Upper bounds on weight hierarchies of extremal non-chain codes (Q5951954)

The weight hierarchy of a linear \([n,k,d]\) code \(C\) over \(\text{GF}(q)\) is the sequence \(\{d_1,d_2,\dots, d_k\}\) where \(d_r\) is the smallest support of an \(r\)-dimensional subcode. Linear codes are classified according to a set of chain and non-chain conditions, the extreme being the chain condition and the extremal non-chain condition.NEWLINENEWLINENEWLINE\textit{W. Chen} and \textit{T. Kløve} [Appl. Algebra Eng. Commun. Comput. 8, 379-386 (1997; Zbl 0877.94055); and IEEE Trans. Inf. Theory 45, 276-281 (1999; Zbl 0947.94017)] found tight upper bounds for non-binary four-dimensional extremal non-chain codes. The author generalizes and improves their upper bounds to arbitrary dimensions. These bounds are the best possible in dimensions 5 and lower.NEWLINENEWLINENEWLINEThe difference sequence of a code is defined by \(\{\delta_0,\delta_1,\dots, \delta_{k-1}\}\) where \(\delta_j= d_{k+1-j}- d_{k-j}\). Several bounds on these difference sequences are given. As an example the author shows that, for any extremal non-chain code, \(\delta_m\leq q^m\delta_0- \sum^m_{i=0} q^i\) for \(1\leq m\leq k-2\). An extremal non-chain code is said to be \(m\)-optimal if the bound is satisfied with equality. The author finds necessary conditions for an extremal non-chain code to be optimal.