The sum of vectors with the same color in the same square is allways the null vector.

The vectors of ${\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{4}}$ are represented by subsets of {ABCD}.

Example: The green fields in the undermost square (number 18): ^{(zoom in)}

• In terms of sets: ${\displaystyle ~\Delta }$ stands for the symmetric difference:

${\displaystyle \{~~B~~~~\}~\Delta ~\{AB~~~~\}~\Delta ~\{~~B~~D\}~\Delta ~\{AB~~D\}~~~=~~~\{~~~~~~~~\}}$

• In terms of vectors: ${\displaystyle ~\oplus }$ stands for the addition modulo 2 (1⊕1=0):

${\displaystyle {\begin{pmatrix}0\\1\\0\\0\\\end{pmatrix}}\oplus {\begin{pmatrix}1\\1\\0\\0\\\end{pmatrix}}\oplus {\begin{pmatrix}0\\1\\0\\1\\\end{pmatrix}}\oplus {\begin{pmatrix}1\\1\\0\\1\\\end{pmatrix}}~~~=~~~{\begin{pmatrix}0\\0\\0\\0\\\end{pmatrix}}}$

Related files:

• The colors and gray numbers are taken from File:Boolean partition matrices colored.svg. There are 35 matrices colored only in red, green, yellow and blue.

• The 4-ary Boolean functions marked with red squares in this graphic are also listed in the central matrix in File:Boolean partition matrices top lines by digit sum.svg, where they are displayed by linear bit patterns, instead of squares.

• If the content of this file remains strange to you, you may take a look at the colored squares in File:Jupiter square magic sums numerical.svg, and convince yourself, that fields with the same color in the same square always sum up to 30. ^{(zoom in ...magic sums numerical.svg)} Here it works in the same way, but the elements are vectors, and fields with the same color in the same square always sum up to the null vector.

• The vectorial magic squares can be found in Category:Jupiter magic squares (vectorial).

Related topics:

• In each of these 35 matrices are three nonzero numbers on red background. ^{(zoom in)}
  

These triples are the blocks of a Steiner triple system of order 15.
They appear at the top of page 10 in
Mats Näslund     Steiner Triple Systems and Perfect Codes     Master of Science thesis, February 1993     [1]
in this arrangement:

{1  2  3}  {1  4  5}  {1  6  7}  {1  8  9}  {1 10 11}  {1 12 13}  {1 14 15}
{2  4  6}  {2  5  7}  {2  8 10}  {2  9 11}  {2 12 14}  {2 13 15}  {3  4  7}
{3  5  6}  {3  8 11}  {3  9 10}  {3 12 15}  {3 13 14}  {4  8 12}  {4  9 13}
{4 10 14}  {4 11 15}  {5  8 13}  {5  9 12}  {5 10 15}  {5 11 14}  {6  8 14}
{6  9 15}  {6 10 12}  {6 11 13}  {7  8 15}  {7  9 14}  {7 10 13}  {7 11 12}

• These 35 matrices are related to the square model of PG(3,2).

• The red fields in these matrices denote the elements of the 35 Klein four-subgroups of the 16 element group Z₂⁴.

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