5    16     7     4     1
5    16    10     3     1
5    16    11     5     0
5    16    12     2     1
5    16    16     3     0
5    16    19     1     0
5    16    21     0     0
5    17    12     4     0
5    17    14     3     0
5    17    16     2     0
5    17    18     1     0
5    17    20     0     0
5    18    13     3     0
5    18    14     2     0
5    20     8     4     0
5    20    10     3     0
5    20    13     2     0
5    20    14     1     0
5    20    18     0     0
5    21    10     2     0
5    21    15     0     0
5    22    13     0     0
6     8    12     8     0
6    10    11     4     1
6    11    12     7     0
6    12    10     7     0
6    12    13     5     0
6    12    18     4     0
6    13    16     4     0
6    14     9     4     1
6    14     9     7     0
6    14    12     6     0
6    14    16     3     0
6    14    19     1     0
6    14    21     0     0
6    15     7     4     1
6    15    10     3     1
6    15    10     6     0
6    15    11     2     1
6    15    12     5     0
6    15    15     4     0
6    15    20     0     0
6    16     7     3     1
6    16     8     6     0
6    16     9     2     1
6    16    10     5     0
6    16    12     1     1
6    16    13     4     0
6    16    14     3     0
6    16    18     2     0
6    16    19     0     0
6    17     9     5     0
6    17    10     4     0
6    17    13     3     0
6    17    15     2     0
6    17    17     1     0
6    17    18     0     0
6    18    13     2     0
6    18    16     1     0
6    18    17     0     0
6    19     9     4     0
6    19    12     3     0
6    19    15     1     0
6    20     7     4     0
6    20     9     3     0
6    20    12     2     0
6    20    13     1     0
6    20    15     0     0
6    21     8     3     0
6    21     9     2     0
6    21    12     1     0
6    21    14     0     0
6    22     7     3     0
6    22     8     2     0
6    22    10     1     0
6    23     9     1     0
6    24     7     2     0
6    24     8     1     0
6    24    12     0     0
6    25     9     0     0
6    26     7     0     0
7     8     6     8     0
7    11     9     4     1
7    11    12     6     0
7    12     8     4     1
7    12     8     6     0
7    12    12     3     1
7    12    12     5     0
7    12    13     4     0
7    12    15     3     0
7    12    17     2     0
7    13     7     4     1
7    13    10     3     1
7    13    11     5     0
7    13    12     2     1
7    13    12     4     0
7    13    14     3     0
7    13    16     2     0
7    14     6     4     1
7    14     6     7     0
7    14     9     5     0
7    14    10     2     1
7    14    12     1     1
7    14    17     1     0
7    14    19     0     0
7    15     7     5     0
7    15     8     3     1
7    15     9     2     1
7    15    11     1     1
7    15    11     4     0
7    15    13     3     0
7    15    16     1     0
7    16     6     3     1
7    16     6     6     0
7    16     8     2     1
7    16    10     1     1
7    16    10     4     0
7    16    12     0     1
7    16    12     3     0
7    16    15     2     0
7    16    17     0     0
7    17     6     5     0
7    17     7     4     0
7    17    11     3     0
7    17    13     2     0
7    17    14     1     0
7    17    16     0     0
7    18    10     3     0
7    18    13     1     0
7    18    15     0     0
7    19     9     3     0
7    20     6     4     0
7    20    11     2     0
7    20    12     1     0
7    20    14     0     0
7    21     8     2     0
7    21    10     1     0
7    21    12     0     0
7    22     9     1     0
7    22    11     0     0
7    23     6     3     0
7    23     7     1     0
7    23    10     0     0
7    24     6     2     0
7    24     9     0     0
7    25     6     1     0
7    25     8     0     0
7    26     3     1     0
7    28     6     0     0
7    29     3     0     0
7    30     1     0     0
8     8     0     8     0
8     8     9     7     0
8     8    12     6     0
8     9     9     4     1
8     9    10     6     0
8     9    12     3     1
8     9    12     5     0
8     9    13     4     0
8     9    15     3     0
8    10     7     4     1
8    10    10     3     1
8    10    10     5     0
8    10    12     2     1
8    10    12     4     0
8    10    13     3     0
8    10    15     2     0
8    11     6     4     1
8    11     9     6     0
8    11    10     2     1
8    11    11     4     0
8    12     7     6     0
8    12     9     3     1
8    12     9     5     0
8    12    10     4     0
8    12    12     1     1
8    12    14     2     0
8    12    16     1     0
8    12    18     0     0
8    13     7     3     1
8    13     7     5     0
8    13     9     2     1
8    13    12     0     1
8    13    12     3     0
8    14     0     7     0
8    14     6     6     0
8    14     7     2     1
8    14     8     1     1
8    14     9     4     0
8    14    11     0     1
8    14    11     3     0
8    14    13     2     0
8    14    15     1     0
8    14    17     0     0
8    15     6     3     1
8    15     6     5     0
8    15     7     1     1
8    16     0     6     0
8    16     4     3     1
8    16     4     5     0
8    16     6     2     1
8    16     8     4     0
8    16     9     0     1
8    16    10     3     0
8    16    12     2     0
8    16    14     1     0
8    16    16     0     0
8    17     0     5     0
8    17     3     4     0
8    17     8     3     0
8    17    10     2     0
8    17    12     1     0
8    17    14     0     0
8    18     9     2     0
8    18    11     1     0
8    18    12     0     0
8    19     6     3     0
8    19     8     2     0
8    20     0     4     0
8    20     4     3     0
8    20     7     2     0
8    20     9     1     0
8    20    11     0     0
8    21     4     2     0
8    21     7     1     0
8    22     3     2     0
8    22     6     1     0
8    22     9     0     0
8    23     0     3     0
8    23     4     1     0
8    24     0     2     0
8    24     3     1     0
8    24     8     0     0
8    25     1     1     0
8    25     6     0     0
8    26     0     1     0
8    26     4     0     0
8    28     3     0     0
8    32     0     0     0
9     8    10     4     0
9     9     9     4     0
9     9    12     3     0
9    10     8     4     0
9    10    10     3     0
9    10    12     2     0
9    10    13     1     0
9    10    15     0     0
9    11    11     2     0
9    12     7     4     0
9    12     9     3     0
9    12    12     1     0
9    12    14     0     0
9    13     7     3     0
9    13    10     2     0
9    14     9     2     0
9    14    11     1     0
9    14    13     0     0
9    15     6     3     0
9    16     0     4     0
9    16     4     3     0
9    16     8     2     0
9    16    10     1     0
9    16    12     0     0
9    17     3     3     0
9    17     6     2     0
9    17     8     1     0
9    17    10     0     0
9    18     2     3     0
9    18     4     2     0
9    18     7     1     0
9    18     9     0     0
9    19     0     3     0
9    19     3     2     0
9    19     6     1     0
9    20     1     2     0
9    20     5     1     0
9    20     8     0     0
9    21     4     1     0
9    21     6     0     0
9    22     1     1     0
9    22     5     0     0
9    24     4     0     0
9    25     2     0     0
9    28     0     0     0
10     8     6     4     0
10     8     8     3     0
10     9     7     3     0
10     9    10     2     0
10     9    11     1     0
10     9    13     0     0
10    10     5     4     0
10    10     9     2     0
10    10    12     0     0
10    11     6     3     0
10    12     4     4     0
10    12     5     3     0
10    12     7     2     0
10    12    10     1     0
10    12    11     0     0
10    13     6     2     0
10    13     8     1     0
10    13    10     0     0
10    14     3     3     0
10    14     5     2     0
10    14     9     0     0
10    15     2     3     0
10    15     7     1     0
10    16     4     2     0
10    16     6     1     0
10    16     8     0     0
10    17     4     1     0
10    17     6     0     0
10    18     2     1     0
10    18     5     0     0
10    20     4     0     0
10    21     2     0     0
10    22     1     0     0
10    24     0     0     0
11     4     6     4     0
11     6     5     4     0
11     7     6     3     0
11     8     4     4     0
11     8     5     3     0
11     9     6     2     0
11     9     8     1     0
11     9    10     0     0
11    10     3     3     0
11    10     5     2     0
11    10     9     0     0
11    11     2     3     0
11    11     7     1     0
11    12     4     2     0
11    12     6     1     0
11    12     8     0     0
11    13     4     1     0
11    13     6     0     0
11    14     2     1     0
11    14     5     0     0
11    16     4     0     0
11    17     2     0     0
11    18     1     0     0
11    20     0     0     0
12     4     3     3     0
12     6     2     3     0
12     6     5     2     0
12     6     7     1     0
12     6     9     0     0
12     8     4     2     0
12     8     6     1     0
12     8     8     0     0
12     9     4     1     0
12     9     6     0     0
12    10     2     1     0
12    10     5     0     0
12    12     4     0     0
12    13     2     0     0
12    14     1     0     0
12    16     0     0     0
13     6     5     0     0
13     8     4     0     0
13     9     2     0     0
13    10     1     0     0
13    12     0     0     0
14     4     3     0     0
14     5     2     0     0
14     6     1     0     0
14     8     0     0     0
15     4     0     0     0
16     0     0     0     0


Using qhull, 58 extremals were found:

    3     16     24      0      0
4     14     19      2      0
4     15     24      0      0
4     16      8      4      1
4     16     14      4      0
4     18     19      0      0
5     12     12      4      1
5     12     19      2      0
5     13     21      1      0
5     15      9      4      1
5     15     12      3      1
5     15     18      3      0
5     16      7      4      1
5     16     10      3      1
5     16     12      2      1
5     20      8      4      0
5     20     18      0      0
5     21     10      2      0
6      8     12      8      0
6     10     11      4      1
6     12     18      4      0
6     14      9      4      1
6     14      9      7      0
6     14     12      6      0
6     14     21      0      0
6     15      7      4      1
6     16     18      2      0
7     12     12      3      1
7     14      6      4      1
7     14      6      7      0
7     16     12      0      1
7     30      1      0      0
8      8      0      8      0
8      8      9      7      0
8      8     12      6      0
8      9      9      4      1
8      9     12      3      1
8      9     15      3      0
8     10      7      4      1
8     10     12      2      1
8     11      6      4      1
8     11     10      2      1
8     12     12      1      1
8     12     18      0      0
8     13     12      0      1
8     14      0      7      0
8     14      6      6      0
8     14      8      1      1
8     15      6      3      1
8     15      7      1      1
8     16      4      3      1
8     16      6      2      1
8     16      9      0      1
8     16     16      0      0
8     26      0      1      0
8     32      0      0      0
11      8      4      4      0
16      0      0      0      0


A search for the linear equations bounding these points gave the following 145: Each row is of the form ax1 + bx2 + cx3 + dx4 + ex5 <= x6

         3          12          12          23          72         480
1           4           4           4          35         160
1           4           4           1          41         160
0           0           1           0          12          24
3          12          19          36         117         648
0           0           2           3          12          48
3          12          17          33         102         600
0           3           2           5          12          96
12          57          48          95         288        2064
4          19          16          19         134         688
4          19          16           4         164         688
0           3           4           9          24         144
0           3           2           3          18          96
0           3           2           0          24          96
3           4           4           7          24         168
1           1           1           1           8          43
2           2           2           3          13          86
3           2           4           6          27         138
6           2           9          12          60         270
6           6           7          12          42         282
20          14          15          18         106         650
9           6           8          12          51         318
27          18          20          24         141         858
34          16          19          15         110         832
7           2           4           4          20         154
6           0           4           3          15         120
18           6          11          12          60         426
18           6          10           9          57         402
11           4           6           5          35         248
9           3           5           3          30         201
3           0           2           0          12          60
16           6           9           8          54         370
0           0           0           0           1           1
0           0           0           1           4           8
0           1           1           3           8          44
1           2           1           3          16          72
3           8           5          15          40         280
1           2           1           9          26         106
7          14           7          29          80         504
0           2           1           2          16          64
0           1           0           0          16          32
0           1           0           6          16          56
0           7           0          18          40         224
0           5           3           9          20         160
10          15          14          25          84         602
1           1           1           3           9          50
9           6           8          18          57         342
9           6           7          18          57         330
1           0           1           3          13          42
9           3          10          18          69         342
5           2           3          10          37         162
3           1           2           6          23          98
2           2           1           3          19          80
3           3           2           6          18         120
2           2           1           9          27         112
14          14           7          31          93         560
6           9           7          15          42         336
1           1           1           2           6          44
2           2           1           2          22          80
6           6           5           6          42         240
30          21          19          27         138         912
3           2           2           4          13          94
15          10           8          19          62         440
6           4           3           7          29         176
21          14          12          25          82         616
81          54          49          72         357        2376
18          12           9          16         102         528
3           3           1          15          44         174
6           6           1          36         107         384
21          21           7          51         146         840
1           1           0           6          19          64
1           1           0           3          10          43
42          42           7         108         317        1680
7           7           1          18          54         280
2           1           1           1           6          48
9           3           4           9          24         198
2           1           1           3           9          56
3           2           1           9          27         118
30          12          11          39         117         720
15           6           5          21          63         366
3           1           1           3           8          62
6           5           1          30          90         328
6           2           1          12          36         160
9           3           2          15          45         222
12           3           2          18          51         264
3           1           1           2           5          56
4           1           2           2           9          80
2           0           1           1           3          34
16           3           6           8          21         272
12           6           5          15          45         306
39          24          17          51         153        1080
3           2           1           5          15          90
17          12           6          28          84         520
15           9           7          18          57         408
9           4           3           9          23         200
33          24          11          57         169        1032
21          14           7          33          97         616
19           6          10           9          55         408
15           5           8           5          47         328
8           2           4           3          21         160
4           1           2           1          11          80
6           5           1          14          42         216
66          63          11         162         478        2544
8           3           4           4          24         176
7           3           3           4          19         152
6           2           3           3          17         128
48          18          17          24         132         960
8           3           3           4          20         160
6           2           2           3          13         112
9           3           2          12          36         201
6           2           1           8          24         132
12           3           2          12          33         222
6           2           1           6          16         118
18           7           3          18          46         368
5           2           1           5          13         104
3           1           1           1           7          56
33          12          11          15          93         648
6           3           2           3          30         144
21           3           2          18          51         336
1           0           0           1           4          16
14           3           3           9          24         226
56          11          12          36          97         896
46           9           6          36          99         746
56           9           6          46         129         896
9           0           4           4          12         144
18           3           6           8          21         288
4           1           1           2           5          64
1           0           0           0           8          16
24           5           5          15          42         384
16           3           2          12          35         256
9           0           4           0          24         144
12           2           4           3          21         192
8           2           2           3          13         128
21           4           2          16          48         336
84          16           9          64         186        1344
6           1           2           1          11          96
4           1           1           1           7          64
48          12          11          15          93         768
26           6           3          18          48         418
28           7           3          18          46         448
28           6           3          20          56         448
8           2           1           5          13         128
8           2           1           2          22         128
12           3           2           3          30         192
4           1           0           0          16          64