Codes associated with generalized polygons (Q1109752)

For any finite thick (generalized) \(n\)-gon \(X\) \((n=4,6,8)\) and any field \(F\), the authors study the code \(C\) of \(X\) over \(F\). They obtain an upper bound on the dimension of \(C\). It holds with equality except possibly when \(\text{char}(F)\) belongs to an explicitly given finite set of primes. A notion of regularity of polygons is introduced. Five of the classical series of polygons are regular in this sense. For such polygons, the supports of the minimum weight words of \(C\) are precisely the lines of \(X\) -- for all fields \(F\). This is in strange contrast with the case, say, of projective geometries where a similar phenomenon occurs only when \(\text{char}(F)\) equals the natural characteristic of the geometry.