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

On components of a Kerdock code and the dual of the BCH code $C_{1,3}$