## Upper bounds on Weight Hierarchies of Extremal Non-Chain Codes

#### Abstract

The weight hierarchy of a linear $[n,k;q]$ code \code\ over \GF\ is the sequence $(d_1,d_2,\ldots,d_k)$ where $d_r$ is the smallest support size of an $r$-dimensional subcode of \code. The difference sequence \DS\ is defined by $\delta_i=d_{k-i}-d_{k-i-1}$. An $[n,k;q]$ code is extremal non-chain if for any $r$ and $s$, where \$1\le rWe give upper bounds on difference sequences of such codes, give some general results about codes meeting these bounds with equality, and finally construct five-dimensional codes meeting the bounds with equality.