THE WEIGHT HIERARCHY OF HADAMARD CODES

The support of an $(n, M, d)$ binary code $C$ over the set $\mathbf{A}=\{0,1\}$ is the set of all coordinate positions $i$, such that at least two codewords have distinct entry in coordinate $i$. The $r$th generalized Hamming weight $d_r(C)$, $1\leq r\leq 1+log_2n+1$, of $C$ is defined as the minimum of the cardinalities of the supports of all subset of $C$ of cardinality $2^{r-1}+1$. The sequence $(d_1(C), d_2(C), \ldots, d_k(C))$ is called the Hamming weight hierarchy (HWH) of $C$. In this paper we obtain HWH for $(2^k-1, 2^k, 2^{k-1}$ binary Hadamard code corresponding to Sylvester Hadamard matrix $H_{2^k}$ and we show that $$d_r=2^{k-r} (2^r -1).$$ Also we compute the HWH of all $(4n-1, 4n, 2n)$ Hadamard code for $2\leq n\leq 8$.