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Published online by Cambridge University Press: 27 June 2025
The losing positions of certain combinatorial games constitute linear error-detecting and -correcting codes. We show that a large class of games, which can be cast in the form of annihilation games, provides a potentially polynomial method for computing codes (anncodes). We also give a short proof of the basic properties of the previously known lexicodes, which were defined by means of an exponential algorithm, and are related to game theory. The set of lexicodes is seen to constitute a subset of the set of anncodes. In the final section we indicate, by means of an example, how the method of producing lexicodes can be applied optimally to find anncodes. Some extensions are indicated.
1. Introduction
Connections between combinatorial games (simply games in the sequel) and linear error-correcting codes (codes in the sequel) have been established in [Conway and Sloane 1986; Conway 1990; Brualdi and Pless 1993], where lexicodes, and some of their connections to games, are explored. Our aim is to extend the connection between games and codes to a large class of games, and to formulate a potentially polynomial method for generating codes from games. We also establish the basic properties of lexicodes by a simple, transparent method.
Let Γ, any finite digraph, be the groundgraph on which we play the following general two-player game. Initially, distribute a positive finite number of tokens on the vertices of Γ. Multiple occupation is permitted. A move consists of selecting an occupied vertex and moving a single token from it to a neighboring vertex, occupied or not, along a directed edge.
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