EvenQuads Game and Error-Correcting Codes

TL;DR

This paper introduces EvenQuads, a card game related to error-correcting codes, and explores how game configurations correspond to linear binary codes, providing insights into code properties and combinatorial structures.

Contribution

It establishes a novel connection between the EvenQuads game and error-correcting codes, analyzing code properties and combinatorial configurations within the game.

Findings

Error-correcting codes can be constructed from EvenQuads game configurations.

The paper estimates the number of quads in decks of various sizes.

Properties of codes based on magic and semimagic quad squares are discussed.

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. Given $\ell$ cards from the deck of EvenQuads, we can build an error-correcting linear binary code of length $\ell$ and Hamming distance 4. The quads correspond to codewords of weight 4. Error-correcting codes help us calculate the possible number of quads when given 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.