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Notes on generalized Hamming weights of some classes of binary codes. (English) Zbl 1457.94235
The generalized Hamming weights (GHW) of a code \(\mathcal{C}\) are defined as the ordered vector
\[(\omega_1^{\min},\omega_2^{\min},\ldots,\omega_k^{\min}),\]
where \(\omega_i^{\text{min}}\) is the minimum weight of all \(i\)-dimensional subcodes.
This research is a continuation of [Y. Liu and Z. Liu, Adv. Math. Commun. 12, No. 2, 415–428 (2018; Zbl 1414.94927)].
For any pair \((a,b)=(a,0)\) for \(a\in\mathbb{F}_q^\ast\) an improved formula for the minimum weight \(\omega_\theta^{\min}\) for any \(\theta, 1\leq \theta\leq \phi(r^m)/2\) is shown and the condition \(\text{wt}(a^{(0)})\) to be even, from previous results, is not needed. Furthermore, a formula for the minimum weight \(\omega_\theta^{\min}\) for any \(\theta, \phi(r^m)/2\leq \theta\leq \phi(r^m)\) is also established, thus it is proved that GHWs of any code \(\mathcal{C}_{\mathcal{A}}\) can be determined for any pair \((a,0), a\in\mathbb{F}_q^\ast.\)
In the case of \((a,b),\) where \(b\neq 0,\) assuming \(c=ab^{-\frac{q-1}{r^m}}\) and \(\text{wt}(c^{(0)})\) is even, two theorems are proved providing formulas for \(\omega_\theta^{\min}\) for any \(\theta, 1\leq \theta\leq\phi(r^m)/2\) and \(\theta, \phi(r^m)/2\leq \theta\leq \phi(r^m)\), respectively.
MSC:
94B05 Linear codes (general theory)
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