Abstract
Can coding theory help you be a better gambler? Do unexpected combinatorial patterns (called Steiner systems) arise naturally in codes? This chapter attempts to address these types of questions. (The books by Assmus and Key (Designs and Codes. Cambridge University Press, Cambridge, 1992) and Conway and Sloane (Sphere Packings, Lattices and Groups, 3rd edn. Springer, Berlin, 1999) do into more detail in these matters and are highly recommended.)
This chapter gives an exposition of some ideas of Hadamard and Mathieu, as well as ideas of Conway, Curtis, and Ryba connecting the Steiner system S(5,6,12) with a card game called mathematical blackjack. An implementation in SAGE is described as well. Then we turn to one of the most beautiful results in coding theory, the Assmus–Mattson Theorem, which relates certain linear codes to combinatorial structures called designs. Finally, we describe an old scheme for placing bets using Golay codes (the scheme was in fact published in a Finnish soccer magazine years before the error-correcting code itself was discovered (Hämäläinen et al., Am. Math. Mon. 102:579–588, 1995)).
Open questions which arise in this chapter include conjectures on Hadamard matrices and on which block designs “arise” from an error-correcting code.
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Notes
- 1.
- 2.
Hadamard determined the maximum value of |det(A)|, where the entries of A range over all complex numbers |a ij |≤1, to be n n/2 and that this maximum was attained by the Vandermonde matrices of the nth roots of unity.
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- 4.
That is to say, the following cannot occur: some column has 0 entries, some column has exactly 1 entry, some column has exactly 2 entries, and some column has exactly 3 entries.
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Joyner, D., Kim, JL. (2011). Kittens, Mathematical Blackjack, and Combinatorial Codes. In: Selected Unsolved Problems in Coding Theory. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8256-9_3
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