of its nonzero coordinates: ${\rm supp} ( x) = \{ i \mid x_{i} \neq 0 \}$. The support design of a code of length $n$ for a given nonzero weight $w$ is the design with $n$ points of coordinate indices, and blocks the supports of all codewords of weight $w$.

The following lemma is easily seen.

\[lem: divisible\] Let $\lambda(S)$ be the number of blocks containing a given set $S$ of $s$ points in a $t$-$(v,k,\lambda)$ design, where $0\leq s\leq t$. Then $$\lambda(S)\binom{k-s}{t-s}
=
\lambda\binom{v-s}{t-s}.$$ In particular, the number of blocks is $$\frac{v(v-1)\cdots(v-t+1)}{k(k-1)\cdots(k-t+1)}\lambda.$$

Harmonic weight enumerators {#sec:Har}
---------------------------

In this section, we extend a method of the harmonic weight enumerators which were used by Bachoc [@Bachoc] and Bannai et al. [@Bannai-Koike-Shinohara-Tagami]. For the readers convenience we quote from [@Bachoc; @Delsarte] the definitions and properties of discrete harmonic functions (for more information the reader is referred to [@Bachoc; @Delsarte]).

Let $\Omega=\{1, 2,\ldots,n\}$ be a finite set (which will be the set of coordinates of the code) and let $X$ be the set of its subsets, while, for all $k= 0,1, \ldots, n$, $X_{k}$ is the set of its $k$-subsets. We denote by ${{\mathbb R}}X$, ${{\mathbb R}}X_k$ the free real vector spaces 