Weight hierarchies of 7-ary linear codes of dimension 3 and improved genetic algorithm (Q2757118)

Consider an \([n,3;7]\)-linear code \(C\). Let \(d_r\) (\(1 \leq r \leq 3\)) denote the \(r\)th generalized Hamming weight of \(C\). The sequence \((d_1 , d_2, d_3)\) is called the weight hierarchy of \(C\). Without loss of generality, assume that \(n=d_3\). The difference sequence (DS) \((i_0, i_1, i_2)\) of \(C\) is defined by \(i_0 = d_3 -d_2\), \(i_1 = d_2 - d_1\) and \(i_2 =d_1\). There is a one-to-one correspondence between the weight hierarchies and difference sequences. NEWLINENEWLINENEWLINELet \(V_2\) denote the projective plane \(\text{PG}(2,7)\). A value assignment is a function \(m: V_2 \rightarrow \{ 0,1, 2, \ldots \}\) and, for a subset \(S\) of \(V_2\), let \(m(S) = \sum_{p \in S} m(p)\). NEWLINENEWLINENEWLINEThe problem of describing explicitly the weight hierarchies of \([n,3;7]\)-linear codes can be reformulated into the problem of determining whether a given triple \((i_0, i_1, i_2)\) is a DS, or equivalently, whether there is a value assignment \(m\) such that \(\max \{ m(p) \mid p \in V_2 \} = i_0\), \(\max \{ m(L) \mid L \text{ is a line in }V_2 \} = i_0 +i_1\), and \(m(V_2) = i_0 + i_1 + i_2\). NEWLINENEWLINENEWLINEIf there exist a line \(V_1\) in \(V_2\) and a point \(V_0\) on \(V_1\) such that \(m(V_0) = i_0\) and \(m(V_1) = i_0 + i_1\), then the corresponding DS is called a chain difference sequence (CDS). If, for any point \(p\) on any line \(L\) of \(V_2\), we have \(m(p) < i_0\) or \(m(L) < i_0 + i_1\), then the corresponding DS is called a non-chain difference sequence (NDS). NEWLINENEWLINENEWLINEIn this paper, the authors exhibit a set of sufficient conditions for a DS to be NDS. Using results in [\textit{W. Chen} and \textit{T. Kløve}, Weight hierarchies of linear codes of dimension 3, J. Stat. Plann. Inference 94, 167-179 (2001)], it is determined that, with the possible exception of 193 triples, all positive triples satisfying \(i_1 \leq 7 i_0\) and \(i_2 \leq 7 i_1\) are CDS. Then, applying an improved genetic algorithm, all but 29 of the remaining 193 triples were determined to be CDS. Of these remaining 29 triples, it was determined that the triple \((1,1,7)\) is not CDS.