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Erschienen in:
04.04.2020
The subfield codes of several classes of linear codes
Erschienen in: Cryptography and Communications | Ausgabe 6/2020
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Abstract
Let \(\mathbb {F}_{2^{m}}\) be the finite field with 2m elements, where m is a positive integer. Recently, Heng and Ding in (Finite Fields Appl. 56:308–331, 2019) studied the subfield codes of two families of hyperovel codes and determined the weight distribution of the linear code
for f(x) = x2 and f(x) = x6 with odd m. Let v2(⋅) denote the 2-adic order function. This paper investigates more subfield codes of linear codes and obtains the weight distribution of \(\mathcal {C}_{a,b}\) for \(f(x)=x^{2^{i}+2^{j}}\), where i, j are nonnegative integers such that v2(m) ≤ v2(i − j)(i ≥ j). In addition to this, we further investigate the punctured code of \(\mathcal {C}_{a,b}\) as follows:
and determine its weight distribution for any nonnegative integers i, j. The parameters of these binary linear codes are new in most cases. Some of the codes and their duals obtained are optimal or almost optimal.
$$ \mathcal{C}_{a,b}=\left\{((\text{Tr}_{1}^{m}(a f(x)+bx)+c)_{x \in \mathbb{F}_{2^{m}}}, \text{Tr}_{1}^{m}(a), \text{Tr}_{1}^{m}(b)) : a,b \in \mathbb{F}_{2^{m}}, c \in \mathbb{F}_{2}\right\}, $$
$$ \mathcal{C}_{a}=\left\{((\text{Tr}_{1}^{m}(a x^{2^{i}+2^{j}}+bx)+c)_{x \in \mathbb{F}_{2^{m}}}, \text{Tr}_{1}^{m}(a)) : a,b \in \mathbb{F}_{2^{m}}, c \in \mathbb{F}_{2}\right\}, $$