The weight hierarchies of the projective codes from nondegenerate quadrics
Let \(C\) be a linear code over the finite field \(F_ q\). \(C\) is called a projective code if the columns of a generator matrix of \(C\) are not proportional. Each column can be regarded as a point in the \((k-1)\)- dimensional projective space over \(F_ q\). Projective codes whose generator matrix columns satisfy a nondegenerate quadric are considered here. For a linear subcode \(D\) of \(C\) let \(\chi(D)\) denote the support of \(D\) defined by \(\chi (D)= \{i\mid x_ i\neq 0\) for some \((x_ 1, x_ 2, \dots, x_ n)\in D\}\). The \(r\)-th generalized Hamming weight of \(C\), denoted by \(d_ r(C)\), is \[ d_ r(C)= \min\{ |\chi(D)| \mid D \text{ is an \(r\)-dimensional subcode of } C\}. \] The weight hierarchy of \(C\) is \(\{d_ 1 (C), d_ 2(C), \dots, d_ r(C)\}\). The code \(C\) satisfies the chain condition if there exists an \(r\)-dimensional subcode \(D_ r\) of \(C\) such that \(|\chi (D_ r)|= d_ r(C)\), \(r=1,2,\dots, k\) and \(D_ 1 \subseteq D_ 2 \subseteq \cdots \subseteq D_ k\). The weight hierarchies of the projective codes from nondegenerate quadrics in projective spaces are calculated here. It is shown the codes satisfy the chain condition.
scientific article; zbMATH DE number 597836