The Set of Sets in SET

BitXor gives the bitwise XOR of a set of integers in binary form. This uses the finite field . Is there something similar for ?
The ternary representations of are ; transposing, , which gives . Similarly, in ternary is with transpose . In each transpose, the triples of numbers are either all the same or all different. Under a ternary form of BitXor, the triples have a ternary bitwise sum of zero.
In the game of SET, there are 81 cards, each having four features: number of shapes (one, two or three), shape (diamond, squiggle or oval), shading (solid, striped or open) and color (red, green or purple). A set of three cards is called a "set" if the four features are pairwise identical or completely different. This is the same as a bitwise sum of zero. SET can be thought of as projective geometry .
There are 1080 possible sets in SET. This Demonstration shows them all, using 40 decks of SET to show 27 sets at a time. This is also a Steiner system: each pair of numbers from 0 to 80 is represented in a unique set. This is also a resolvable design.

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References
[1] Wikipedia. "Projective Plane." (Feb 7, 2020) en.wikipedia.org/wiki/Projective_plane.
[2] Wikipedia. "Set (Card Game)." (Feb 7, 2020) en.wikipedia.org/wiki/Set_(card_game).
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