Permutations and codes: Polynomials, bases, and covering radius

We will be considering sets of n-tuples over an alphabet A, in two important cases: A ¡ ¢ 0£ 1¤ (binary code); n¤ , all entries of each word distinct (set of permutations). We often impose closure conditions on these sets, as follows: A binary code is linear if it is closed under coordinatewise addition mod 2. A set of permutations is a group if it is closed under composition. x£ yïs the number of coordinate positions where two words differ. It is a metric on the set of words. In the binary case, d § x£ y¨© ¡ wt § x y¨£ so for a linear code, minimum distance equals smallest number of non-zero coordinates of a non-zero element (minimum weight). In the permutation group case, d § x£ y¨© ¡ n fix § x 1 y¨£ so, for a permutation group, minimum distance equals smallest number of points moved by a non-identity element (minimal degree). This is not really graph theory: the distance between permutations is not a graph distance, because there do not exist two permutations at distance 1. However, it is closely related to the distance d in the Cayley graph of the symmetric group with respect to the set of transpositions: we have d § g£ h¨¦ 2 d § g£ h¨d § g£ h¨1 for g ¡ h. Also, we will be considering the size of the smallest dominating set in the graph G n k with vertex set S n , two permutations joined if they agree in at least k places.