  In the paper we investigate the structure of $i$-components of two classes of
codes: Kerdock codes and the duals of the primitive cyclic BCH code with
designed distance 5 of length $n=2^m-1$, for odd $m$. We prove that for any
admissible length a punctured Kerdock code consists of two $i$-components and
the dual of BCH code is a $i$-component for any $i$. We give an alternative
proof for the fact presented by De Caen and van Dam in 1999 that the
restriction of the Hamming scheme to a doubly shortened Kerdock code is an
association scheme.
