Randomness, arrays, differences and duality

Random variables that take on values in the finite field of q elements are considered. It is shown that joint distributions of such random variables are equivalently described by the individual distributions of their linear combinations. Random vectors X that are equally likely to take on any row of an arbitrary q-ary rectangular array as their value are treated extensively, together with the random vector /spl Delta/X defined as the difference between two independent versions of such a random vector. It is shown that linear combinations of exactly /spl tau/ of the components of X are always biased toward 0. A quantitative measure /spl beta//sub /spl tau//, of this bias is introduced and shown to be given by a sum of Krawtchouk polynomials. The vanishing of /spl beta//sub /spl tau// is shown to be equivalent to the maximal randomness of linear combinations of exactly /spl tau/ of the components of X as well as of /spl Delta/X. When the rows of the original array are the codewords of a q-ary linear code, then the bias /spl beta//sub /spl tau// coincides with the number of codewords of Hamming weight /spl tau/ in the dual code. The results of this article generalize certain well-known results such as the MacWilliams' (1977) identities and Delsarte's (1973) theorem on the significance of the "dual distance" of nonlinear codes.