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.

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    
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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