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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)
Full Text:
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