ASSOCIATION SCHEMES AND CODING THEORY

This paper contains a survey of association scheme theory (with its al gebraic and analytical aspects) and of its applications to coding theo ry (in a wide sense), It is mainly concerned with a class of subjects that involve the central notion of the distance distribution of a code . Special emphasis is put on the linear programming method, inspired b y the MacWilliams transform, This produces upper bounds for the size o f a code with a given minimum distance, and lower bounds for the size of a design with a given strength. The most specific results are obtai ned in the case where the underlying association scheme satisfies cert ain well-defined ''polynomial properties;'' this leads one into the re alm of orthogonal polynomial theory. In particular, some ''universal b ounds'' are derived for codes and designs in polynomial type associati on schemes. Throughout the paper, the main concepts, methods, and resu lts are illustrated by two examples that are of major significance in classical coding theory, namely, the Hamming scheme and the Johnson sc heme, Other topics that receive special attention are spherical codes and designs, and additive codes in translation schemes, including Z(4) -additive binary codes.