Evaluation of bounds of codes defined over hexagonal and honeycomb lattices via linear programming based on association schemes

IICQI 2007

While we can say much about codes without talking about association schemes, the algebraic structure they provide, gives us a more complete picture of how codes interact with the set they belong to. So far, Delsarte has used this algebraic structure to give one of the strongest bounds on the size of codes. Typically, the most of works on Delsarte's linear programming bound have been developed for the underlying codes of distance regular association schemes (distance-regular codes). In this work, by using underlying graphs of association schemes and Delsarte's linear programming bound, a way has been introduced for calculating some new bounds for more general codes which are not distance regular. Some upper bounds on underlying codes of association schemes with two commuting generators derived from finite root lattices corresponding to the lie algebra SU(n), have investigated by using Delsarte's linear programming bound. In these cases, despite the distance regular ones, distance between any two arbitrary code- words is not equal to the number of steps needed for going from one codeword to the other. Hence, for achieving upper bound for a given code with minimum distance d, the number of steps needed for covering all code-words in distance d from an arbitrary codeword (called reference codeword), is determined via the structure of the corresponding underlying graph of association scheme. Also in these cases, some new perfect codes have been introduced. Similarly, we have introduced some upper bounds for GF(4) quantum codes.