This paper provides an alternative proof to the MacWilliams identity for linear block codes over GF(q). The intermediate results of our alternative proof offer some insights of the vector-space structure. Our vector-space-based methodology can be used to establish the relation between weight distributions of dual codes for other metrics.

We introduce a class of exible conditional probability models and techniques for classi cation regression problems Many existing methods such as generalized linear models and support vector machines are subsumed under this class The exibility of this class of techniques comes from the use of kernel functions as in support vector machines and the generality from dual formulations of stan dard re...