On the weight hierarchy of the semiprimitive codes (Q1917493)

Let \(\mathbb{F}=\text{GF}(2^k)\) be the finite field with \(2^k\) elements. Let \(h(x)\in \text{GF}(2)[x]\) be irreducible with degree \(k\) and period \(n\), and let \(C=\{c(a)=(\text{Tr}(a), \text{Tr} (a\beta),\dots, \text{Tr} (a\beta^{n-1})) \mid a\in \mathbb{F}\}\), where \(h(\beta)=0\) and Tr is the trace function from \(\text{GF}(2^k)\) to GF(2). Let \(\Psi\) be a generator of \(F\setminus \{0\}\), and write \(\beta=\Psi^N\). \(C\) is semi-primitive if \(N>2\) and \(N|(2^j+1)|2^k-1\) for some \(j\geq 1\). Note that in this case, \(2j|k\). The weight distribution of \(C\) is known. The authors determine, for the case that \(k/2j\) is odd, the weight hierarchy of \(C\), that is, for \(1\leq r\leq k\), they find \(d_r=\min\{|\chi(D)|\mid D\) is an \(r\)-dimensional subcode of \(C\}\), where \(\chi(D)\) is the set of coordinate positions in which not all words of \(D\) are zero. With their results, the authors obtain the weight hierarchy of the dual of a primitive BCH code of length \(2^k-1\) and designed distance \(N+2\).