EvenQuads Game and Error-Correcting Codes

Abstract EvenQuads is a new card game that is a generalization of the SET game, where each card is characterized by three attributes, each taking four possible values. Four cards form a quad when, for each attribute, the values are the same, all different, or half and half. For any $$\ell$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> cards selected from the deck of EvenQuads, it is possible to construct an error-correcting linear binary code of length $$\ell$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> and Hamming distance 4, where quads correspond to codewords of weight 4. Using error-correcting codes, we calculate the number of possible quads that can be formed with up to 8 cards. We also estimate the number of cards that do not contain quads for decks of different sizes. In addition, we discuss properties of error-correcting codes built on semimagic, magic, and strongly magic quad squares. This highlights a rich interplay between recreational mathematics games and coding theory and encourages others to explore similar combinatorial games for hidden connections!