It is possible to save the results of LatNet Builder in a folder. This folder contains two files:

input.txt which contains the summary of the input parameters of LatNet Builder
output.txt which contains an easily parsable formatting of the results

The contents of the output.txt file depend on the point set type (ordinary lattice rule, polynomial lattice rule, sobol digital net, and explicit digital net). For each point set type, several formats are available. The user can specify the desired format using the --output-style option.

Below, we describe the different available output formats.

Summary of available formats

For ordinary lattice rules: lattice format
For polynomial lattice rules: lattice format (default) and net format
For Sobol digital net: Sobol format (default), Joe & Kuo format, net format
For explicit digital net: net format

Lattice format

It is obtained with the option
--output-style lattice. For ordinary lattice rules, the output file looks like follows:

# Parameters for a lattice rule
8       # 8 dimensions
65536   # modulus = n = 65536 points
# Coordinates of generating vector, starting at j=1
1
19463
17213
5895
14865
31925
30921
26671

The first two lines of the file give the dimension of the net and the modulus (number of points). The following lines give the generating values, one per line.

For polynomial lattice rules, the format is very similar, as follows:

# Parameters for a polynomial lattice rule in base 2
8       # s = 8 dimensions
16      # n = 2^16 = 65536 points
45781   # polynomial modulus
# Coordinates of generating vector, starting at j=1
1
17213
5895
14865
31925
30921
26671
17213

There are two differences with the previous case: first, the polynomial modulus is in general different from the number of points, so we give both information on two separate lines. Second, the modulus and the generating values are now polynomials. However, we represent them as integers with the (standard) convention that the integer is obtained by replacing the formal variable by 2. For example, if the polynomial is \(Q(z) = z^4 + z^3 + 1\), its integer representation is \(2^4 + 2^3 + 1 = 25\).

Net format

The most general format for digital nets is the --output-style net format, which looks as follows.

# Parameters for a digital net in base 2
8       # s = 8 dimensions
10      # n = 2^10 = 1024 points
31      # r = 31 digits
# The columns of gen. matrices C_1, ..., C_s, one matrix per line :
1073741824 536870912 268435456 134217728 ...
2012537125 1382645254 ...

The first three lines give: \ns : the number of dimensions \nk : the number of points, which equals \(2^k\) with \(k\) the number of columns. \nr : the number of rows in the binary generating matrices. In general, we have \(r = 31 \geq k\).

The following lines give the representation of the generating matrices, one matrix per line. Each matrix is represented by \(k\) integers encoding its columns in the following format: the binary representation of the integer gives the r bits in the column, with the bit on the first row of the matrix (row \(0\)) being the most significant, and the one on the last row (row \(r − 1\)) being the least significant.

For example: \nIf \(r = 31\) and the first column has a 1 in the first row and 0 in all other rows, then the integer representation of this column will be \(2^30= 1 073 741 824\). \nIf there is a 1 in the last row and 0 elsewhere, the representation will be \(2^0= 1\). \nIf all 31 elements of the column are 1, the representation will be \(2^31 - 1\).

Formats for Sobol digital nets

Joe & Kuo format

This output format, which can be optained with --output-style soboljk is the one from Joe and Kuo (2008). The output file looks as follows:

# Parameters for Sobol points, in JK format
# 8 dimensions
#       d       a       m_{j, c}
2       1       0       1
3       2       1       1 3
4       3       1       1 3 1
5       3       2       1 1 1
6       4       1       1 1 3 3
7       4       4       1 3 5 13
8       5       2       1 1 5 5 17

The only difference with the Joe & Kuo format is that there are more comment lines at the beginning. After these comment lines, the other lines of the file specify the primitive polynomials and the initial direction numbers for each coordinate \(j \geq 2\), one line per coordinate.

For coordinate \(j=1\), the generating matrix is the identity and is not given in the file (it is implicit).

Then for each following coordinate:

The first number on each line is the coordinate j.
The second number is the degree \(c_j\) of the primitive polynomial \(p_j(x)\) used for this coordinate.
The third number is the integer that corresponds to the binary representation of the inner coefficients of this polynomial (the first and last coefficients are ignored, as they are always 1). For example : \nIf the polynomial is \(p_{j}(x) = x^4 + x^3 + 1\), the coefficients are “1 1 0 0 1”, and after removing the first and last “1”, we get 100 in base 2, which is 4, so the third column contains the number 4. (Without removing the first and last “1”, the number would be 25 instead.)
After these three numbers, there are \(c_j\) integers \(m_{j,1}, \dots, m_{j,c_j}\) where \(m_{j,c}\) is the cth (real-valued) initial direction number for this coordinate, multiplied by \(2^c\) to obtain an integer. This \(m_{j,c}\) is the integer formed by taking the bits in row 1 to row c of column c, in this order. The last bit is the bit on the diagonal, which is always 1, so all \(m_{j,c}\)’s are odd integers.

More details are given at :
http://umontreal-simul.github.io/ssj/docs/master/classumontreal_1_1ssj_1_1hups_1_1SobolSequence.html

Simplified Sobol format

This format, which can be used with the --output-style sobol option is a simplification of the JK format. The first three columns are removed of the original Joe and Kuo format, because this numbers never change. We only give the (initial) direction numbers, starting from the second coordinate, as previously.

Example of simplified Sobol output format:

# Parameters m_{j, c} for Sobol points
# 8 dimensions
# m_{j, c} starting from the second coordinate
1        
1 3
1 3 1
1 1 1
1 1 3 3
1 3 5 13
1 1 5 5 17

Interlaced point sets

For interlaced point sets, the output is the underlying point set in dimension \(ds\). The digit interlacing function should be applied to this point set in dimension \(ds\) to obtain the interlaced point set in dimension \(s\).

Two comment lines are added to signal that we are working with an interlaced point set, for instance:

# Parameters for a digital net in base 2
2    # 2 dimensions
2    # Interlacing factor
4    # Number of components = interlacing factor x dimension
10   # k = 10,  n = 2^10 = 1024 points
31   # r = 31 binary output digits
# Columns of gen. matrices C_1,...,C_{ds}, one matrix per line
1073741824 536870912 268435456 1476395008 1543503872 1644167168 1728053248 511705088 180355072 1625292800
1073741824 1610612736 1879048192 134217728 1543503872 503316480 1325400064 2021654528 71303168 1839202304
1073741824 1610612736 805306368 134217728 1811939328 369098752 1828716544 1753219072 943718400 568328192
1073741824 536870912 1879048192 134217728 1811939328 167772160 1795162112 595591168 809500672 828375040