## Suboptimal covering of multidimensional unit sphere

### Short description

Each of the text files given below provides a covering of the unit sphere,
which is the surface of the **d**-dimensional unit ball.

The covering was constructed by using the Stepwise Covering Replenishment (SCR) method
that applies adaptive approximating of the unit ball by a polytope.

In the SCR method, vertices of the approximating polytope are used as the covering base,
i.e. a system of points of the sphere, which **epsilon**-neighborhood covers the sphere.

In a text file, any list of points, which begins with the first line of the file, provides a covering base.

In addition to a covering base, information of a different type is provided in a text file.

It can be used for calculating the value of **epsilon**, which is guaranteed by the covering base.

The value of **epsilon** can be found by using the formula

**epsilon = square root of (2 h),**

where the value **h** describes the deviation of the unit ball from the approximating polytope given
by its vertices that are the points listed above the information concerning **h**.

Such information is given after those points of the text files that start with multiple dashes.

The achieved value of **h** is given only after those points that change the value of **h**.

The text files given below contain the covering bases with a certain number of points
(the number depends upon the dimension **d**).

To obtain a covering with a desired number of points **N**
(which is not greater than the total number of points of the covering base in the file),
one has to take the first **N** points.

Alternatively, one can start with a desired value of **epsilon**,
then calculate a related value of **h**,
and select the points that are above the calculated value of **h**.

### Downloadable covering bases

- Covering base (85984 points) for d=3 (9222K text file)

- Covering base (49118 points) for d=4 (6372K text file)

- Covering base (23484 points) for d=5 (3592K text file)

- Covering base (10208 points) for d=6 (2085K text file)

- Covering base (3195 points) for d=7 (701K text file)

- Covering base (1122 points) for d=8 (258K text file)

- Covering base (400 points) for d=9 (92K text file)

- Covering base (150 points) for d=10 (30K text file)